Paper Title: A note about high-order semi-implicit differentiation: application to a numerical integration scheme with Taylor-based compensated error.


Fast and scalable finite-element based approach for density functional theory calculations using projector-augmented wave method
    Phani Motamarri
    August 5, 2024
================================================================================================================================





Abstract: In this brief, we discuss the implementation of a  third order semi-implicit differentiator as a complement of the recent work by the author that proposes an interconnected semi-implicit Euler double differentiators algorithm through Taylor expansion refinement. The proposed algorithm is dual to the interconnected approach since it offers alternative flexibility to be tuned and to be implemented in real-time processes. In particular, an application to a numerical integration scheme is presented as the Taylor refinement can be of interest to improve the global convergence. Numerical results are presented to support the rightness of the proposed method.



Preprint - Work in progress



§ INTRODUCTION


Acary et al. <cit.> introduced an implicit discretization technique from which differentiation techniques, for implicit differential inclusion, have emerged recently (see e.g. a survey in <cit.>). The principle of this method was initially to replace the classical sign function by an implicit projector, then it has been combined with explicit (homogeneous) terms to define the semi-implicit differentiation <cit.>; the overall goal is to reduce the effects of chattering. 
Based on the latest work from the author et al.  <cit.> that proposes a semi-implicit cascade differentiator including Taylor refinement to reduce the estimation error, in this brief, an alternative version of this latter algorithm is proposed that could possibly offer more flexibility towards practical implementation.

The brief is organized as followed. The first Section defines the third order semi-implicit differentiator as a natural extension of the previous works <cit.> <cit.> including numerical results, the second Section illustrates an application towards a numerical integration scheme and the third Section gives some concluding remarks and perspectives.
Simulation code can be found in the repository: 



§ SEMI-IMPLICIT DIFFERENTIATION ALGORITHM INCLUDING TAYLOR REFINEMENT




 §.§ Presentation


Considering a real signal x_1(t) to differentiate, denote x_1(k δ) its corresponding discretization where k is an integer and δ the sampling period. Moreover, x_1^+ stands for x_1((k+1) δ).
From the cascaded differentiator  <cit.>, we derive a third-order Semi-Implicit Differentiator (SIHD-3), including correction terms (in bold) that refine the Taylor expansion of the two first estimation lines:


    {[                               z_4^+= z_4+  E_1^+ E_2^+ E_3^+ h ( λ_4 μ^4|e_1|^4 α-3𝒩_4  );                                  z_3^+= z_3+ E_1^+ E_2^+ h (z_4^++λ_3 μ^3|e_1|^3 α-2𝒩_3 );                 z_2^+= z_2+ E_1^+ h (z_3^+ -E_3^+ h 1/2 z_4^+  + λ_2 μ^2|e_1|^2 α-1𝒩_2  ); z_1^+= z_1+ h ( z_2^+ -E_2^+  h/2 z_3^+ + E_3^+ E_2^+ h^2/3! z_4^+  +  λ_1 μ |e_1|^α𝒩_1 );                                                                                           ].

 
 or, equivalently, introducing the SIHD related operator:
 
 
    z_i = D_f ^(i)(z, z^+, x_1),    i = 1...4

 
 where z_i is the estimation associated to the ith line of the differentiator (<ref>).
 
 Denote y = x_1 the signal to differentiate; ẏ = x_2,  ÿ = x_3 and y^(4) = x_4 are respectively the first, second and third order differentiation;  z_2,   z_3 and z_4 are respectively the first, second and third order related estimations; e_1 = x_1 - z_1 is the estimation error.

The projector[The projector behaves either as a sign function, or as a nonlinear "slope" depending on the value of the error e_1 <cit.>.] 𝒩_q for q = 1...4  is defined by:

 
    𝒩_q(ϵ_1):= {[ ϵ_1 ∈ SD →𝒩_q=⌈ϵ_1⌋^q(1-α)λ_q (μ h)^q,  E_q^+=1;                                                ;           ϵ_1 ∉ SD →𝒩_q=   sign(ϵ_1),   E_q^+=0 ].

with the convergence domain[Note that, according to <cit.>, it has been shown that the error e_2, corresponding to the second estimation line of (<ref>), converges asymptotically to e_ 2^+ = e_1^+ / h, hence 'propagating' the errors of estimation through the differentiation orders by 'increment' of h.] SD={ϵ_1   /   |ϵ_1|≤ (λ_1 μ h)^1/q(1-α)}.

In eq. (<ref>), the variable E^+_q allows activating the upper (q+1)th line if the convergence domain has been reached at the qth line. This prevent from getting unexpected behavior between lines like oscillations that could destabilize the differentiator.

The corrections terms have been introduced by the author et al.  in <cit.> with respect to the second order differentiation and a study of the convergence has been proposed in <cit.>. The complete derivation of the refined Taylor expansion in (<ref>)  is described in Appendix.



 §.§ Numerical results


The  differentiator (<ref>) is simulated over 10 sec. considering a sine function signal 


    y(t) = sin(2 π t )
 

under the following initial conditions (that match the derivatives): x_1(0) = 0,   x_2(0) = 1,   x_3(0) = 0 and x_4(0) = -1.  The gains and parameters  are set as follows: λ_1 = 10^3, λ_2 = 10^6, λ_3 = 10^9, λ_4 = 10^9 and α = 0.95.



Figures <ref> and <ref> illustrate respectively the evolution of the estimated z_1, z_2, z_3 and z_4 with respect to the corresponding exact differentiation orders of the sine function without extra noise on y.

Figures <ref> and <ref> illustrate respectively the evolution of the estimated z_1, z_2, z_3 and z_4 with respect to the corresponding exact differentiation orders of the sine function including extra noise of very small amplitude on y (obtained by a random generator[The noise is computed under Matlab® using the instruction  where  is a factor that multiplies the noise.] that is multiplied by η_0 = 10^-6).












Low estimation errors are observed, thanks to the correction terms, without extra noise. In particular, the error is of order < e_1 > = 2.96   10^-12  on the z_1 estimation, of order < e_2 > = 4.69   10^-9 on the estimation of the first derivative z_2 and of order < e_3 > = 4.49   10^-6 on the estimation of the second derivative z_3.
In presence of noise, the accuracy of the estimation remains correct for the two first orders (< e_1 > = 7.55   10^-7 and < e_2 > = 7.47   10^-4) but the noise becomes predominant for the third order, which makes the estimation very noisy. Note that the θ projector <cit.>, which has shown interesting performances with respect to the rejection of the noise, has not been implemented in the current version of the proposed SIHD3 differentiation algorithm. It constitutes an important issue to deal with noisy signals for future investigations.



§ APPLICATION TO NUMERICAL INTEGRATION SCHEME




 §.§ Observer-based numerical scheme



Consider an ordinary differential equation (ODE) such as:

    d   y(t)/d   t = f(y(t), u(t)),      t > t_0,    y(0) = y_0.

The quantities u and y represent respectively the input and the solution of (<ref>). The corresponding usual discrete explicit Euler scheme reads:


    y_k+1 - y_k/h = f(y_k, u_k),      k ∈ℕ,    y(0) = y_0


where k is the sampled time, h is the time-step, and y_k, y_k+1 are respectively the solution of (<ref>) at the discrete instants t_k and t_k+1. The sampled are supposed equally distributed i.e. t_k - t_k-1 is constant for all k.

In the context of solving ODE, an observer-based numerical integration scheme is proposed to provide a better precision of the estimation of the ODE solution. The proposed scheme is close to the Taylor-based numerical schemes <cit.> for which we point out the refined Taylor based estimation of f(y_k, u_k) via our proposed SIHD algorithm. 

Our proposed differentiation method can be  associated to the nonstandard finite-difference (NSFD) methods <cit.>, described as "powerful numerical methods that preserve significant properties of exact solutions of the corresponding differential equations" (a survey can be found e.g. in <cit.>)[Applying a particular differentiator algorithm to the ODE discretization in the context of NFSD was proposed by the author in <cit.>].

The NSFD & SIHD3 based numerical integration scheme reads:


    {[ z_1^+  = z_1 + ψ D_f^(1)(z, z^+, ŷ );                              ŷ = z_1 ], .   k ∈ℕ,   y(0) = y_0   and  ψ = h + O(h^2)


where the SIHD3 differentiator is used to estimate the f part of the ODE and ψ = h + O(h^2) that do satisfy h → 0 according to the NSFD rules[For example, ψ can be chosen linear of h or exponential of h <cit.>.].

In this work, the purpose is to illustrate the convergence of the proposed NSFD &  SIHD3-based algorithm by comparing the evolution of the error with the classical forward Euler scheme and the Runge-Kutta (RK) scheme according to the time.
Note that in this scheme, the semi-implicit differentiator  (<ref>) appears as a closed-loop observer which tends to adjust and propagate the estimation of f through the numerical integration scheme.




 §.§ Numerical examples and discussion


Three examples are presented to illustrate some first convergence properties  through the evolution of the error | y_exact(t) - y_ODE   scheme(t) | with the classical forward Euler scheme and the Runge-Kutta (RK) scheme according to the time. y_exact(t) denotes the exact solution of the ODE (<ref>) and y_ODE   scheme(t) denotes the approximated solution of the ODE (<ref>) using either the classical Euler forward method or the RK method.
 The internal parameters (λ, α) of  SIHD3 are the same as in the first Section (λ_1 = 10^3, λ_2 = 10^6, λ_3 = 10^9, λ_4 = 10^9 and α = 0.95).



  
Example 1 Consider the basic first order equation:


    d   y(t)/d   t = -y(t) + 1,    y(0) = 10^-2,     t > 0


whose exact solution is y_exact = 1 - exp( - t ). Fig. <ref> illustrates the comparaison of the errors between the schemes.




  
Example 2 Consider the logistic equation <cit.>:


    d   y(t)/d   t = 2 y ( 1 - y),     y(0) = 1/5,     t > 10


whose exact solution is y_exact = 1/1 + 4 exp( 2 (10-t  )).  Fig. <ref> illustrates the comparaison of the errors between the schemes.





  
Example 3 Consider the simple model of a water-discharge problem <cit.>:


    d   y(t)/d   t = -10y^2 + 20,     y(0) = 10^-2,     t > 0


whose exact solution is y_exact = √(2)tanh(√(300) t).  Fig. <ref> illustrates the comparaison of the errors between the schemes.


These preliminary numerical experiments show that the proposed NSFD-SIHD3 numerical scheme provides a slightly reduced error compared with the Euler and Runge-Kutta schemes, especially when the solution is converged towards the asymptotic solution. It is not easy to claim that the transient behavior is more precise than the other scheme since SIHD3 performs a dynamic estimation that depends on the (λ, α) parameters, inducing delays during the estimation. Moreover, the NSFD approach appears naturally to adjust the rate of convergence of SIHD3 thanks to the ψ time-step: indeed, ψ can be considered as an additional SIHD3 'external' gain that can 'virtually' increase the speed of convergence of the scheme, hence giving some flexibility between the SIHD3 parameters (λ, α) from the estimation side, and ψ from the numerical scheme side.




§ CONCLUSION AND FUTURE WORKS


We have derived a   differentiator scheme of order three based on semi-implicit techniques that uses Taylor expansion in an implicit way. This differentiator allows reducing the estimation error in comparaison with exact derivatives. 
Future works include the implementation in real-time in the context of robotics applications.
We have also proposed a numerical scheme based on NSFD scheme including the  differentiator as a refined estimator of the f term in the right side of the ODE.
Further investigations concern a complete study of convergence of the proposed scheme, as well as a comparison of the global error with existing algorithms and possibly an extension toward a semi-implicit scheme.




§ APPENDIX


In this appendix, the corrections terms of the differentiator (<ref>) are calculated from the assumption that the discretized signal x_1 can be differentiated until the third order (inducing a constant fourth order derivative).

Assuming that the discretized signal x_1 can be differentiated though the following 'multi-step' Taylor expansion:


    x_1^+   =    x_1+h x_2 + h^2/2! x_3 + h^3/3! x_4 + h^4/4! x_5 
    x_2^+   =    x_2+h x_3 + h^2/2! x_4 + h^3/3! x_5 
    x_3^+   =    x_3+h x_4 + h^2/2! x_5 
    x_4^+   =    x_4 + h x_5 
    x_5^+   =    x_5


and assuming that the fifth order remains constant x_5 = x_5^+, the retro-propagation of the next-step predictions x_q^+, q = 2...4 terms allows refining the first Taylor expansions by substituting each x_q terms by their such corresponding predictions.

To begin with, the third line:

    x_3^+ = x_3+h x_4 + h^2/2! x_5


can be rewritten easily and substituting both x_4 = x_4^+ - h x_5 and x_5^+ = x_5, allows rewriting:

    x_3^+    =    x_3+h ( x_4^+ - h x_5^+ ) + h^2/2! x_5^+


that gives:

    x_3^+ = x_3+h  x_4^+ - h^2/2! x_5^+.


In the same way, substituting x_3 =  x_3^+ -h x_4^+ - h^2/2! x_5 and x_4 = x_4^+ - h x_5 in the second line gives after few calculations:











    x_2^+= x_2 + h   x_3^+ - h^2/2! x_4^+ + R( x_5 )


The term R( x_5 ) denotes the "residual" term associated to the useless fifth order.

Lastly, in the same way, substituting x_3, x_4 and  x_2 into x_1 gives: 


    x_1^+= x_1+h  x_2^+ -  h^2/2!  x_3^+ + h^3/3!x_4^+   + R( x_5^+ )


The bold terms in (<ref>) and (<ref>) correspond indeed to the correction terms that refine the Taylor expansion of the two first lines in (<ref>), taking into account the refinement of the precision of the Taylor terms thanks to the "retro-propagation" of the high order implicit terms.



§ ACKNOWLEDGMENTS

This work was supported partially by the ANR project DigitSlid ANR-18-CE40-0008-01 (https://anr.fr/Project-ANR-18-CE40-0008). Jean-Pierre Barbot is supported with (Région Pays de la Loire) Connect Talent GENYDROGENE project.


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 Paper Title: Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory.


Modeling stochastic eye tracking data: A comparison of quantum generative adversarial networks and Markov models
    Pedro Lind
    August 5, 2024
================================================================================================================



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	[anchor=north east,inner sep=20pt] at (current page.north east)
	
    < g r a p h i c s >
;


§ ABSTRACT 
We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. 
The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. 
By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. 
The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.


0.9
 AMS classification: 65M99, 81-08




Keywords: absorbing boundary conditions, variable potential, Schrödinger equation,
Titchmarsh-Weyl m-function, unbounded domain






§ INTRODUCTION

In this paper we consider the linear Schrödinger problem of the form

    u_t+∂_x^2u   =V(x)u,   (x,t)∈×(0,T],
    
       u(x,0)   =u_0(x),   x∈,

where T denotes the finite evolution time, and u_0 is an initial wave packet supported in a finite interval Ω_ int=[x_-,x_+] with x_-<x_+. 
It is well known that under mild conditions the Cauchy problem (<ref>) has a unique solution
u∈ C(^+,L^2()), cf. <cit.>, e.g.:

Let u_0∈ L^2() and real-valued potential V∈ L^∞().
Then the problem (<ref>) has a unique solution u∈ C(^+,L^2()).
Moreover, the “energy” is preserved, i.e.
    u(.,t)_L^2()=u_0_L^2(),  ∀ t ≥ 0.



The Schrödinger problem (<ref>) is defined on an unbounded domain x∈.
To numerically simulate its solution, it is common practice to truncate the domain to a bounded one,
for example, Ω_ int,T=Ω_ int×(0,T].
Absorbing boundary conditions (ABCs) are thus necessary for well-posedness at the two artificially introduced boundaries, Σ_±,T={x_±}×(0,T].

Numerical simulation of the linear Schrödinger equation on
unbounded domains with an external potential has been a hot
research area for nearly thirty years, cf. the concise review article <cit.>.
An ABC is called exact if the solution of the truncated domain problem remains the same
as that of the original unbounded domain problem.
The exact ABC is guaranteed to exist due to the well-posedness
of the linear Schrödinger problem (<ref>),
but it can only be formulated analytically for some special potentials,
such as constant potential <cit.>, linear potential <cit.>, symmetric periodic potential <cit.>,
isotropic free particle potential,
Morse potential, harmonic potential, and Bargeman potential,
cf. e.g. <cit.>.

In the more general case, i.e., for general variable potential problems
one is led to design approximate analytical ABCs for a given frequency regime with respect to some a priori criterion.
Methods in this category include the pseudo-differential calculus method <cit.>, the perfectly matched layer (PML) method <cit.>, 
and the operator splitting method <cit.>.
To the authors' knowledge, all of them are essentially based on the high frequency approximations.
For low-frequency problems, the ABCs would be less accurate by these methods.

This paper proposes a new approach to the design of ABCs for the Schrödinger problem.
Inspired by the work of Alpert, Greengard, and Hag­strom <cit.> on the fast evaluation of nonreflecting boundary kernels for time-domain wave propagation,
we approximate the Titchmarsh-Weyl m-function (equivalently, the exact DtN operator) in the frequency domain by a rational function with respect to an appropriate spectral parameter.
In the time domain, the nonreflecting boundary kernels are thus approximated by a sum of exponentials, which makes the approximate ABCs easy to implement.

The rationality of the above treatment is due to the analyticity property and the asymptotic behavior of the m-function.
Since our approximation is performed in the whole frequency regime, the proposed ABCs are expected to be more versatile and accurate, 
especially in the low-frequency regime, thus overcoming the typical high-frequency restriction.
Note that the Titchmarsh-Weyl m-function is nothing else but the so-called total symbol in microdifferential calculus,
which is treated by an asymptotic expansion to obtain a hierarchy of ABCs, cf. <cit.>.
Note also that Titchmarsh-Weyl theory is used in the analysis of initial value problems for Schrödinger equations operator-valued potentials <cit.> or strongly singular potentials <cit.>. It is also used in practical applications in quantum mechanics <cit.> and in option pricing in mathematical finance <cit.>.

This paper is organized as follows.
 In Section <ref> we review the basic facts of the Titchmarsh-Weyl theory for Schrödinger operators in one dimension for ease of later reference.
Then, in Section <ref> we discuss the Titchmarsh-Weyl m-function (i.e. the exact Dirichlet-to-Neumann operator) and explain the algorithm used to compute the m-function numerically.
Thus, at least from a numerical point of view, the exact ABC is explicitly known, see Section <ref>. 
However, when simulating the Schrödinger equation (<ref>), the difficulty does not lie in the computation of the m-function in a frequency domain method, 
which is presented in Section <ref>, but in its inverse Laplace transformation, which is too expensive.
For this reason, in Section <ref> we introduce a rational approximation of the m-function in the frequency domain
to obtain an approximate ABC that can be computed efficiently using a fast evaluation technique <cit.> in the time domain.
We discuss practical implementation issues and finally in Section <ref> we conclude with numerical results
illustrating that our new approach leads to an efficient and reliable algorithm for the time-dependent Schrödinger equation with a general variable potential.






§ THE TITCHMARSH-WEYL THEORY

We will review here, for convenience of later work, the essentials of
Titchmarsh-Weyl (TW) theory for Schrödinger operators in one dimension.
The interested reader may consult <cit.> or <cit.> for a more detailed presentation.

First, consider the Schrödinger operator L on the real line given by

    L=-∂_x^2+V(x),     x∈,

with a real-valued, locally integrable potential V∈ L^1_ loc().
Let x_0∈ be an arbitrarily chosen point, called reference point.
In the following we will study how the solutions depend on this parameter x_0.

To do this, we consider θ(x;x_0,λ) and ϕ(x;x_0,λ) to be the fundamental solutions of the Schrödinger eigenvalue problem.

    -u_xx+V(x)u=λ u,     x∈,  λ∈,

with the following initial conditions at the reference point x_0:
 

    θ(x_0;x_0,λ)   =1,    θ_x(x_0;x_0,λ)=0,
    ϕ(x_0;x_0,λ)   =0,    ϕ_x(x_0;x_0,λ)=1.


It can be shown that under these assumptions
θ(x;x_0,λ) and ϕ(x;x_0,λ) exist on the whole real axis,
and they are entire functions of λ and real for λ∈.
Now, as a basic fact of TW theory, the equation (<ref>) has at least one solution ψ_±, called Weyl's solution with


    ψ_±(x_0;x_0,λ)=1,

and


    ψ_±(x;x_0,λ)∈ L^2(^x_0_±)

for any λ∈_+.

Here ^x_0_± stands for the interval [x_0,±∞) and _+ denotes the upper half of the complex plane, i.e. _+={z∈  | z>0}.
A potential V(x) is said to be in the limit-point case at ±∞ if and only if there exists only one Weyl's solution in the corresponding L^2 space.
The reader will immediately realize that the assumption of V(x) in the limit-point case is necessary for the well-posedness of the Schrödinger problem (<ref>) in a more general setting.
At positive infinity, a standard sufficient condition for the limit-point case is given by <cit.>:


Let V(x) be a continuous real-valued function on (x_0,∞) and suppose that there
exists a positive differentiable function M(x) such that

  
  (i) V(x)≥-M(x) if x>x_0;
  
  (ii) ∫_x_1^∞(M(x))^-1/2 dx=∞ for any x_1>x_0;
  
  (iii) M'(x)/(M(x))^3/2 is bounded near ∞.

Then V(x) is in the limit-point case at  ∞.

An analogous result can be given at negative infinity point.

According to this theorem, a potential V(x) is in the limit-point case
provided that V(x)≥-k x^2 for some constant k and for all sufficiently large x.
This implies that the restriction on the potential for the limit-point case is very weak:
 It only excludes some especially strange potentials, which may not be physically relevant at all.
Roughly speaking, the limit case does not admit potentials that tend too fast
(faster than quadratically) to -∞ for x→±∞.

Due to the boundary conditions (<ref>) we can write

    ψ_±(x;x_0,λ)= θ(x;x_0,λ)+m_±(x_0,λ) ϕ(x;x_0,λ),

with some uniquely determined coefficient,
the Titchmarsh-Weyl m-function m_±(x_0,λ).
This function plays a fundamental role in the spectral theory of
the Schrödinger operator (<ref>) on the half-line ^x_0_±.

We will now summarize some of the most important properties of the
Titchmarsh-Weyl m-function. First,

    m_±(x_0,λ)  is analytic with respect to λ to \ and  m_±_+→_+

and is therefore called a Herglotz function (or Nevanlinna or Pick function),
cf. <cit.>.
It is easy to show that this Herglotz-property is directly related to the positive type of the DtN-map in the sense of memory equations,
cf. <cit.> for the corresponding case of constant external potential.
Thus it is an essential ingredient of the stability w.r.t. the L^2 norm.

We also have the symmetry property

    m_±(x_0,λ)=m_±(x_0,λ̅)

and the local singularities of m

are real and and most of them are first order, i.e.

    lim_ϵ→0+(-ϵ) m_±(x_0,λ+ϵ)   ≥0,  λ∈,


cf. <cit.>.

Another important property is given by the Borg-Marchenko theorem <cit.>, 
which states that the Titchmarsh-Weyl m-function m_±(x_0,λ) 
uniquely determines the potential V(x) at x>x_0 (or x<x_0).
Moreover, since ψ_±(x;x_0,λ) changes with a simple multiplication 
when the reference point x_0 changes, one has

    m_±(x,λ)= ∂_xψ_±(x;x_0,λ)/ψ_±(x;x_0,λ).

Thus, it is easy to verify that the m function satisfies the following Riccati equation:

    ∂_xm_±(x,λ)=-m_±^2(x,λ)+V(x)-λ.







§ THE EXACT ABC BY TITCHMARSH-WEYL THEORY

We apply the Laplace transform

    (x,s) = ℒ(u(x,t))(s) = ∫_0^+∞ u(x,t)  e^-st dt,    s>0,

to the Schrödinger equation (<ref>) on the right exterior domain Ω_+={x∈| x>x_+} and on the left exterior domain Ω_-={x∈| x<x_-}.
In the frequency domain, the Schrödinger equation is a second order homogeneous ODE

    -_xx+V(x)=λ,   x∈Ω_±,

with λ= s∈_+. The exact absorbing boundary condition of the DtN form in the frequency domain is thus

    û_x(x_±,λ) = m_±(x_±,λ)û(x_±,λ).



Only in some special cases, the m-function has a closed analytic form <cit.>.
For example, in the case of a constant potential V≡ V_0 one gets

    m_+(x_+,λ)=-√(-λ+V_0).

If the potential represents a harmonic oscillator, i.e. V(x)=x^2 on the interval [0,∞),
one obtains a meromorphic m-function given by the ratio of two gamma functions:

    m_+(0,λ)=-2Γ(3/4-1/4λ)/Γ(1/4-1/4λ).










Finally, for the Bargmann potential

    V(x)=-8β^2 β-γ/β+γe^-2β x/(1+β-γ/β+γe^-2β x)^2,  β>0,  γ≥ 0,

one obtains the m-function

    m_+(0,λ)=-√(-λ)-γ^2-β^2/√(-λ)+γ.


In the general case, however, numerical methods must be considered.
This problem has been studied in many papers, e.g. <cit.>.
In this paper we simply compute the m-function by evolving the Riccati equation (<ref>) 
with the classical fourth-order Runge-Kutta scheme and setting an initial data m_±(x_±,λ,λ)=∓√(-λ) 
at a sufficiently distant point x_±,λ=± 200. 
This treatment is reasonable since the potentials in our numerical tests actually decay to zero for x→∞.







§ THE FREQUENCY-DOMAIN METHOD

The solution of the time-dependent Schrödinger equation could then be computed 
using the following frequency-domain method:


  * Fix σ>0.
For each s=σ+μ with μ∈, solve the

Laplace-transformed Schrödinger equation in the bounded interval 
[x_-,x_+]:

    -_xx+V(x)   =  s  - u_0(x),   x∈[x_-,x_+],
    
    
      _x(x_-)   =m_-(x_-, s) (x_-),
    _x(x_+)   =m_-(x_+, s) (x_+).


  * Perform the inverse Laplace transformation

    u(x,t)    = ℒ^-1(û(s,t))(x)
       =1/2π∫_σ-∞^σ+∞e^st(s,t) ds
    =
    e^σ t/2π∫_-∞^∞e^ ftu(x,σ+ f) df,


to derive the wave function u(x,t) for any t∈(0,T].


In the numerical implementation, some parameters have to be tuned.
The function û is smoother for larger damping factor σ, but the evolution time span is then limited
because an exponential term is involved in the inverse Laplace transformation. 
As a common practice, we set σ=1/T, where T is a prescribed evolution time. 
The integration domain is unbounded in (<ref>) and must be truncated. 
We introduce a cut-off frequency f_c and confine the integration to the interval [-f_c,f_c]. 
In addition, to get rid of high frequency oscillations in the inverse-transformed function, we should introduce another filtering function χ,
which remains 1 over a sufficiently large frequency band with zero frequency as its center, and vanishes smoothly near the endpoints of [-f_c,f_c]. 
A good candidate (empirically) is

    χ=exp(-(1.2f/f_c)^20).

After these treatments, we then derive an approximate inverse transformation as

    e^σ t/2π∫_-∞^∞e^ ftu(x,σ+ f) df≈e^σ t/2π∫_-f_c^f_cχ(f) e^ ftu(x,σ+ f) df.

The right side is computed with an appropriate quadrature scheme.




§ THE TIME-DOMAIN METHOD

It follows from Section <ref> that the exact ABC we are looking for is now explicitly known, at least from a numerical point of view.
But this is not the whole story for simulating the solution of the time-dependent Schrödinger equation.
The difficulty lies not in computing the m-function itself, but in computing its inverse Laplace transformation.
Of course, a numerical inverse transformation is possible, but it would be too expensive.

Therefore, in this section we design an approximate ABC based on the rational approximation of the m-function.
The kernel functions are of exponential type with respect to the half-order time derivative operator, 
so the fast evaluation technique proposed in <cit.> (cf. Appendix) is applicable.
For some alternative fast evaluation methods, we refer the reader to <cit.> and the references therein.
The rational approximation is realized by solving a least squares problem, 
an analogous technique to that used in <cit.> for fast evaluation of the boundary kernel functions of the hyperbolic wave equation.

In the time domain, the truncated Schrödinger problem reads

    u_t+∂_x^2u    = V(x)u,   (x,t)∈[x_-,x_+]×(0,T],
    
      u(x,0)   =u_0(x),   x∈[x_-,x_+],
    
      u_x(x_±,t)    = ℒ^-1(m_±(x_±, s) û(x_±,s))(t),   t∈ (0,T].

To simplify the notation, we will focus on the right boundary at x=x_+.
Let us recall that the DtN map in the frequency domain is

    û_x(x_+,s) = m_+(x_+, s) û(x_+,s).

Returning to the time domain we have to consider the convolution

    u_x(x_+,t) = K(t)*u(x_+,t),
        with  K(t) = ℒ^-1(m_+(x_+, s))(t).

There are two major difficulties here.
First, it is generally hard to compute K(t)
and second, the convolution involved naturally leads to a nonlocal-in-time DtN map.

To get an idea, let us first consider two specific simple examples.
In the case of the free Schrödinger (V≡ 0), cf.(<ref>), we have

    m_+( s)=-√(- s)   and thus   ℒ^-1(m_+( s))=-e^-π/4 ∂_t^1/2,

with the half-order time derivative defined as

    ∂_t^1/2 v(t) = 1/√(π)  d/ d t∫_0^t v(τ)/√(t-τ)  dτ,

which can be efficiently evaluated with some existing methods,
e.g. <cit.>.
Second, we consider the Bargmann potential (<ref>), and in this case
the m-function is

    m_+(0,is)=-√(- s)-γ^2-β^2/√(- s)+γ,

thus we have

    ℒ^-1(m_+(0, s)) = -e^-π/4 ∂_t^1/2
        -(γ^2-β^2)(e^-π/4 ∂_t^1/2+γ)^-1.

This operator can then be evaluated efficiently by introducing an unknown function and using the fast methods for ∂_t^1/2.

Inspired by these two examples it naturally leads us to think about the possibility of
approximating the m-function with a rational function with respect to a new spectral parameter k=√(- s) (NOT s), i.e.,

    m_+(x_+, s)≈m̃_+(x_+, s)
      =-√(- s)+∑_n=1^dα_n/√(- s)+β_n.

Once this is done, we can then replace the exact m-function with the approximate alternative m̃,
which leads to the approximate kernel function

    ℒ^-1(m̃_+(x_+, s)) 
       = -e^-π/4 ∂_t^1/2+∑_n=1^dα_n(e^-π/4 ∂_t^1/2+β_n)^-1.

The analogous idea for the Bargmann potential (<ref>) can then be used to handle this kernel function.

The answer for the possibility is affirmative considering that the asymptotic expansion has been given in <cit.> as

    m_+(x_+,λ)=-√(-λ)+(1/√(r)),   r→∞,

where λ=μ r, r∈ and the convergence is uniform for μ in any compact subset of _+, cf. <cit.>.

Now putting

    _+(λ)=m_+(x_+,λ)+√(-λ),

in view of (<ref>) we know that _+(λ) is analytic in _+ with respect to λ and
it tends to zero for λ→∞.
We then use the method of Alpert, Greengard and Hagstrom  <cit.> to approximate _+(λ) with a rational function with respect to √(-λ) (NOT λ).
In terms of (<ref>) we consider the following nonlinear least squares problem

    ϵ=min_P,Q∫_-∞+σ^∞+σ|P(√(-λ))/Q(√(-λ))-_+(λ)|^2 |d√(-λ)|,

where P, Q are polynomials with (P)+1=(Q)=d, and d is determined by making ϵ≤ϵ_0, where ϵ_0 is a prescribed tolerance number.
This nonlinear problem (<ref>) is then solved using the technique of
 linearization and orthogonalization <cit.>.
Finally, by expressing P/Q as a sum of poles, we arrive at

    m_+(x_+,λ)≈m̃_+(x_+,λ)=-√(-λ)+∑_n=1^dα_n/√(-λ)+β_n.

Note that the coefficients α_n and β_n should appear as conjugate pairs due to the symmetry property (<ref>).
Unfortunately, it is not clear whether the rational approximation m̃_+(x_+,λ) in (<ref>) still has the important Herglotz-property of the m-function.
Moreover, the Herglotz-property cannot be checked by some conditions on the poles due to the leading square root in (<ref>).



Applying the same idea to the m-function m_-(x_-,λ) and we get
the approximate boundary condition in the frequency domain

    _x(x_±,s) = (∓√(- s)+∑_n=1^d_±α_n,±/√(- s)+β_n,±)(x_±,s).

If we introduce new unknowns ŵ_n,± as

    ŵ_n,± = (x_±,s)/√(- s)+β_n,±,

then we can rewrite (<ref>) as


    _x(x_±,s)±√(- s)    = ∑_n=1^d_±α_n,±ŵ_n,±,
    √(- s) ŵ_n,±+β_n,±ŵ_n,±   = (x_±,s),     n=1,…,d_±.


In the time domain, the approximate boundary condition is


    u_x(x_±,t)± e^-π/4 ∂_t^1/2u(x_±,t)
         = ∑_n=1^d_±α_n,±w_n,±(t),
    
      e^-π/4 ∂_t^1/2w_n,±(t)+β_n,±w_n,±(t)   =u(x_±,t),     n=1,…,d_±.






The final approximate truncated time-domain problem is formulated as

    u_t+∂_x^2u   =V(x)u,   (x,t)∈Ω_ int×(0,T],
    
      u(x,0)   =u_0(x),   x∈Ω_ int,
    
      u_x(x_±,t)   ± e^-π/4 ∂_t^1/2u(x_±,t)
      = ∑_n=1^d_±α_n,±w_n,±(t),
    
      e^-π/4 ∂_t^1/2w_n,±(t)   +β_n,±w_n,±(t)
    =u(x_±,t),     n=1,…,d_±.






§ NUMERICAL RESULTS

In this section, we present some numerical results to test the accuracy of the proposed methods.
In each example, the standard Crank-Nicolson scheme for time discretization is used.
The fast evaluation of the half-order time derivative operator (<ref>) is performed with the method of Zheng <cit.>.
The computational domain is chosen to be Ω_ int=[x_-,x_+]=[-5.5] and the initial data is a Gaussian beam: u_0(x)=e^-x^2+4 x.
We use an 8^ th-order FEM method with 1024 elements for the spatial discretization and a uniform time step of size Δ t=10^-4.





 §.§ The Free Schrödinger Equation

The exact solution of the free Schrödinger equation (V(x)≡0) is

    u_ exa(x,t) = √(/-4t+)exp(- x^2-4x+16t/-4t+).

We set the cut-off frequency f_c in (<ref>) to be f_c=256 and used a filtering function χ(f)=exp(-(1.2f/f_c)^20).
The following Table <ref> shows the relative L^2-errors to the exact solution at certain time points when using 8097 quadrature points with Simpson's rule.

Using this simple example, we can see that the frequency method with truncation and filtering works quite well: the magnitude of the relative L^2-errors is at most on the order of 10^-7.





 §.§ The Coulomb-like Potential

In the second example we test the time-domain method with a Coulomb-like potential

    V(x)=1/√(1+x^2).

We set σ for Step 1 to σ=1 and the tolerance number for the nonlinear least squares problem (<ref>) to ϵ_0=10^-8. 
Here we get 4 poles.
Figure <ref> shows the time evolution in a colored contour plot. 
One can see how the initial beam spreads out as time increases.

Figure <ref> shows how the relative L^2-error evolves in time.
In this example with a varying external potential the magnitude
of the relative L^2-errors remains below 10^-5.






 §.§ The Gaussian Barrier

Next, we change the potential to a Gaussian barrier

    V(x)=30e^-36(x-8)^2

with a height of 30, located in the exterior domain x>5 and centered at x=8. 
We set σ=1 and ϵ_0=10^-4 and get 21 poles with the nonlinear least squares algorithm. 
Figure <ref> shows the time evolution of the solution. 
One can clearly see how the initial beam propagates, spreads, and is (partially) reflected by the Gaussian barrier (<ref>). 
The time evolution of the corresponding relative L^2-error is shown in Figure <ref>.
The relative L^2 errors remain below 5× 10^-4.

Unfortunately, the nonlinear least squares algorithm used in this paper failed to produce a rational approximation within an error tolerance much smaller than ϵ_0. 
A more efficient algorithm is still desired, and this problem is currently under investigation.












§ CONCLUSION AND OUTLOOK

In this work we presented a new approach to simulate the solution to the Schrödinger equation with a general space-dependent potential in unbounded domains. 
Both frequency-domain and time-domain methods have been developed.

Future work will consist of implementing a more sophisticated algorithm for computing the m-function.
Instead of solving the Riccati equation (<ref>), we will consider computing the Weyl circles <cit.>
or the boundary control approach of Avdonin, Mikhaylov and Rybkin <cit.>.
We will also look for a more stable algorithm for its rational approximation.
It will also be clarified how this rational approximation can be made to preserve the essential 
Herglotz-property of the analytic m-function.
This study will allow us to perform a rigorous stability analysis of this new approach.
Finally, as a future goal, we want to extend our approach to the multi-dimensional Schrödinger problem, following the idea of Amrein and Pearson <cit.>.






§ APPENDIX: FAST EVALUATION METHOD

Here we present a short description of the method in <cit.> for evaluating the
half-order time derivative ∂_t^1/2. 
For any smooth function v=v(t) with v(0)=v'(0)=0, 
it is known that the semi-discrete half-order derivative

    D_t^1/2v(t_n):=√(2/Δ t)∑_m=0^nα_m  v(t_n-m)

with

    α_m=β_k=(2k)!/2^2k(k!)^2   , m=2k,
    
    -β_k   , m=2k+1,

gives a second-order approximation of ∂_t^1/2v(t_n) (see <cit.>).
Suppose there exists a sum of decaying exponentials satisfying

    β̃_k = ∑_j=1^Lw_j e^-s_jk,   s_j>0,   |β_k-β̃_k|≤ϵ, k=0,1,…,[N/2].

Here N is the total number of time steps. 
If ϵ is small enough, it is reasonable to approximate (<ref>) with

    D̃_t^1/2v(t_n) := √(2/Δ t)(v(t_n)-v(t_n-1)) + √(2/Δ t)∑_m=2^nα̃_mv(t_n-m),

where

    α̃_m=
    β̃_k   , m=2k,
    
     -β̃_k   , m=2k+1.

Set v_k=v(t_k), 𝐯=(v_0,v_1,…), and define

    ℱ_ odd(w,s;𝐯,k)
    
    :=∑_m=1^kwe^-smv_2k+1-2m

and

    ℱ_ even(w,s;𝐯,k)
    
    :=∑_m=1^kwe^-smv_2k-2m.

Thus,
ℱ_ odd(w,s;𝐯,0)=ℱ_ even(w,s;𝐯,0)=0.
In addition, we have the following recursions

    ℱ_ odd(w,s;𝐯,k)
       =   e^-s[wv_2k-1+ℱ_ odd(w,s;𝐯,k-1)],
    ℱ_ even(w,s;𝐯,k)
       =   e^-s[wv_2k-2+ℱ_ even(w,s;𝐯,k-1)].

The summation (<ref>) is then computed within O(L) operations
as

    ∑_m=2^nα̃_m v_n-m=
    ∑_j=1^Lℱ_ even(w_j,s_j;𝐯,k)-∑_j=1^Lℱ_ odd(w_j,s_j;𝐯,k-1)   , n=2k,
    ∑_j=1^Lℱ_ odd(w_j,s_j;𝐯,k)-∑_j=1^Lℱ_ even(w_j,s_j;𝐯,k)   ,
    n=2k+1.

In <cit.> for N=1,000,000, the authors found a sum of 81
decaying exponentials that approximates β_k with an error of less than 5.0× 10^-11.



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 Paper Title: Tree-Cotree-Based Tearing and Interconnecting for 3D Magnetostatics: A Dual-Primal Approach.


    
	

		

		cemaddress,cceaddress]Mario Mally
		mario.mally@tu-darmstadt.de

		epfladdress]Bernard Kapidani
		bernard.kapidani@epfl.ch

		cemaddress,cceaddress]Melina Merkel
		melina.merkel@tu-darmstadt.de

		cemaddress,cceaddress]Sebastian Schöps
		sebastian.schoep@tu-darmstadt.de

		[cemaddress]Computational Electromagnetics Group, Technische Universität Darmstadt, 64289 Darmstadt, Germany
		[cceaddress]Centre for Computational Engineering, Technische Universität Darmstadt, 64289 Darmstadt, Germany
		[epfladdress]Chair of Numerical Modelling and Simulation, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

		§ ABSTRACT 
			The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree-cotree decomposition. The classical tree-cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable.
			Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual-primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence.
		

		
			 IGAIETIDual-PrimalTree-Cotree GaugingMagnetostatics
		

	

	

§ INTRODUCTION

    Simulation-based design workflows for engineering products are ubiquitous in academia and industry. Their success relies on accurate and efficient solvers. A suitable method to achieve this is IsoGeometric Analysis (IGA), due to its ability to accurately represent complex geometries and to provide higher continuity in the solution space. This is achieved by using NURBS (Non-Uniform Rational B-Splines) to represent the geometry and B-Splines to approximate the solution <cit.>. These advantages are particularly attractive for the magnetostatic simulation of permanent magnet synchronous machines due to their cylindrical shape <cit.> and the computation of sensitive quantities, such as torque <cit.>. This is made possible by the introduction of spline spaces suitable for the representation of electromagnetic fields <cit.>.

However, as the complexity and size of simulation models increase, it becomes essential to exploit parallel computing architectures to effectively handle the computational load. We follow the Domain Decomposition (DD) paradigm <cit.> and in particular the Tearing and Interconnecting (TI) methods originally proposed by  in <cit.>. These methods are probably best known as Finite Element Tearing and Interconnecting (FETI), while the IGA counterpart is called Isogeometric Tearing and Interconnecting (IETI) <cit.>. Both decompose the computational domain into subdomains, couple them weakly at conformal interfaces using Lagrange multipliers, and thus promise scalability for large problems. However, solvability is a fundamental issue of decomposed Laplace-type problems due to the lack of essential boundary conditions for `floating' subdomains. Two main approaches address this problem: The dual-primal approach, see e.g. <cit.> or the total/all-floating approach <cit.>. 

For high-frequency electromagnetic problems, FETI has been investigated for scattering and wave propagation, as discussed in <cit.> and references therein. In the case of magnetism, it was applied to the scalar potential formulation, i.e. a Laplace-type problem, in <cit.>. The vector potential formulation suffers from non-uniqueness issues due to the kernel of the curl operator. This formulation is considered in <cit.> and <cit.>. The first two contributions regularize the problem by perturbing the operator. We follow the idea of  and employ a tree-cotree decomposition <cit.>.

Let us state our contributions. We provide a mathematical justification of the approach from  which applies only to problems with full Dirichlet boundary conditions and low-order finite elements. We give a new explicit algorithm for constructing a tree-cotree decomposition that also applies to problems with mixed boundary conditions and high-order isogeometric discretizations. Finally, we validate our approach with numerical experiments using the IGA-library  <cit.>.

The paper is structured as follows: <ref> introduces the relevant Maxwell equations and provides an overview on the weak FETI/IETI formulation for magnetostatics as well as its discretization. <ref> explains the fundamental concepts of tree-cotree decompositions and presents an explicit algorithm for constructing appropriate trees.
In <ref>, we examine and analyze the constructed tree within the context of dual-primal TI, detailing how to correctly select primal degrees of freedom (DOFs) to ensure local invertibility in general settings. Finally, in <ref> the proposed approach is validated with numerical experiments and its scalability is examined with respect to relevant properties for large-scale problems.

    
    
	

§ MAGNETOSTATICS ON DECOMPOSED DOMAINS

    In the low-frequency regime, e.g. for the simulation of permanent magnet synchronous machines <cit.>, the magnetostatic approximation of Maxwell's equations <cit.>
[left = ]align
	()    = 

	   = 0

is most relevant. The system is formulated in terms of the magnetic flux density  and the excitation current density . For the underlying material parameter  (reluctivity) we assume 0<<∞. Furthermore, we assume that these laws are prescribed on a bounded, simply-connected domain Ω⊂ℝ^3 (this assumption will be relaxed later). We introduce a magnetic vector potential via =, which automatically satisfies (<ref>). Inserting it into (<ref>) results
[left=]align
    ()   =    in  Ω,

    ×   =   on  Γ_D,

    ()×   =   on  Γ_N,

which includes inhomogeneous Neumann boundary conditions on Γ_N and inhomogeneous Dirichlet boundary conditions on Γ_D. Here,  denotes the outward normal vector of ∂Ω. Furthermore, we assume that ∂Ω=Γ_N∪Γ_D (where D denotes the closure of open set D) as well as Γ_N∩Γ_D=∅ hold. For brevity, we consider only =0 in all derivations, but more general  can be considered by using homogenization techniques. The vector potential formulation (<ref>)-(<ref>) has a major intricacy originating from the non-trivial kernel of the curl-operator. To summarize the descriptions of <cit.>, we can write

    (ϕ + )=

and characterize the kernel with gradient fields ϕ and harmonic fields  which arise if Ω is not simply-connected, i.e. the first Betti number b_1∈ℕ_0 of Ω is greater than zero <cit.>. The identity (<ref>) implies that only _⊥∈ker(curl)^⊥, which is orthogonal to both gradient and harmonic fields, can be uniquely determined from (<ref>)-(<ref>). The remaining part 𝐀_0=ϕ+ is undetermined, i.e., there is the gauge freedom

    =  = (_⊥ + _0) = _⊥.

By fixing the kernel component _0, which does not affect our quantity of interest , and reformulating (<ref>), one can compute a unique _⊥ if the current density is compatible, i.e. ∈ker(curl)^⊥, e.g. see <cit.>. This procedure is denoted as gauging in the literature, e.g. <cit.>. One particular example, for simply-connected domains, is the Coulomb gauge where =_0=0 is imposed as an additional constraint. In this paper, we focus on the tree-cotree gauge, which is applied in the discrete setting for the TI approach. Consequently, we need to discuss our weak formulation, the discretization, and the extension to domain decomposition before we can elaborate more on the gauging technique.



 §.§ Weak Formulation and its Discretization

Let us state (<ref>)-(<ref>) in its weak form as: Find ∈Ω such that

    Ω()·()=Ω·+Γ_N·

for all ∈Ω where Ω denotes the adequate Hilbert space for vector potentials with vanishing tangential component on Γ_D <cit.>. We follow the typical Ritz-Galerkin procedure and introduce a finite-dimensional basis {_i}_i=1^n_g such that 𝕍span(_i)_i=1^n_g⊂Ω.

We approximate  by a linear combination of finitely many basis functions, i.e.

    _h=∑_i=1^n_ga_i_i∈𝕍.

In this context, we call a_i a degree of freedom (DOF) and introduce _g∈ℝ^n_g as the vector of all degrees of freedom (DOFs). Inserting the discrete field _h into (<ref>) leads to

    (_g)_ijΩ(_j)·(_i)

for the entries of stiffness matrix _g∈ℝ^n_g× n_g and to

    (_g)_i Ω·_i + Γ_N·_i

for the right hand side _g∈ℝ^n_g. Here, we added the subscript g to emphasize that this is the global problem. 
After incorporating the Dirichlet boundary conditions, <cit.>, the linear system

    _g_g=_g,

arises where _g, _g and _g are appropriately modified into _g, _g and _g, respectively.

Note that while we expect our findings to apply to any higher-order FEM discretization, e.g. tetrahedral <cit.>, we focus on the multipatch IGA approach with tensor product, high-order, edge-element basis functions. The adequate spline space is 𝕍=S^1_p(Ω) as defined in <cit.> and references therein. Details are omitted for brevity, but note that we assume degree p for each directional component of the vector-valued basis functions _i. The associated degrees of freedom a_i can be identified with the edges of the control mesh <cit.>, which we interpret as a graph and denote by

    =(,).

We write  and  for the sets of nodes and edges of the graph, respectively. In the lowest-order case, i.e. p=1, this control mesh corresponds to the mesh of a conventional FEM discretization with lowest-order Nedelec-type hexahedral elements.



 §.§ Nonoverlapping Domain Decomposition

We assume that a non-overlapping decomposition of domain Ω into N bounded and simply-connected subdomains Ω^(i) is given with i∈{1,…,N}. Again, for brevity, we assume simply-connectedness. This decomposition shall satisfy Ω=⋃_iΩ^(i) and the subdomains are mutually disjoint, i.e., Ω^(j)∩Ω^(k)=∅ for j≠ k. 

Each boundary ∂Ω^(i) shall consist of open facets Γ^(i,k) such that ∂Ω^(i)=⋃_kΓ^(i,k) and Γ^(i,k)∩Γ^(i,ℓ)=∅ for k≠ℓ. A facet is either an interface, i.e., there are i<j and k,l such that Γ_ij:=Γ^(i,k)=Γ^(j,l) or a subset of the Dirichlet or Neumann boundary, i.e., Γ^(i,k)⊂Γ_D or Γ^(i,k)⊂Γ_N, respectively. Finally, each facet boundary consists of edges λ^(i,j,k) such that ∂Γ^(i,j)=⋃_kλ^(i,j,k) and λ^(i,j,k)∩λ^(i,j,ℓ)=∅ for k≠ℓ. The union of all edges (including their endpoints) defines the wire basket 𝒲^(i):=⋃_j,kλ^(i,j,k), which is visualized in <ref>.

Following <cit.>, we assume that the subdomains are unions of shape-regular elements for which the nodes and edges of the local control meshes, associated with the local bases 𝕍^(i)span(_j^(i))_j=1^n_i⊂Ω^(i), match on the pairwise, non-empty intersections ∂Ω^(j)∩∂Ω^(k) of the respective boundaries. Under mild restrictions as in <cit.>, the multipatch setting in IGA naturally yields a decomposition that satisfies these assumptions.

For the local (control) meshes, we introduce graphs again. Analogous to the previous notation, ^(i) = (^(i),^(i)) denotes the (sub)graph formed by the nodes ^(i) and edges ^(i) related to the basis functions on Ω^(i). Furthermore, Γ^(i) = (Γ^(i),Γ^(i)) and λ^(i) = (λ^(i),λ^(i)) are the graphs for the basis functions on the boundaries and local wire baskets, respectively.
Note that these subdomain-wise definitions can be extended to the global domain, i.e. Γ and λ, by creating unions due to the consistency of the decomposition. A visualization of these local and global subgraphs is provided in <ref> and <ref>.





  §.§.§ The Local Discrete Problems and Their Coupling


Again, with each 𝕍^(i), we can exploit ^(i)_h=∑_j=1^n_ia^(i)_j^(i)_j. This leads to stiffness matrices ^(i)∈ℝ^n_i× n_i, for which the components are given by

    (^(i))_jkΩ^(i)^(i)(_k^(i))·(_j^(i)).

For the local ^(i)∈ℝ^n_i, we write

    (^(i))_kΩ^(i)·^(i)_k + Γ_N^(i)·^(i)_k

such that ^(i)^(i)=^(i) defines the locally independent problems for the DOFs ^(i)∈ℝ^n_i without Dirichlet boundary contribution. Summarizing all local linear systems in a block diagonal structure yields

    =[ ^(1)          ;         ⋱     ;           ^(N);      ][ ^(1);    ⋮; ^(N);      ]=[ ^(1);    ⋮; ^(N);      ]=

as a system. Here, all local problems are still independent from each other and it is necessary to introduce coupling to recover the original global context. Due to the conforming decomposition of Ω and of space 𝕍, coupling constraints are defined in the following form

    =,

where ∈{-1,0,1}^m× n is a signed boolean matrix. Here,
n is given by
n=∑_i=1^Nn_i and each row of  encodes a constraint of the form

    a_i_j^(j) - a_i_k^(k) = 0,

which enforces tangential continuity of the discrete solutions ^(i)_h on all interfaces. The constraints are then incorporated by using an approach based on Lagrange multipliers which we denote here as ∈ℝ^m. This results in

    [    ;   0;   ][ ; ;  ]=[  ; 0;   ],

where the Dirichlet boundary conditions are already imposed. The underlying modification affects not only the local systems, i.e. ^(i), ^(i) and ^(i), but also the coupling constraints. At the intersection between the Dirichlet boundary parts and interfaces, the basic coupling between interface DOFs produces zero rows when the Dirichlet DOFs are eliminated from the system. After removing the corresponding constraints and multipliers, we write  and  instead of  and , respectively.



  §.§.§ From Tearing and Interconnecting to the Global Problem

Let us establish the equivalence of the saddlepoint formulation (<ref>) and the original (multipatch) formulation (<ref>). We reorder the equations and unknowns such that _p and _r are related to all globally coupled basis functions and the remaining ones, respectively. This gives rise to the system

    [ _rr _rp   0; _pr _pp  _p;   0  _p   0;     ][ _r; _p;   ;    ]=[ _r; _p;  0;    ].

The constraint _p_p=0 can be expressed in another way if we have a matrix ∈{0,1}^n_p× n_gp such that ker(_p)=im() or equivalently _p=0 with n_gp=ker(_p). Then, we can express the constrained DOFs as _p=𝐩 with 𝐩∈ℝ^n_gp which automatically satisfies the constraints. Substituting _p and eliminating  in (<ref>) yields

    [ _rr _rp; _pr _pp;     ][ _r;  𝐩;    ] = [ _r; _p;    ]   Reord.⟺  _g_g=_g

as a rearranged representation of (<ref>). In the context of quadratic programming under equality constraints, this approach is called a null-space method <cit.>. Details in the context of FEM are given, for example in <cit.> and for multipatch IGA in <cit.>. Typically, the challenge of this approach is to compute the kernel representation  due to inefficient numerical algorithms. However, due to the signed boolean coupling structure,  can be constructed explicitly and efficiently in the TI setting.

    
    

§ TREE-COTREE GAUGING FOR TEARING AND INTERCONNECTING

    An impasse for efficiently finding a solution to (<ref>) is the rank deficit of the global and local problems. In this section, we aim to provide an algorithm that consistently regularizes both levels, eventually allowing the use of direct sparse solvers for the subdomain problems.

There are several culprits for the rank deficit: First, there are gradient fields in the nullspace of the curl-operator. Secondly, harmonic forms introduce an additional rank deficit per hole in Ω. Additional problems arise due to `cross-edges', i.e., the edge-element equivalent to `cross points'. These are edges that are shared by more than two subdomains or equivalently, belong to more than one interface. We denote them by II^(i) and II. They can introduce redundancy in the coupling matrix  and thus destroy the well-posedness of (<ref>).

We follow <cit.> and employ a tree-cotree decomposition to deal all the culprits. The idea behind this approach is to split the DOFs into determined _c (pivot or cotree DOFs) and undetermined variables _t (zero-pivots or tree DOFs) that need gauging <cit.>. This decomposition can be obtained from topological mesh information alone by growing an adequate tree on the corresponding graph. The gauging consists of setting the tree DOFs to an arbitrary value, e.g., _t=0, reformulating and solving (<ref>) for the remaining _c <cit.>. This takes care of the (discrete) gradient fields while the harmonic ones require additional tree modifications, <cit.>.



 §.§ Tree Construction for Dual-Primal Approach

Let us start with a wish list of properties of the tree-cotree decomposition so that we can apply our dual-primal approach. First, our tree needs to be able to preserve the global information, i.e. we need a full tree on the subgraph D belonging to the Dirichlet boundary. We define a full tree on a subset of nodes ∙ by the property ∙ t=∙-1. Second, we want to have matching tree elements on separated interface sides. The full tree on D is necessary because eliminating the Dirichlet DOFs removes exactly ker(_g)-ker(_g)=Dt kernel components, where DtD∩t denotes the edges of a full tree on Γ_D, see <cit.>. The matching condition on interfaces automatically preserves the coupling conditions. If two coupled DOFs are both set to zero, we do not violate the coupling constraint. Howevery, one needs to remove zero rows as for Dirichlet boundary conditions again. These conditions motivate us to grow the tree on the global graph  and obtain the local trees for gauging (<ref>) by tearing the global mesh into local ones.

For now, we only have conditions related to the local boundaries ∂Ω^(i), but we want to grow the tree on λ first. If we do this correctly, we can preserve the first two conditions because λ⊂Γ. This step is relevant for two reasons. First, we can separate the facets from each other (from a tree construction point of view) which enables extending the tree into the facets independently of the other facets. The second reason is that we want to have as many edges as possible in II∩t which makes focusing on the wire basket necessary due to II⊂λ. Having matching trees on each interface implies that we are able to completely remove the corresponding redundant coupling for II∩t.



To summarize, we start growing the tree on λ, then extend it into Γ before considering the full graph . This procedure is highlighted on the left side (gray blocks) in <ref>. The tree hierarchy ensures  consistency with boundary/interface information: we classify the edges into Γ and λ depending on which boundary parts they contribute to. The edges of Γ belong to  either an interface I, the Dirichlet boundary D, or to the Neumann boundary N. The edges λ in the wire basket are separated into intersections of different facet types, i.e. DID∩I, DND∩N, NIN∩I, and the cross-edges II. If there are multiple, local facets which share the same BC type, we denote their (non-empty) intersection as either DD or NN. Note that we are able to ignore the difference between the local and global subgraphs for all edge subsets that do not contain contributions from interfaces. Altogether, the relations

    λ=DI∪DN∪NI∪II∪DD∪NN  and  Γ=D∪N∪I

are valid. To preserve the desired tree hierarchy, we employ Kruskal's algorithm <cit.> with weights summarized in the right part of <ref>. The algorithm works by selecting an unvisited edge with the lowest weight and checking if it closes a loop with edges that are already marked as tree components. The selected edge is added to the tree only if it does not close a loop. This is a simple and explicit way to obtain the desired hierarchy. We have visualized the weights and the resulting tree for an example configuration in <ref>.



We observe that the number of cotree elements on the wire basket λ cλt chosen by the algorithm does not depend on the local discretization in each subdomain. This follows from

	Let t be a set of edges which forms a full tree on the subgraph λ of the wire basket. If it holds
	
    λ=λ+c

	with c∈ℕ does neither depend on λ^(i) nor λ^(i), then λ c does not depend on λ^(i) or λ^(i), i.e. the local subdomain discretization.


    For a full tree on the wire basket, λ∩t=λ-1 needs to hold such that,
    
    λ c=λt=λ-λ∩t=c+1,

    i.e., <ref> follows automatically.


    Note that (<ref>) is a technical assumption that can be proven for common hexaedral decompositions of the computational domain, e.g. see <ref>.


    
    

§ DUAL-PRIMAL FORMULATION FOR 3D MAGNETOSTATICS

    In this section, we derive our dual-primal method and discuss the choice of primal DOFs. Here, `primal' denotes DOFs for which continuity is strongly enforced, as in <ref>. But unlike there, in a typical dual-primal approach, only selected interface DOFs are primal. The details of our choice will be introduced in <ref>, but first that we examine and reformulate the arising discrete system in a general context.

Starting with (<ref>), where the Dirichlet boundary is already eliminated, we further split  into tree DOFs _t, primal DOFs _p and the remaining ones _r. Note that we chose the same notation as in (<ref>) to highlight the similarities, although we removed the tree DOFs and our selection of _p is different. Incorporating _t=0 and reordering the linear system leads to

    [ _rr _rp  _r; _pr _pp  _p;  _r  _p   0;     ][ _r; _p;   ;    ]=[ _r; _p;  0;    ].

As above, we can define and construct a matrix ∈{0,1}^n_p× n_cp which is the kernel representation of _p, i.e., ker(_p)=im() needs to hold. Note that we use n_cp instead of n_gp from <ref> to emphasize that not all interface DOFs are used. In this context, g stands for the global and c for the coarse problem. The size of the remaining DOFs _r is given as n_r. Substituting _p=𝐩 in (<ref>), removing the constraints associated with the zero rows in _r and multiplying the second line of (<ref>) with  yields

    [ _rr _rp  _r; _pr _pp   0;  _r   0   0;     ][ _r;  𝐩; _r;    ]=[ _r; _p;  0;    ]

as the global dual-primal system with remaining multipliers _r∈ℝ^m_r.

To enable parallelism in a typical dual-primal TI method, one may require that

    _rr=diag(_rr^(1), … , _rr^(N))

is nonsingular. This implies that every local _rr^(i) is invertible, or in other words, that fixing the tree DOFs and the selected primal DOFs removes all local kernels. Assuming that this holds, one can eliminate _r by local Schur complements and write

    _r = _rr^-1(_r - _rp𝐩 - _r_r)

for the first line of (<ref>). Inserting (<ref>) in the second and third line of (<ref>) results in

    [ _pr_rr^-1_rp-_pp      _pr_rr^-1_r;      _r_rr^-1_rp       _r_rr^-1_r;                  ][  𝐩; _r;    ] = [ _pr_rr^-1_r - _p;       _r_rr^-1_r ]

after some algebra. We shorten the notation by introducing 

    _pr_rr^-1_rp-_pp, 
    𝐆   _r_rr^-1_rp, 
    𝐖   _r_rr^-1_r, 
    𝐝   _pr_rr^-1_r - _p,
    𝐞   _r_rr^-1_r,

which leads to

    [   𝐆; 𝐆 𝐖;   ][  𝐩; _r;    ] = [ 𝐝; 𝐞;   ].

For this, we apply yet again a Schur complement to eliminate 𝐩 and obtain

    _r = (𝐆^-1𝐆-𝐖)_r   = 𝐆^-1𝐝 - 𝐞
    𝐚_p   = ^-1(𝐝 - 𝐆_r)
    _r   = _rr^-1(_r - _rp𝐚_p - _r_r).

Note that the inherent parallelism becomes evident when all matrices are split into local contributions. Details are omitted but an interested reader is referred to <cit.>.



 §.§ Selection of Primal DOFs and Invertibility

In this subsection, we discuss the choice of appropriate primal DOFs. From (<ref>) we know that the global problem can be considered as a special dual-primal approach where all interface DOFs are chosen as primal. Consequently, after eliminating a globally consistent tree as constructed in <ref>, the systems in (<ref>) are solvable because no cross-edges are relevant and global gradients as well as discrete harmonic forms are taken care of. This implies that global invertibility of the system matrix in (<ref>) only depends on our choice of primal DOFs _p. For global invertibility it is sufficient to select all DOFs belonging to edges in the set II^(i)t^(i) because only these are still redundantly coupled after eliminating the tree DOFs.



While this is enough for global invertibility, it is not sufficient for local invertibility of _rr. An example is given in <ref> where the subdomains in the middle and the lower left are not locally invertible. The reason is that the underlying local graphs e^(i)=(^(i),e^(i)) are not connected, where e^(i)t^(i)∪p^(i)∪D^(i) denotes edges belonging to all locally eliminated DOFs. In this case, connectedness means that it is possible to reach all nodes of the graph from all other nodes. This motivates us to formulate <ref>.

    The matrix _rr is invertible if all local graphs e^(i) are connected for 1≤ i≤ N.


    We follow the arguments from <cit.>. If the tree DOFs are removed from a stiffness matrix belonging to the curl-curl-operator, it is invertible. A tree has the fundamental property that it connects all nodes with a minimal amount of edges. If we add more edges to the tree, i.e., remove more DOFs from the system, the stiffness matrix stays invertible and the underlying graph remains connected.

Note that <ref> and its proof ignore the consistency of the original problem. In other words, DOFs which are not in the tree but are selected to be eliminated, must be eliminated consistently. They are determined by the linear system and therefore cannot be simply be set to zero. However, in our setting, consistency is preserved because the additional DOFs belong either to the Dirichlet boundary or are primal DOFs. In particular, the consistency for the primal DOFs stems from the fact that they are only eliminated from the local context but computed in a global setting.

As a consequence of <ref>, we need to show that the combination of our tree construction and our final selection of primal DOFs yields connected graphs e^(i) on every subdomain. We  select 

    p^(i)=(^(i)∪NI^(i))t^(i)

and investigate local connectedness in three separate steps. First, we need to show that all nodes on the local wire basket λ^(i) are connected by edges in λ e^(i)λ^(i)∩e^(i). By reformulating λ e, one obtains

    λ e^(i)=DD^(i)∪DI^(i)∪DN^(i)∪NI^(i)∪II^(i)_=λ^(i)NN^(i)∪(NN^(i)∩t^(i)).

This implies that all nodes in λ^(i)NN^(i) are already connected because all edges λ^(i)NN^(i) are contained in λ e^(i). Now it remains to show that all nodes in NN^(i) are connected to the others in λ^(i). With the priorization shown in <ref>, Kruskal's algorithm grows the tree from λ^(i)NN^(i) onto NN^(i) by construction. Consequently, all nodes λ^(i) are connected by edges in λ e^(i). The second and third step of the proof follow a similar reasoning. For the second one, we start with connected nodes on the wire basket and Kruskal's algorithm, with the proposed weights, extends the tree to the interior of all subdomain facets and connects all nodes in Γλ to those in λ. On Dirichlet facets even more DOFs are selected to be eliminated which does not destroy connectedness. The same principles apply to the extension from the local facets to the local interiors which is the third step. In conclusion, all graphs e^(i) are connected and _rr is invertible due to <ref>.


    Our choice of p^(i) is both straightforward to implement and efficient. It is simple because we do not need an algorithm to search for edges that are needed for local connectedness. At the same time, we may select more edges than needed for local connectedness but we know that
    
    p⊂λt

    holds which implies p≤λt. Thus, p^(i) are independent of the local subdomain discretization if the underlying assumptions of <ref> are satisfied. This is related to efficiency, since the size of the coarse problem is a major factor for scalability. 


    To the best of our understanding, our procedure corresponds to the approach from  in <cit.> when considering a full Dirichlet problem. Their construction is less explicit but Kruskal's algorithm will eventually satisfy the same conditions. In addition, we can extend the underlying idea to problems with mixed boundary conditions by introducing additional edge weights and proving solvability.

    
    

§ NUMERICAL EXPERIMENTS

    In this section, we verify that the overall algorithm correctly solves general 3D magnetostatic problems and investigate some performance indicators. We focus on supporting our theoretical findings rather than high-performance computing. We employ the IGA-library  <cit.> in  2023a <cit.> for all tests.

Our numerical implementation assembles all contributions to (<ref>) from the local subdomains  in parallel. In other words, we obtain the actions of _rr^-1 via subdomain-wise Cholesky factorizations. Then it is also possible to factorize - with a Cholesky decomposition. This factorization in combination with forward/backward substitution can be exploited to solve (<ref>) iteratively using a Conjugate Gradient (CG) method for (<ref>). After obtaining _r, one can recover _p and _r from (<ref>) and (<ref>), respectively.

For tests related to numerical accuracy we use the method of manufactured solutions <cit.>. We predefine the magnetic vector potential 

    _ana(x,y,z)=
        [   cos(y)cos(z)sin(x); -2cos(x)cos(z)sin(y);   cos(x)cos(y)sin(z);                      ]  ⇒  _ana(x,y,z)=
        _ana=
        [ -3cos(x)sin(y)sin(z);                    0;  3cos(z)sin(x)sin(y);                      ].

Then we compute the source current density and the boundary conditions by substituting _ana into (<ref>)-(<ref>) for the respective configuration. This allows us to study inhomogeneous Dirichlet and Neumann boundary conditions. For simplicity, e.g. to avoid issues related to bad conditioning due to material jumps, we assume ≡1, but other material distributions are possible.

Numerical errors will be measured by the subdomain-wise Ω seminorm

    √(∑_i=1^N_ana-_numL^2(Ω^(i))^2).

To measure the rate of convergence, we investigate the number of subdivisions in each direction in each subdomain s_h, for which we may expect

    ≤ C h^p=C s_h^-p

for suitable C,C∈ℝ_+. Here, h denotes the element size in the classical setting of FEM methods, see for example <cit.>. Furthermore, p∈ℕ denotes the uniform degree of each local space 𝕍^(i).

Finally, the condition numbers of several matrices are investigated. We use the following definition <cit.>

    𝐗=𝐗/𝐗,

where 𝐗 and 𝐗 are the maximal and minimal absolute eigenvalues of 𝐗, respectively. In practice, these are computed by 's eigenvalue solver .



 §.§ Proof of Concept on Spherical Geometry

In this subsection, we provide experimental proof that our tree construction and the IETI-DP algorithm work as promised. We use a spherical geometry, with radius 2, as depicted in <ref>, and solve a manufactured problem with solution (<ref>) for either Dirichlet Γ_D=∂Ω or Neumann boundary conditions Γ_N=∂Ω.
We visualize the control mesh with tree and primal edges in <ref> and the numerical flux density in <ref>.

The main experiment of this subsection is a convergence study with respect to s_h and . As expected, the system from (<ref>) and all its subdomain-stiffness contributions ^(i) are invertible. The results are visualized in <ref> where one can observe the expected optimal behavior as given in (<ref>). Therefore, we conclude that the combination of our tree construction, the primal DOF selection and IETI-DP can be used to compute correct solutions for the flux density  to (<ref>)-(<ref>).




 §.§ Experiments on Toroidal Geometry

In this subsection, we examine the toroidal geometry with b_1=1 (tunnel through the domain) in <ref> and the associated issues with the harmonic field  in the kernel of the curl-operator. In <ref>, a convergence study for similar settings as in <ref> is visualized. We employ full Neumann conditions, i.e., Γ_N=∂Ω, and the mixed configuration where

    Γ_D{r=r_i,   θ∈(0,2π],  z∈(0,2)}  and  Γ_N:=∂Ω∖Γ_D.

We expect problems related to a non-trivial kernel to occur only for the full Neumann, but not the mixed setting. When the Dirichlet constraints, which form a loop around the hole, are strongly imposed, the harmonic contribution to the kernel is automatically removed. However, in the full Neumann setting, we need to add an edge to the tree that closes a loop around the hole <cit.>.

In practice, the predicted issue is not immediately visible, see e.g., <ref>. 's direct solver can cope with the linear system, but issues warnings. However, when examining the condition number, we see that  is very large, see <ref> for N=6, in the full Neumann boundary case,  indicating a rank deficit. Nevertheless,  and the error  remain comparable to the problem with mixed boundary conditions due to the implicit gauge of the applied factorizations. The configuration denoted with `Mod.' in <ref> refers to the Neumann problem with an additional tree edge. As expected, this repairs the rank deficit and  behaves normally. A visualization is given in <ref> where the violet edge is added to the blue tree. We include a second simpler decomposition with N=3 where the rank deficit can be observed in the local context, i.e., in _rr. This may be surprising since harmonic fields are a global issue, but a closer look at <ref> explains this behavior: without the additional edge, the graph e^(i) of eliminated DOFs is not connected in one subdomain. However, as before, the  solvers are able to deal with the rank deficit such that the corresponding ,  and  stay in appropriate bounds. But again, adding an edge that closes a loop around the hole eliminates the additional kernel element and its associated difficulties. This verifies our assumptions and emphasizes that our algorithm can correctly handle both mixed boundary conditions and geometries with b_1≠ 0. In conclusion, we recommend using a cohomology-aware tree construction, such as <cit.>, to automatically create appropriate trees.









 §.§ Numerical Scalability

To theoretically evaluate the behavior of our algorithm for large problems, we follow the tests and explanations of  in <cit.> and references therein. They determine scalability by looking at the sequential bottlenecks in the dual-primal algorithm. In the context of parallel performance, the sequential part is relevant in both Amdahl's law (strong scalability) <cit.> and Gustafson's law (weak scalability) <cit.>.

For the following test, we use the unit cube geometry with side length 1, which is decomposed into N equally sized cubes. For the boundary, we employ mixed boundary conditions with a non-simply connected Dirichlet boundary

    Γ_D{[x_1, x_2, x_3]^⊤,  x_1,x_3∈(0,1), x_2∈{0,1}},
    	  and  Γ_N:=∂Ω∖Γ_D.

which is handled automatically by our algorithm. 
Due to this special decomposition, we can introduce the coarse problem size (body diagonal of each subdomain) as H=√(3)N^-1/3 and the corresponding number of coarse subdivisions s_H=√(3)H^-1. If the local subdivisions are uniform in each subdomain, the relation h=√(3)(s_hs_H)^-1 can be derived. Consequently, the well-known FETI-ratio, e.g., discussed in  <cit.> and references therein, can be expressed as H/h=s_h. The quantities s_h and s_H allow a concise discussion of the following four test configurations:

    
  * The first test consists of setting H=const., i.e., s_H=const. and vary s_h. In other words, we only change the size of the local problems.
    
  * The second test consists of setting h=const., i.e., s_hs_H=const.. In other words, we keep the overall problem size constant while increasing the number of subdomains (i.e., processors). This is equivalent to testing for so-called strong scalability, which follows Amdahl's law.
    
  * In the third test we set H/h=s_h=const. and vary s_H. In other word, we increase both, the problem size and the number of subdomains/processors, while keeping the workload per processor constant. This is a test of the so-called weak scalability based on Gustafson's law.
    
  * In the fourth test we increase both s_h and s_H to examine the number of automatically selected primal DOFs. This serves as a measure of the overhead introduced by the coarse problem.


The results of the first scalability test are summarized in <ref>. We observe the expected convergence orders w.r.t. s_h as in the other numerical tests of this section. The number n_r of remaining local DOFs approaches 66% of the total number of DOFs n for increasing s_h. This limit is consistent with analytical considerations for a single hexahedral mesh. In both cases, the interior cotree elements dominate all other remaining contributions in the limit, implying that our results are reasonable. With increasing local refinement, the conditioning _rr and  deteriorate which is typical for TI methods. As explained earlier with <ref>, we see that the size of 𝐩, i.e., n_cp, remains constant when the subdomains are  refined locally.



The results of the second scalability experiment (strong scaling) are shown in <ref>. We fix the size of the global problem and decompose the corresponding control mesh into an increasing number of subdomain meshes. This means that the local problems become increasingly coarse the more subdomains we have. Therefore, it is not surprising that we observe the same error in each simulation run and that the conditioning of _rr improves with increasing s_H. The same is true for the interface problem, i.e.,  and the number of CG iterations. Conversely, the coarse problems get worse as the number of subdomains increases. This can be observed for both n_cp and .



The results of the third scalability experiment (weak scaling) are shown in <ref>. We observe that the same error is obtained as in experiment 1, see <ref>. This follows from the fact that the number of subdomains is equivalent to a specific local refinement of each subdomain. Furthermore, we see that n_cp and  do not change between <ref> and <ref> which supports the fact that the coarse problem depends only on the subdomain configuration and not on the local discretizations. For the interface problem, the number of CG iterations remains approximately constant and  increases very slowly compared to the results in <ref>. This also applies to the local problems, where we observe that _rr also remains approximately constant.



The fourth scalability experiment analyzes the overhead of the dual-primal approach. As a measure, we examine the amount n_λ=λ c of cotree DOFs on the wire basket, since it is independent of the boundary conditions, with respect to n. In this particular cuboidal setting the limit can be computed in closed-form as

    r_λlim_s_H→∞n_λ/n
    	=2/3s_h(s_h+1)^2,

for which a derivation is given in <ref>. The actual ratios, n_λ/n and n_cp/n are visualized in <ref> for different combinations of s_h and s_H. One observes n_λ≥ n_cp which holds since 𝐩 contains all cotree elements on the wire basket except the ones having a connection to the Dirichlet boundary. Then, the estimated ratio r_λ is close to n_cp/n, which gives a good indication of the coarse problem size. Finally, we conclude that the coarse problem is only a fraction of the overall problem if the local systems are sufficiently refined.


In the experiments of this subsection, we observed that both the local problem and the interface problem depend on the local discretization in each subdomain and improve with a coarser local refinement. We identified that optimizing the coarse problem is a conflicting goal since it is independent of the local discretization and depends only on the subdomain configuration. Determining the optimal trade-off is left for future research, since preconditioning and implementation choices, which are not the focus of this paper, can strongly influence this aspect.

    
    

§ CONCLUSIONS

    In this work, we discussed dual-primal Tearing and Interconnecting for magnetostatic problems using IGA edge-elements and mixed boundary conditions in combination with the tree-cotree gauge. In this context, we introduced an explicit way to construct appropriate trees and how to select primal DOFs to obtain local invertibility to enable parallel computations for each subdomain. Furthermore, we proved that the local subdomain contributions are invertible in general configurations using our approach. Additionally, an extension to non-contractible domains was investigated and all results were verified numerically. Finally, we examined certain properties related to scalability and observed that the coarse problem size, i.e. the number of primal DOFs is independent of the local discretization in every subdomain. Future research may explore this property and its implications for the simulation of industrial applications in more detail.


    

§ ACKNOWLEDGEMENTS

    The work is supported by the joint DFG/FWF Collaborative Research Centre CREATOR (DFG: Project-ID 492661287/TRR 361; FWF: 10.55776/F90) at TU Darmstadt, TU Graz and JKU Linz. We acknowledge the funding of The "Ernst Ludwig Mobility Grant" of the Association of Friends of Technical University of Darmstadt e.V. Furthermore, we thank Rafael Vazquez for the many fruitful discussions regarding tree-cotree gauging in the Tearing and Interconnecting context.

    

§ DECLARATION OF COMPETING INTEREST

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

    

§ DECLARATION OF GENERATIVE AI AND AI-ASSISTED TECHNOLOGIES IN THE WRITING PROCESS

    During the preparation of this work the authors used  and  in order to improve the readability and language of this paper. After using these tools/services, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

    

§ RATIO BETWEEN COARSE AND FULL PROBLEM

To show that the assumption (<ref>) of <ref>, i.e.

    λ=λ+c

holds for the configuration of <ref>. First, we look at a single subdomain Ω^(i) with subdivisions s_h for p=1. Note that the following concepts can be extended to settings with p>1 by adding a constant offset onto to s_h. The wire basket of this cubic subdomain Ω^(i) consists of m_e^(i)=12 macroscopic edges and m_n^(i)=8 vertices, each shared by three macroscopic edges. One can easily check that one of these edges contains s_h element edges and s_h+1 nodes which yields

    λ^(i)   =12s_h-4 = 12(s_h + 1) - 16 = m_e^(i)(s_h + 1) - (3-1)m_n^(i)
    λ^(i)   = 12s_h = m_e^(i)s_h

for the edges and nodes of the corresponding wire basket graph λ^(i). If we look at the broader picture with N=s_H^3 subdomains, we can compute the number of macroscopic edges by

    m_e=.|_s_h=1=.3s_hs_H(s_hs_H + 1)^2|_s_h=1=3s_H(s_H + 1)^2,

which implies

    λ=m_es_h=3s_hs_H(s_H + 1)^2

is the number of edges on λ. Correspondingly, we can state

    λ=m_e(s_h+1)-α m_n=m_es_h + (m_e - α)m_n_=c=λ + c

for the number of nodes on the global wire basket. The only remaining step would be to show that c=(m_e - α)m_n does not depend on s_h. For the global number of endpoints

    m_n=.|_s_h=1=.(s_hs_H + 1)^3|_s_h=1=(s_H + 1)^3

can be calculated. The parameter α only depends on how many edges are connected to each respective endpoint, which is quite technical to compute but clearly independent of the local discretization. Thus (<ref>) is satisfied and the number of cotree elements λ c=c+1 on the wire basket does not depend on s_h. To compute c, we can exploit that λ=m_n needs to be valid for s_h=1. This results in

    m_n=λ=m_e - c   ⇔   c = 3s_H(s_H + 1)^2-(s_H + 1)^3 = 2s_H^3 + 3s_H^2 - 1

and

    n_λλ c=c+1=2s_H^3 + 3s_H^2

for the number of cotree elements on the wire basket. Then, the limit for the ratio n_λ/n is given by

    r_λlim_s_H→∞n_λ/n
        =
        lim_s_H→∞(2s_H^3 
        + 
        3s_H^2/3s_H^3s_h(s_h+1)^2)
        =2/3s_h(s_h+1)^2,

where

    n=N^(i)=3s_H^3s_h(s_h+1)^2

is valid.

    abbrvnat
    



 Paper Title: Space-Time Isogeometric Method for a Nonlocal Parabolic Problem.


	

		Space-Time Isogeometric Method for a Nonlocal Parabolic Problem 
	

	

		
		[Department of Basic Sciences, Institute of Infrastructure, Technology, Research and Management, Gujarat, India, (dr.sudhakarchaudhary@gmail.com)]
		[Department of Basic Sciences, Institute of Infrastructure, Technology, Research and Management, Gujarat, India, (shreyachauhan898@gmail.com)]
    [Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 5, Pavia, Italy, (monica.montardini@unipv.it)]
	

	
§ ABSTRACT In the present work, we focus on the space-time isogeometric discretization of a parabolic problem with a nonlocal diffusion coefficient. The use of a space-time discretization with smooth basis functions yields advantages in the approximation of the solution. 
 
The existence of the unique solution for continuous and discrete space-time variational formulations is proven. We also establish the a priori error estimate for the space-time isogeometric scheme. The non-linear system is linearized through Picard's method and a suitable preconditioner for the linearized system is provided.  Finally, to confirm the theoretical findings, results of some numerical experiments are presented.
	
	

		Keywords: Space-time, Isogeometric Method, Nonlocal, Parabolic, Solver

	



§ INTRODUCTION

In this paper, the following nonlocal problem is studied: Find u(x,t) such that,
	
    ∂_tu -a(l(u))Δ u    = f  Ω×(0,T), 
    
    			u(x,t)    =0  ∂Ω×(0,T),  
    
    			u(x,0)   =0  Ω,

	where ∂_tu=∂ u/∂ t,  Ω is a bounded domain with Lipschitz boundary ∂Ω in ℝ^d, d≥ 1 and f:Ω×(0,T)→ℝ and a:ℝ→ℝ^+ are given functions. The term l(u) is chosen as given in <cit.>:
	
    l(u)=∫_Ωu dΩ .

In recent years, the study of nonlocal problems has received great attention due to its usefulness in real-world applications (see <cit.>). Problem (<ref>) is used to model the population density of bacteria in a given vessel (the domain Ω) subject to spreading or propagation of heat <cit.>. The study of the well-posedness of solution of problem (<ref>) and its stationary version was carried out in <cit.>. In <cit.>, the authors studied the asymptotic behaviour of the solution of (<ref>). In <cit.>, the well-posedness of problem  (<ref>) was studied with the choice a(l(u))=(∫_Ω|∇ u|^2 dΩ).





			
		
	
	
To obtain a numerical solution of problems like (<ref>), the usual time-stepping methods combined with various spatial discretization techniques like the Finite Element Method (FEM) <cit.> and Virtual Element Method (VEM) <cit.> are used. This approach is sequential in time, which complicates parallel solutions algorithm and affects the simultaneously done adaptive refinement strategies in space and time. As an alternative to time-stepping methods, an overall space-time discretized problem can be considered. We refer to <cit.> for more details about space-time finite element methods. In <cit.>, a completely unstructured space-time finite element method was analyzed to solve a linear parabolic problem. The author in <cit.> proposed a stable space-time finite element method with time-upwind test functions for the numerical solution of linear evaluation problems in moving domain. Recently in <cit.>, the space-time finite element scheme was analysed for the nonlinear parabolic p-Laplace problem.
Pioneered by Hughes et.al. in <cit.>,  Isogeometric Analysis (IgA)  is developed on the idea of using the same functions that are used for the parameterization of the domain (typically B-splines or Non-Uniform Rational B-splines (NURBS)) as the basis functions of the discrete computational space.  The main aim behind IgA is to increase the interoperability between computer-aided design industry and numerical simulations.
A full space-time approach with isogeometric method was introduced by Langer et. al. in <cit.>.  In <cit.>, the authors developed a suitable preconditioner and an efficient solver for a Galerkin space-time isogeometric discretization of a linear parabolic problem. An efficient solver for a least-squares discretization of a parabolic problem is provided in <cit.>. A space-time isogeometric method for linear and non-linear electrodynamics problems was investigated in <cit.>. The authors in <cit.> used the spline upwind stabilization method in space-time IgA frame work for the study of cardiac electrophysiology. Although the numerical schemes based on space-time IgA for non-linear parabolic problems have been proposed in the literature, their convergence analysis is not much explored.
Motivated from above, in the present work, we propose a space-time isogeometric discretization to solve the problem (<ref>) numerically. To handle the nonlinear term, we use the Picard's iterative scheme. Each linearized iteration is solved using  GMRES method, coupled with a preconditioner that is a variant of <cit.>.To the best of our knowledge, this is the first work on space-time isogeometric method for nonlocal parabolic problems.
The remainder of the paper is outlined as follows: Some preliminaries on Isogeometric Analysis and function spaces are illustrated in Section <ref>. In Section <ref>, we write the space-time weak formulation of problem (<ref>) and we give some results on the existence and uniqueness of the weak solution.  The space-time isogeometric discretization of (<ref>) is given in Section <ref>. In this section, we also provide a suitable solver for the linearized problem. The error estimate of the numerical scheme is presented in Section <ref>. Finally, we provide some numerical experiments to verify the theoretical findings in Section <ref>.



§ PRELIMINARIES


We begin by defining some function spaces and then we introduce the isogeometric spaces for the numerical approximation of (<ref>). We also recall some results from functional analysis.


 §.§ Function spaces

First, we define some function spaces which will be used in our analysis. Let L^2(Ω) be the Hilbert space of all square integrable functions with norm || v||=(∫_Ω| v(x)| ^2 dΩ)^1/2 and inner product (w,v)=∫_Ωw(x)v(x)  dΩ  ∀ w,v∈ L^2(Ω). Let H^r(Ω) denote the Sobolev space on Ω for r∈ℕ, with norm 
  
    v_r=(∑_0≤|α|≤ r‖∂^αv/∂x^α‖^2)^1/2.

  For 1≤ p≤∞, by L^p(0,T;H^r(Ω)), we denote the space of the vector valued functions v:(0,T)→ H^r(Ω), with norm || v||_L^p(0,T;H^r(Ω))=(∫_0^T|| v(t)||_r^p dt)^1/p for 1≤ p<∞ and || v||_L^∞(0,T;H^r(Ω))=_0≤ t≤ T|| v(t)||_r for p=∞. To derive a space-time weak formulation of (<ref>), we consider the following space-time function spaces: V={v∈  L^2(0,T;H_0^1(Ω)): ∂_tv∈ L^2(0,T;H^-1(Ω)), v(x,0)=0} and W=L^2(0,T;H_0^1(Ω)) with the corresponding norms given by
	
    ||v||^2_V:=||∂_tv||^2_L^2(0,T;H^-1(Ω))+||v||^2_L^2(0,T;H_0^1(Ω))

	and
	 
    ||v||^2_W:=||v||^2_L^2(0,T;H_0^1(Ω)).
 
	Here H^-1(Ω) denotes the dual space of H_0^1(Ω). We note that the dual of the space W is W^*=L^2(0,T;H^-1(Ω)). We also consider the space Z=L^2(0,T;L^2(Ω)) with norm 
 
    ||v||^2_Z:=||v||^2_L^2(0,T;L^2(Ω)).
 


 §.§ Isogeometric spaces

	Some fundamental aspects of B-spline spaces are given in the present subsection. For further discussion on B-splines, we refer to the monograph <cit.>.

	Let n and q be two non-negative integers. A knot vector is a  set of non-decreasing real numbers  
	
    Ξ={ζ_1,⋯,ζ_n+q+1}.

	We set ζ_1=0 and ζ_n+q+1=1. In our discussion, we will consider only open knot vectors, i.e. we set  ζ_1=⋯=ζ_q+1 and ζ_n=⋯=ζ_n+q+1. The j^th B-spline basis function is defined   using Cox-de Boor recursion formula as:
	
    b_j,0(ζ)=
    			1    ζ_j≤ζ<ζ_j+1, 
    
    			0

 and
	
    b_j,q(ζ)=ζ-ζ_j/ζ_j+q-ζ_jb_j,q-1(ζ)+ζ_j+q+1-ζ/ζ_j+q+1-ζ_j+1b_j+1,q-1(ζ) for q≥ 1,

	where we take 0/0=0.  By h, we denote the mesh size which is defined as h:=max{|ζ_j+1-ζ_j|: j=1,⋯,n+q}. Then we define the spline space of functions in one variable as
	
    𝒮_h:=span{b_j,q:j=1,⋯,n}.

	The continuity of the B-Splines at knots depends on the multiplicity of the knots (see <cit.>). 

Multivariate B-splines can be defined by taking the tensor product of the univariate B-splines. Let n_r,q_r for 1≤ r≤ d and n_t,q_t be positive integers. Then we consider the knot vectors Ξ_r:={ζ_r,1,⋯,ζ_r,n_r+q_r+1} for 1≤ r≤ d and Ξ_t:={ζ_t,1,⋯,ζ_t,n_t+q_t+1}. We denote by h_t, the mesh size associated with Ξ_t and h_r the mesh size associated with Ξ_r for 1≤ r≤ d. Let h_s:=max{h_r,1≤ r≤ d} be the maximum of all mesh sizes in each spatial direction. We define the degree vector by 𝐪=(𝐪_s,q_t) with 𝐪_s=(q_1,⋯,q_d). In the present work, for simplicity, we set q:=q_1=⋯=q_d=q_t. 
	We  consider quasi-uniform knot vectors, i.e. we suppose that  there exists 0<α<1, such that α h_s≤ζ_r,j+1-ζ_r,j≤ h_s  for every knot span (ζ_r,j,ζ_r,j+1) with ζ_r,j≠ζ_r,j+1 of Ξ_r and for 1≤ r≤ d and every knot span (ζ_t,j,ζ_t,j+1) with ζ_t,j≠ζ_t,j+1 of Ξ_t satisfies α h_t≤ζ_t,j+1-ζ_t,j≤ h_t.
	The B-splines in multi-variables are defined as:
	
    B_𝐣,𝐪(ζ,τ):=B_𝐣_s,𝐪_s(ζ)b_j_t,q_t(τ),

	where
	
    B_𝐣_s,𝐪_s(ζ)=b_j_1,q_s(ζ_1)⋯b_j_d,q_s(ζ_d),

	with 𝐣_s=(j_1,⋯,j_d), 𝐣=(𝐣_s,j_t) and  ζ=(ζ_1,⋯,ζ_d).

	We define the corresponding spline function space as
	
    𝒮_h,𝐪:=span{B_𝐣,𝐪},

	where h=max{h_s,h_t}.

	We can write 𝒮_h,𝐪 as a tensor product of spatial spline space and spline space in time variable as
	
    𝒮_h,𝐪= 𝒮_h_s,𝐪_𝐬⊗𝒮_h_t,q_t,

	where
	
    𝒮_h_s,𝐪_𝐬:=span{B_𝐣_s,q_s :  j_k=1,… , n_k  and  k=1,…, d}

 and
 
    𝒮_h_t,q_t:=span{b_j_t,q_t  :  j_t=1,…,n_t }.

 We need the following assumption.

We assume that q_t, q_s≥ 1  and that 𝒮_h_s,𝐪_𝐬⊂ C^0(Ω) and 𝒮_h_t,q_t⊂ C^0((0,1)).

 
	Now, we define isogeometric function spaces on the space-time cylinder Ω×(0,T). The following assumption is considered in our analysis.

 
	  Let 𝐅:Ω→Ω be the   parametrization of  the spatial domain Ω.  We suppose that 𝐅∈ [𝒮_h_s,𝐪_𝐬]^d and that the piece-wise derivatives of any order of 𝐅^-1 are bounded.

The space-time cylinder is parameterized by the map 𝐆:Ω×(0,1)→Ω×(0,T), 𝐆(ζ,τ):=(𝐅(ζ),Tτ).

	To incorporate the boundary and initial conditions, we consider the following spline space in parametric coordinates  
	
    V_h:={u_h∈ 𝒮_h,𝐪:u_h=0 on ∂Ω×(0,1) and u_h=0 in Ω×{0}}.

	Due to the tensor product structure, we have V_h=V_s,h_s⊗V_t,h_t,
	where
	
    V_s,h_s={v_h∈𝒮_h_s,𝐪_𝐬: v_h=0 on ∂Ω}=span{B_j,𝐪_𝐬:j=1,⋯,N_s}

	and
	
    V_t,h_t:={v_h∈𝒮_h_t,q_t: v_h(0)=0}=span{b_j,q_t:j=1,⋯,N_t},

 where  N_s=∏_l=1^d n_s,l, n_s,l=n_l-2 for 1≤ l≤ d, N_t=n_t-1 and where we have used a colexicographical reordering of the degrees of freedom.
	Then
	
    V_h=span{B_j,𝐪:j=1,⋯,N_dof},

	where N_dof=N_sN_t.

	Finally, the isogeometric space is given by
	
    V_h=span{B_j,𝐪:=B_j,𝐪∘𝐆^-1:j=1,⋯,N_dof}

 and we can write
 
    V_h=V_s,h_s⊗ V_t,h_t,

where

    V_s,h_s=span{B_j,𝐪_s=B_j,𝐪_s∘𝐅^-1: j=1,⋯,N_s}

and

    V_t,h_t=span{b_j,q_t=b_j,q_t(·/T):j=1,⋯,N_t}.

Next, we state the approximation property of the isogeometric space V_h which will be useful in proving the a priori error estimate.
<cit.>
Let r be an integer such that 1< r≤ q+1 and u∈ H^r(Ω×(0,T)). Then there exists a projection 𝒫_h: V∩ H^r(Ω×(0,T))→ V_h such that 

    || u-𝒫_h u||_V≤ C h^r-1|| u||_H^r(Ω×(0,T)),

  where the constant C is independent of the mesh size h.  



 §.§ Special types of operators

In this subsection, we recall some definitions and results of functional analysis.
By X, we denote a reflexive Banach space and X^* denotes its dual. The duality pairing between X^* and X is denoted by the symbol ⟨· ,·⟩.
<cit.>
	Let A:X→ X^* be an operator. We say that
	
		
  * A is bounded if each bounded subset of X is mapped to a bounded subset of X^*.
		
  * A is demicontinuous if for every sequence (v_k) in X converging to v in X, A(v_k) converges weakly to A(v) in X^*.
		
  * A is monotone if ⟨ A(v_1)-A(v_2),v_1-v_2⟩≥ 0 for every v_1,v_2∈ X.
		
  * A is maximal monotone if A is monotone and if v_1∈ X and b∈ X^* such that ⟨ b-A(v_2),v_1-v_2⟩≥ 0 for every v_2∈ X, then A(v_1)=b.
		
  * A is pseudomonotone if for every sequence (v_k) in X converging to v in X and lim sup_k→∞⟨ A(v_k),v_k-v⟩≤ 0, it follows that A(v_k) converges to A(v) weakly in X^* and lim_k→∞⟨ A(v_k),v_k-v⟩=0.
		
  * A is coercive if lim_||v||_X→∞⟨ A(v),v⟩/||v||_X=+∞.
	


Following <cit.>,  we define the differential operator L: V→ W^*   as
	
    Lv:=∂_tv.


The operator L is maximal monotone, closed and densely defined linear operator <cit.>.

 Now, we recall the results which are used to show the existence of the weak solution of problem (<ref>).
Consider the following problem,

    ∂_tu-∑_i=1^d∂/∂ x_i(a_i(x,t,u(x,t),∇ u(x,t);u))+a_0(x,t,u(x,t),∇ u(x,t);u)=f,

where a_i are real valued functions defined on Ω×(0,T)×ℝ^d+1× W. Problem (<ref>) can be considered as a special case of (<ref>). In <cit.>, the following theorems (Theorems <ref> and <ref>) were used to prove the existence of the weak solution of (<ref>). 

<cit.>
Let A:W→ W^* be the operator defined as 

    ⟨ A(u),v⟩=
       ∫_0^T∫_Ω[∑_i=1^da_i(x,t,u(x,t),∇ u(x,t);u)∂ v/∂ x_i+a_0(x,t,u(x,t),∇ u(x,t);u)v] dΩ dt
 
∀ u,v∈ W.

Assume that the functions a_i satisfy the following conditions:

	
  * The functions a_i:Ω×(0,T)×ℝ^d+1× W→ℝ satisfy the Carathéodory conditions for fixed u∈ W and they have the Volterra property.
	
  * There exists bounded operators h_1:W→ℝ^+and l_1:W→ Z such that
	
    | a_i(x,t,η_0,η;u)|≤ h_1(u)[|η_0|+|η|]+[l_1(u)](x,t), i=0,1,⋯, d,

	for a.e. (x,t)∈Ω× (0,T), each (η_0,η)∈ℝ^d+1 and u∈ W.
	
  * ∑_i=1^d[a_i(x,t,η_0,η;u)-a_i(x,t,η_0,η^*;u)](η_i-η^*)>0 if η≠η^*.
	
  * There exist bounded operators h_2:W→ℝ^+, l_2:W→ L^1(Ω×(0,T)) such that
	
    ∑_i=0^da_i(x,t,η_0,η;u)η_i≥ [h_2(u)][|η_0|^2+|η|^2]-[l_2(u)](x,t)

	for a.e. (x,t)∈Ω×(0,T), all (η_0,η)∈ℝ^d+1, u∈ W and
	
    lim_|| u||_W→∞[h_2(u)|| u||_W-|| l_2(u)||_L^1(Ω×(0,T))/|| u||_W]=+∞.

	
  * There exists θ>0 such that if (u_n) converges to u weakly in W, strongly in L^2(0,T;H^1-θ(Ω)) then for i=0,1,⋯,d
	
    a_i(x,t,u_n(x,t),∇ u_n(x,t);u_n)-a_i(x,t,u_n(x,t),∇ u_n(x,t);u)→ 0

	in Z.

Then the operator A is coercive, bounded, pseudomonotone with respect to V, demicontinuous and of Volterra type.


<cit.>
	If X is a reflexive Banach space, L:X⊇ D(L)→ X^* is a closed, maximal monotone, densely defined  linear operator and A:X→ X^* is  bounded, coercive, demicontinuous and pseudomonotone with respect to D(L), then (L+A)(D(L))=X^*.


	

§ SPACE-TIME WEAK FORMULATION: EXISTENCE-UNIQUENESS RESULTS

	Multiplying the parabolic equation (<ref>) with a test function v∈ W, integrating  over Ω×(0,T) and integrating by parts in the nonlinear elliptic term, the space-time weak formulation of problem (<ref>) reads: Find u∈ V such that
	
    ∫_0^T∫_Ω∂_tu v dΩ dt+∫_0^T∫_Ω a(l(u))∇ u∇ v dΩ dt=∫_0^T∫_Ωfv dΩ dt  ∀ v∈ W.

 In our analysis, we consider the following hypotheses:
	
		
  * There exists two positive real numbers m and M such that,
		
    0< m≤ a(s)≤ M<∞ ∀ s∈ℝ.

		
  * Let R_1=|Ω|^1/2C_p√(M/m^2)|| f||_Z, where C_p is the Poincaré constant. The function a is Lipschitz continuous with Lipschitz constant L_M satisfying the condition
		(m-L_MR_1)>0.
	

Now, in the following theorem, we prove the existence and uniqueness of the weak solution of the problem (<ref>).
	
		Let f∈ W^*. Then there exists a unique  u∈ V  which satisfies (<ref>).
	
	Proof:
	In order to prove the existence of a solution of (<ref>), we define a nonlinear operator A:W→ W^* by
	
    ⟨ A(u),v⟩:=∫_0^T∫_Ω a(l(u))∇ u∇ v dΩ dt  ∀ v∈ W.

	Then A is coercive, bounded, demicontinuous and pseudomonotone with respect to V. The proof follows from Theorem <ref> by taking a_0(x,t,η_0,η;u)=0 and a_i(x,t,η_0,η;u)=a(l(u))η_i for i=1,⋯,d.
	Since the differential operator L defined in (<ref>) is densely defined, closed and maximal monotone operator, from Theorem <ref> it follows that there exists  u∈ V which satisfies (<ref>) .

To show the uniqueness of the weak solution, we follow the idea given in <cit.>.
	
 Suppose u_1,u_2 are two solutions of problem (<ref>). From (<ref>), evidently we have
	
    ⟨∂_tu_1(t),v(t)⟩+ (a(l(u_1))∇  u_1(t),∇  v(t))=⟨ f(t),v(t)⟩

	and 
	
    ⟨∂_tu_2(t),v(t)⟩+ (a(l(u_2))∇  u_2(t),∇  v(t))=⟨ f(t),v(t)⟩
for almost all t∈(0,T).

 Subtracting (<ref>) from (<ref>) and defining
	ψ:=u_1-u_2, we get
	
    ⟨∂_tψ,ψ⟩+a(l(u_1))(∇  u_1,∇ψ)-a(l(u_1))(∇  u_2,∇ψ) 
       = a(l(u_2))(∇  u_2,∇ψ)-a(l(u_1))(∇  u_2,∇ψ).

	Then
	
    1/2d/dt||ψ||^2+a(l(u_1))||∇ψ||^2=    
        (a(l(u_2))-a(l(u_1)))(∇  u_2,∇ψ).

	We also have a(l(u_1))||∇ψ||^2≥ m||∇ψ||^2 and as a is a Lipschitz continuous function, we have
	
    |a(l(u_1))-a(l(u_2))|    ≤ L_M|l(u_1)-l(u_2)| 
       ≤ C||u_1-u_2||.

	So,
	
    d/dt||ψ||^2+2m||∇ψ||^2≤ C||ψ||||∇  u_2||||∇ψ||.

	Thus,  
    d/dt||ψ||^2≤ C(t)||ψ||^2.

	Then by using the Gronwall's lemma, we get
	
    ||ψ(t)||^2≤ 0.

	So, 
    ψ(t)=0.

	Hence u_1=u_2.
 
	
	

§ SPACE-TIME ISOGEOMETRIC DISCRETIZATION

	For the space-time isogeometric discretization of the weak formulation (<ref>), we consider the isogeometric spaces V_h defined in (<ref>) and we take W_h=V_h. Then the isogeometric approximation of (<ref>) is given by: Find u_h∈ V_h such that
	
    ∫_0^T∫_Ω∂_t u_hv_h dΩ   dt+∫_0^T∫_Ω^a(l(u_h))∇ u_h∇  v_h dΩ   dt=∫_0^T∫_Ω^fv_h dΩ   dt  ∀ v_h∈ V_h.

 	We will show the existence of a unique solution of the discrete formulation (<ref>). 
	First, in the following lemma, we prove a priori estimates of the solution u_h of (<ref>).
	
		Let f∈ Z and  u_h be a solution of (<ref>). Then
		
    || u_h||_L^∞(0,T;H_0^1(Ω))≤√(M/m^2)|| f||_Z.

	
	Proof: We define a discrete Laplacian operator Δ_h_s:V_s,h_s→ V_s,h_s as 
	
    -(Δ_h_su_h,v_h)=(a(l(u_h))∇  u_h,∇  v_h)  ∀ v_h∈ V_s,h_s.

	From (<ref>), we have for almost all τ∈(0,T),
	
    (∂_τ u_h(τ),v_h(τ))+(a(l(u_h))∇  u_h(τ),∇  v_h(τ))=(f(τ),v_h(τ)).

	Replacing v_h by -Δ_h_su_h, we get
	
    (a(l(u_h))∇∂_τ u_h(τ),∇  u_h(τ))+(Δ_h_su_h(τ),Δ_h_su_h(τ))=-(f(τ),Δ_h_su_h(τ)).

	Then we have
	
    1/2a(l(u_h))d/dτ(∇  u_h(τ),∇  u_h(τ))+(Δ_h_su_h(τ),Δ_h_su_h(τ))=-(f(τ),Δ_h_su_h(τ)).

 Thus 
    1/2a(l(u_h))d/dτ||∇  u_h(τ)||^2+||Δ_h_su_h(τ)||^2≤|| f(τ)|| ||Δ_h_su_h(τ)||.

 Dividing by a(l(u_h)), we get
 
    1/2d/dτ||∇  u_h(τ)||^2+1/a(l(u_h))||Δ_h_su_h(τ)||^2≤1/a(l(u_h))|| f(τ)|| ||Δ_h_su_h(τ)||.

 Then, using hypothesis (H1), we have
 
      
    1/2d/dτ||∇  u_h(τ)||^2+1/M||Δ_h_su_h(τ)||^2≤1/m|| f(τ)|| ||Δ_h_su_h(τ)||

and

    1/2d/dτ||∇  u_h(τ)||^2+1/M||Δ_h_su_h(τ)||^2≤M/2m^2|| f(τ)||^2+1/2M||Δ_h_su_h(τ)||^2.

	Integrating from 0 to t, we get

    ∫_0^td/dτ||∇  u_h(τ)||^2 dτ+1/M∫_0^t||Δ_h_su_h(τ)||^2  dτ≤M/m^2∫_0^t|| f(τ)||^2 dτ.

Hence

    ||∇  u_h(t)||^2≤M/m^2|| f||^2_Z.

Taking the supremum over almost all t in (0,T), we get the required estimates.


In the following, we prove the existence and uniqueness of the solution of (<ref>) <cit.>.
	
		Suppose f∈ Z. Then there exists a unique solution u_h∈ V_h of (<ref>).
	
	Proof: We consider V_h as a finite dimensional subspace of the Hilbert space W with inner product
	
    (u_h,v_h)_W=∫_0^T∫_Ω∇  u_h∇  v_h dΩ dt.
  Fix w_h∈ V_h. Define Q_w_h:V_h→ℝ by
	
    Q_w_h(v_h)=∫_0^T∫_Ω∂_tw_hv_h dΩ dt+∫_0^T∫_Ωa(l(w_h))∇  w_h∇  v_h dΩ dt-∫_0^T∫_Ωfv_h  dΩ dt.

	Then 
    | Q_w_h(v_h)|   ≤||∂_tw_h||_Z|| v_h||_Z+ M||∇  w_h||_Z||∇  v_h||_Z
       +|| f||_Z|| v_h||_Z
       ≤ (C|| w_h||_W+C_p|| f||_Z)|| v_h||_W.

Thus, Q_w_h is a continuous linear map and hence by the Ritz representation theorem, we get R(w_h)∈ V_h such that

    (R(w_h),v_h)_W=Q_w_h(v_h) ∀ v_h∈ V_h.
 
We show that the function R:V_h→ V_h is continuous.

Fix w_h∈ V_h and let ϵ>0. Then for any v_h,ϕ_h∈ V_h, we have

    (R(w_h)-R(ϕ_h),v_h)_W=   (R(w_h),v_h)_W-(R(ϕ_h),v_h)_W
    
    						=   ∫_0^T∫_Ω(∂_tw_h-∂_tϕ_h)v_h dΩ dt
       +∫_0^T∫_Ω(a(l(w_h))∇  w_h-a(l(ϕ_h))∇ϕ_h)∇  v_h dΩ   dt.

Now,

    ∫_0^T∫_Ω(a(l(w_h))∇  w_h-a(l(ϕ_h))∇ϕ_h)∇  v_h dΩ dt
       =∫_0^T∫_Ω(a(l(w_h))-a(l(ϕ_h)))∇  w_h∇  v_h dΩ dt+∫_0^T∫_Ωa(l(ϕ_h))(∇  w_h-∇ϕ_h)∇  v_h dΩ dt
       ≤ C'||∇  w_h-∇ϕ_h||_Z||∇  w_h||_Z||∇  v_h||_Z+M||∇  w_h-∇ϕ||_Z||∇  v_h||_Z.

So,

    (R(w_h)-R(ϕ_h),v_h)_W≤ (C+C'|| w_h||_W+M) || v_h||_W|| w_h-ϕ_h||_W.

Replacing v_h by R(w_h)-R(ϕ_h), we get

    || R(w_h)-R(ϕ_h)||^2_W
       ≤ (C+C'|| w_h||_W+M)	|| R(w_h)-R(ϕ_h)||_W	|| w_h-ϕ_h||_W.

Hence 
    || R(w_h)-R(ϕ_h)||_W≤ (C+C'|| w_h||_W+M)		|| w_h-ϕ_h||_W.

Taking δ=ϵ/(C+C'|| w_h||_W+M), the continuity of R is proved.


Now for w_h∈ V_h,

    (R(w_h),w_h)_W   =∫_0^T∫_Ω∂_tw_hw_h dΩ dt+∫_0^T∫_Ωa(l(w_h))∇  w_h∇  w_h dΩ dt-∫_0^T∫_Ωfw_h dΩ dt
       ≥(m|| w_h||_W-|| f||_Z)|| w_h||_W.

Then  (R(w_h),w_h)_W>0 for w_h∈ V_h having the property || w_h||_W=ρ, where ρ is any number such that

    ρ>|| f||_Z/m.

Then the existence of the solution u_h of (<ref>) follows from the Brouwer's fixed point theorem.


We now show the uniqueness of the solution of (<ref>). Suppose that there are two solutions u_h_1 and u_h_2 of (<ref>). Then
 
    ∫_0^T∫_Ω∂_tu_h_1v_h dΩ dt+∫_0^T∫_Ω^a(l(u_h_1))∇ u_h∇  v_h dΩ dt=∫_0^T∫_Ω^fv_h dΩ dt  ∀ v_h∈ V_h

 and
  
    ∫_0^T∫_Ω∂_t u_h_2v_h dΩ dt+∫_0^T∫_Ω^a(l(u_h_2))∇ u_h∇  v_h dΩ dt=∫_0^T∫_Ω^fv_h dΩ dt  ∀ v_h∈ V_h.

 Subtracting (<ref>) from (<ref>), we get
 
    ∫_0^T∫_Ω(∂_t u_h_1- ∂_t u_h_2)v_h dΩ dt+∫_0^T∫_Ω^(a(l(u_h_1))∇ u_h_1-a(l(u_h_2))∇ u_h_2)∇  v_h dΩ dt=0.

 Let ψ_h:=u_h_1-u_h_2. By replacing v_h by ψ_h in (<ref>), we obtain
 
    || u_h_1(T)-u_h_2(T)||^2+∫_0^T∫_Ω^(a(l(u_h_1))∇ u_h_1-a(l(u_h_2))∇ u_h_2)∇ψ_h dΩ dt=0.


 
 
 
 Adding and subtracting ∫_0^T∫_Ω a(l(u_h_1))∇  u_h_2∇ψ_h dΩ dt in above equation, we get
 
    || u_h_1(T)-u_h_2(T)||^2+∫_0^T∫_Ω^a(l(u_h_1))∇ψ_h∇ψ_h dΩ dt+
       ∫_0^T∫_Ω(a(l(u_h_1))-a(l(u_h_2)))∇  u_h_2∇ψ_h dΩ dt=0.

 Then
 
    ∫_0^T∫_Ω^a(l(u_h_1))∇ψ_h∇ψ_h dΩ dt≤∫_0^T∫_Ω(a(l(u_h_2))-a(l(u_h_1)))∇  u_h_2∇ψ_h dΩ dt.

 Using the Lipschitz continuity of a and the a priori estimate proved in Lemma <ref>, we get

    m||ψ_h||_W^2≤ L_MR_1||ψ_h||^2_W.
 
Then 

    (m-L_MR_1)||ψ_h||^2_W≤ 0.

Since (m-L_MR_1)>0, we get ||ψ_h||^2_W=0. Hence, ψ_h=0 and  the solution u_h of (<ref>) is unique.



 §.§ Discrete system and linear solver



The nonlinear system associated with the discrete formulation (<ref>) is given by
	
    𝐀( 𝐮)𝐮=𝐟,

	where 
	
    [𝐀( 𝐮)]_j,k=∫_0^T∫_Ω( ∂_tB_k,𝐪B_j,𝐪+a(l(𝐮))∇  B_k,𝐪∇  B_j,𝐪) dΩ  dt,   [𝐟]_j=∫_0^T∫_ΩfB_j,𝐪  dΩ dt

	and where, with a little abuse of notation, we defined l(𝐮):=∫_Ω∑_i=1^N_dof𝐮_iB_i,𝐪 dΩ.
	Thanks to the tensor product structure of the space V_h and the fact that a(l( 𝐮)) does not depend on the space variables, the matrix 𝐀( 𝐮) can be written as the sum of Kronecker products of matrices 
	
    𝐀( 𝐮)=𝐖_t⊗𝐌_s+𝐌_t( 𝐮)⊗𝐊_s,

	where 
    [𝐖_t]_j,k=∫_0^Tb'_k,q_t(t)b_j,q_t(t) dt,   j,k=1,⋯,N_t,

	
    [𝐌_s]_j,k=∫_ΩB_j,𝐪_s(x)B_k,𝐪_s(x)  dΩ ,   j,k=1,⋯,N_s,

	
    [𝐌_t(𝐮)]_j,k=∫_0^Ta(l( 𝐮))b_k,q_t(t)b_j,q_t(t) dt,   j,k=1,⋯,N_t,

	and
	
    [𝐊_s]_j,k=∫_Ω∇  B_j,𝐪_s(x)·∇  B_k,𝐪_s(x)  dΩ ,   j,k=1,⋯,N_s.

	We solve the nonlinear system (<ref>) by using Picard's method, which reads as follows: For given initial guess 𝐮^0, we solve
	
    𝐀( 𝐮^(n-1))𝐮^(n)=𝐟,
 for n=1,2,⋯ until 
    ||𝐮^(n+1)-𝐮^(n)||_∞≤ϵ
 for a given tolerance ϵ>0.

  We now propose a solving strategy for the linear system (<ref>). For the sake of simplicity, we drop the dependence of 𝐀(𝐮^(n-1)) and 𝐌_t(𝐮^(n-1)) from 𝐮^(n-1), and we write, instead, 𝐀 and 𝐌_t. Following <cit.>, we build a stable factorization of the time pencil (𝐖_t,𝐌_t) for n≥0: we find 𝐔_t,Δ_t∈ℂ^N_t× N_t, depending on 𝐮^(n-1), such that
  
    𝐔_t^*𝐌_t 𝐔_t=1_N_t  and  𝐔_t^*𝐖_t𝐔_t=Δ_t,

  where 1_m∈ℝ^m× m denotes the identity matrix and Δ_t has an arrowhead structure (only the diagonal, that last row and the last column could be non-zero). Then, the linear system matrix 𝐀 factorizes as
  
    𝐀=(𝐔_t^-*⊗1_N_s)(Δ_t⊗𝐌_s+1_N_t⊗𝐊_s)(𝐔_t^-1⊗1_N_s).
 
Differently than in <cit.>, we do solve  iteratively the linear system  (<ref>), but we first exploit the decomposition of the system matrix.
Thus, to compute the solution of (<ref>), we use  Algorithm <ref>.
 

 

 
  
   
    



The computational cost of Step 1 is O(N_t^3), while Step 2 and Step 4 yield a computational cost of 4N_dofN_t FLOPs. 
We solve the linear system in Step 3 iteratively and we use as a preconditioner the matrix

    𝐏:=Δ_t⊗𝐌_s+1_N_t⊗𝐊_s

where [𝐌_s]_j,k:=∫_ΩB_k,q_sB_j,q_s dΩ and [𝐊_s]_j,k:=∫_Ω∇B_j,q_s·∇B_k,q_s  dΩ are the mass and stiffness matrices discretized in the parametric domain.  Thanks to the tensor product structure of the basis functions, we have that

    𝐌_s=⊗_i=1^d𝐌_i   and  𝐊_s=∑_i=1^d𝐌_d⊗…⊗𝐌_i+1⊗𝐊_i⊗𝐌_i-1⊗…⊗𝐌_1,

where 𝐌_i and  𝐊_i are the one-dimensional mass and stiffness matrices, respectively. As the matrices 𝐊_i and 𝐌_i are positive definite, they admit a generalized eigendecomposition: we can find 𝐔_i matrix of generalized eigenvectors, and Λ_i matrix of generalized eigenvalues, such that

    𝐔_i^T𝐌_i𝐔_i=1_n_s,i  and  𝐔_i^T𝐊_i𝐔_i= Λ_i.

Factorizing the common terms, the preconditioner (<ref>) can be written as

    𝐏= (1_N_t⊗𝐔^-T_d⊗…⊗𝐔_1^-T)(Δ_t⊗1_N_s+1_N_t⊗Λ) (1_N_t⊗𝐔^-1_d⊗…⊗𝐔^-1_1),

where Λ:=∑_i=1^d⊗1_n_s,d…⊗1_n_s,i+1⊗Λ_i⊗1_n_s,i-1⊗…⊗1_n_s,1.

Finally, the application of the preconditioner (<ref>) is given by the solution of the following equation: given 𝐫, find 𝐬 such that 

    𝐏𝐬=𝐫

and it is performed by Algorithm <ref>. 
  
 We briefly analyze the computational cost of the preconditioner. We suppose for simplicity that all the spatial directions have the same number of degrees of freedom, i.e. n_s,l = n_s for l=1,…,d.  The setup of the preconditioner is performed by step 1 in Algorithm <ref> and has a computational cost of O(dn_s^3) FLOPs, which is optimal for d=2 and negligible for d≥ 3. Step 2 and step 4 require a computational cost of 4N_dofdn_s, while step 2 can be easily computed exploiting an LU decomposition, yielding a computational cost of O(N_dof) (see <cit.>  for details).  
 
 In our computational tests, we use a variant of the preconditioner (<ref>), that includes some information on the geometrical mapping 𝐅 in the spatial matrices 𝐊_s and 𝐌_s. 
 The idea is to make a separable approximation of the coefficients of the geometry and incorporate the result in the stiffness and mass matrices build in the univariate spline spaces. Then, a diagonal scaling is also added. We
 use techniques similar to the ones described in <cit.>. We name such modified preconditioner 𝐏_G. The computational cost of the setup and application of 𝐏_G is the same as the ones of 𝐏. We remark that the setup step has to be done only once in each non-linear iteration.



 



§ CONVERGENCE ANALYSIS

In this section, we derive a priori error estimate for the numerical scheme (<ref>). We begin with proving the following Cea's type lemma.

	Let u be the solution of (<ref>) and u_h be the solution of (<ref>). Then
	
    || u-u_h||_W≤ C|| u-v_h||_V  ∀ v_h∈ V_h,
	
	where C is a constant independent of mesh size h.

Proof: Let v_h∈ V_h. Then we have

    || u-u_h||_W≤|| u-v_h||_W+	|| v_h-u_h||_W.

Let ϕ_h=v_h-u_h.

Then, we have that 





    ∫_0^T∫_Ω∂_t ϕ_hϕ_h dΩ dt+∫_0^T∫_Ωa(l(u))∇ϕ_h∇ϕ_h dΩ dt
       =∫_0^T∫_Ω(∂_tv_h-∂_tu_h)ϕ_h dΩ dt+∫_0^T∫_Ωa(l(u))∇ (v_h-u_h)∇ϕ_h dΩ dt
       =∫_0^T∫_Ω∂_t v_hϕ_h dΩ dt+∫_0^T∫_Ωa(l(u))∇ (v_h-u)∇ϕ_h dΩ dt
       +∫_0^T∫_Ωa(l(u))∇  u∇ϕ_h dΩ dt+∫_0^T∫_Ωa(l(u_h))∇  u_h∇ϕ_h dΩ dt
       -∫_0^T∫_Ωfϕ_h dΩ dt-∫_0^T∫_Ωa(l(u))∇  u_h∇ϕ_h dΩ dt

    
    =∫_0^T∫_Ω(∂_tv_h-∂_tu)ϕ_h dΩ dt+∫_0^T∫_Ωa(l(u))(∇  v_h-∇  u)∇ϕ_h dΩ dt
       +∫_0^T∫_Ω(a(l(u_h))-a(l(u)))∇  u_h∇ϕ_h dΩ dt
       = I_1+I_2+I_3.



We now estimate each term I_1, I_2, I_3.

We have

    I_1=   ∫_0^T∫_Ω(∂_tv_h-∂_tu)ϕ_h dΩ   dt
    ≤   ||∂_tv_h-∂_tu||_Z||ϕ_h||_W
    ≤   || v_h-u||_V||ϕ_h||_W.

For I_2, we have

    I_2=   ∫_0^T∫_Ωa(l(u))(∇  v_h-∇  u)∇ϕ_h dΩ dt
    ≤   M|| v_h-u||_W||ϕ_h||_W
    ≤    M|| v_h-u||_V||ϕ_h||_W.

And for the term I_3, we work as follows,

    I_3   =∫_0^T∫_Ω(a(l(u_h))-a(l(u)))∇  u_h∇ϕ_h dΩ dt
       =∫_0^T∫_Ω(a(l(u_h))-a(l(v_h)))∇  u_h∇ϕ_h dΩ dt
       +∫_0^T∫_Ω(a(l(v_h))-a(l(u)))∇  u_h∇ϕ_h dΩ dt
       =I_31+I_32.

Now,

    I_31=   ∫_0^T∫_Ω(a(l(u_h))-a(l(v_h)))∇  u_h∇ϕ_h dΩ dt
       ≤∫_0^T|(a(l(u_h))-a(l(v_h)))|||∇  u_h||||∇ϕ_h|| dt
       ≤ L_MR_1||ϕ_h||_W^2.

Similarly,

    I_32≤ L_MR_1||ϕ_h||_W|| v_h-u||_W.

So,

    1/2||ϕ_h(T)||^2+   ∫_0^T∫_Ωa(l(u))∇ϕ_h∇ϕ_h dΩ dt
    ≤   || v_h-u||_V||ϕ_h||_W+M|| v_h-u||_V||ϕ_h||_W
       +L_MR_1||ϕ_h||^2_W+L_MR_1|| v_h-u||_V||ϕ_h||_W.

Hence,

    m||ϕ_h||^2_W
    		≤ (1+M+L_MR_1)|| v_h-u||_V||ϕ_h||_W+L_MR_1||ϕ_h||^2_W.


Then

    (m-L_MR_1)||ϕ_h||_W≤ C|| v_h-u||_V,

where C is a constant independent of the mesh size h.
And hence

    ||ϕ_h||_W≤ C|| v_h-u||_V.

Combining all the estimates, we get

    || u-u_h||_W   ≤|| u-v_h||_W+||ϕ_h||_W
       ≤ C|| u- v_h||_V,

which proves the required inequality given in (<ref>).

 In the following theorem, we prove the a priori error estimate in ||·||_W norm.
 
	Let r be an integer such that 1< r≤ q+1.
	Given the solution u∈ V∩ H^r(Ω×(0,T))  of (<ref>) and u_h∈ V_h the solution of (<ref>),   we have

	
    ||u-u_h||_W≤ Ch^r-1||u||_H^r(Ω×(0,T)),

 where the constant C does not depend on mesh size h.


 Combining the Lemma <ref> and the approximation estimates given in (<ref>), we can prove the inequality given in (<ref>). We outline the main steps here.

We take v_h=𝒫_h u in (<ref>). So, we get

    || u-u_h||_W
    		≤ C|| u- 𝒫_h u||_V.

Then using the estimates of (<ref>), we get the required estimates of (<ref>).   






 







§ NUMERICAL EXPERIMENTS

 In this section, first we present some numerical experiments to confirm the convergence behaviour of the approximation method given in Section <ref> and then we show some numerical results to analyze the performance of the preconditioner.  All tests are performed with Matlab R2024a and GeoPDEs toolbox <cit.>, on a Intel Xeon processor running at 3.90 GHz, with 256 GB RAM.
 In each example, we consider the final time T=1 and fix the tolerance ϵ=10^-10 for stopping the Picard's iterations in (<ref>). We use the preconditioned GMRES method to solve the linear system (<ref>) with tolerance 10^-12, the null vector as an initial guess and the preconditioner 𝐏_G described in Section <ref>.  We set the mesh sizes as h_s=h_t=:h and the degree of the basis functions as q_s=q_t=:q.
   
 

 §.§ Orders of convergence: quarter of annulus domain

 We consider the space domain Ω, a quarter of annulus, given by Ω={(x,y):1< x^2+y^2<4} (see Figure <ref>). The nonlocal term for this example is taken to be a(l(u))=2-1/1+(l(u))^2. We set the exact solution u(x,t)=(x^2+y^2-1)(x^2+y^2-4)xysin(t)  and compute the corresponding f from (<ref>). We have calculated the numerical solution with basis functions of degrees q=1, 2, 3, 4 on meshes with size h=1/4, 1/8, 1/16,  1/32. In Figure <ref>, the errors in L^2(0,T;L^2(Ω)) and L^2(0,T;H_0^1(Ω)) norms are plotted with respect to h. From Figure <ref>, it can be observed that the order of convergence in L^2(0,T;H_0^1(Ω)) is O(h^q) which coincides with the theoretical estimates given in the Theorem <ref>. We also compute the order of convergence in L^2(0,T;L^2(Ω)) norm and find that it is of order O(h^q+1) which is still optimal.		
  




	

		
 

 §.§ Orders of convergence: thick ring shaped domain

In this example, we consider problem (<ref>) on a thick ring shaped domain shown in Figure <ref>. The right hand function f is taken so that the exact solution is u(x,y,z,t)=-(x^2+y^2-1)(x^2+y^2-4)sin(t)sin(π z)xy^2 for the nonlocal term  a(l(u))=3+sin(l(u)). We have calculated the numerical solution with basis functions of degrees q=1, 2, 3, 4 on meshes with size h=1/4, 1/8, 1/16,  1/32. In Figure <ref>, we have reported the convergence rates in L^2(0,T;L^2(Ω)) and L^2(0,T;H_0^1(Ω)) norms for different degrees of B-spline basis functions. We see that the order of convergence in L^2(0,T;L^2(Ω)) norm is O(h^q+1) and the order of convergence in L^2(0,T;H_0^1(Ω)) norm is O(h^q).
  






 §.§ Performance of preconditioner: igloo-shaped domain 

In this example, we consider problem (<ref>) with  f=1, the nonlocal term a(l(u))=3+sin(l(u)), zero initial condition, homogeneous Dirichlet boundary conditions and domain Ω as shown in Figure <ref>. Note that, in this example the exact solution is unknown and the domain Ω is not trivial. In this example, we are interested in studying the number of Picard's and the GMRES iterations which are required in finding the numerical solution. We have calculated the numerical solution with basis functions of degrees q=1, 2, 3, 4 on meshes with size h= 1/8, 1/16,  1/32,  1/64. Table <ref> shows the number of Picard's iterations and the maximum number of GMRES iterations at any one of the Picard's iterations. We also have noticed that we get the same number of GMRES iterations (up to 1 more/ 1 less) in every Picard's iteration. We can see that the number of GMRES iterations is almost independent of q and h.




	




§ CONCLUSIONS

In this work, we have proposed a space-time isogeometric method for a parabolic problem with nonlocal diffusion coefficient. We have shown the existence and uniqueness of space-time weak formulations both at continuous and discrete levels. Theoretically, we have derived the error estimate in L^2(0, T; H_0^1(Ω)) norm and we have confirmed it through the numerical experiments. Although the error estimate in L^2(0, T; L^2(Ω)) norm is not proven theoretically, the optimal order of convergence in L^2(0, T; L^2(Ω)) norm is observed in the numerical experiments.
We have also provided a preconditioner suited for the linearized space-time isogeometric discretization of the nonlocal problem, based on <cit.>. At each non-linear iteration, the matrix 𝐀 changes and this causes to build the preconditioner from scratch at each iteration. However, the set-up and application cost of preconditioner is almost optimal as discussed in Section <ref>. Numerical experiments have confirmed that the proposed preconditioner is robust with respect to q and h.


Acknowledgment: The second author would like to express gratitude to the Department of Science and Technology (DST), New Delhi, India for providing INSPIRE Fellowship to carry out the research work. The third author is member of the Gruppo Nazionale Calcolo
Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INDAM).


	
	
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 Paper Title: High-order quasi-interpolation with generalized Gaussian kernels restricted over tori.





[label1]The first author was supported in part by NSFC (No. 12271002). The second author was supported in part by NSFC (No. 12101310), NSF of Jiangsu Province (No. BK20210315), and the Fundamental Research Funds for the Central Universities (No. 30923010912).



wenwugao528@163.com


zhengjiesun@njust.edu.cn
[cor1]Corresponding author



dieck-w-mathematics@outlook.com

 [label2]School of Big Data and Statistics, Anhui University, Hefei, China
 [label3]School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China

§ ABSTRACT 
The paper proposes a novel and efficient quasi-interpolation scheme with high approximation order for periodic function approximation over tori. The resulting quasi-interpolation takes the form of Schoenberg's tensor-product generalized Gaussian kernels restricted over circles. Notably, theoretical analysis shows that it achieves the highest approximation order equal to the order of the generalized Strang-Fix condition satisfied by the generalized Gaussian kernels. This is in sharp contrast to classical quasi-interpolation counterparts, which often provide much lower approximation orders than those dictated by the generalized Strang-Fix conditions satisfied by the kernels. Furthermore, we construct a sparse grid counterpart for high-dimensional periodic function approximation to alleviate the curse of dimensionality. Numerical simulations provided at the end of the paper demonstrate that our quasi-interpolation scheme is simple and computationally efficient.



Quasi-interpolation; Gaussian kernel; Generalized Strang-Fix conditions; Periodic function approximation; Periodic Strang-Fix conditions.

AMS Subject Classifications: 41A30, 41A25, 42B05, 65D15







§ INTRODUCTION

Quasi-interpolation is a well-established tool in function approximation and its related areas <cit.>. It provides an approximation via a weighted average of sampling data without the need to solve any large-scale linear system of equations <cit.>. In addition, it possesses optimality and regularization properties <cit.>. More importantly, one can even construct a quasi-interpolant such that it preserves   inner structures of the target function, for example, Wu and Schaback <cit.> constructed quasi-interpolation preserving positivity,  monotonicity, convexity, while reference <cit.> constructed divergence-free quasi-interpolation for vector-valued function approximation. 

Given a target  function f, a set of functions Φ_j and sampling data λ_j(f), j∈J, quasi-interpolation takes a general form 
Qf=∑_j∈Jλ_j(f)Φ_j. Usually,   Qf is required to reproduce polynomial of certain order to  derive corresponding approximation orders. This  can be either imposed conditions on  Φ_j or λ_j(f) or even both <cit.>.  The best investigated quasi-interpolation maybe the Schoenberg's model taking the form <cit.>
Q_hf=∑_j∈^df(jh)Ψ(·/h-j)   
with point evaluation functionals λ_j(f)=f(jh) and a set of functions  generated by dilation and translation of a single kernel Ψ, i.e., Φ_j=Ψ(·/h-j). Moreover, Jia and Lei <cit.> even showed that approximation order of  Q_h is completely characterized by  Strang-Fix conditions of Ψ <cit.>. More precisely, Q_h provides an approximation error bounded by powers of  𝒪(h^ℓ) if and only if Ψ satisfies Strang-Fix conditions of order ℓ for some positive integer ℓ. There have been many pioneering works on constructing  kernel Ψ with high-order Strang-Fix conditions, see, <cit.>, <cit.> for example. 
However, Strang-Fix conditions rules out many popular kernels for quasi-interpolation. In particular, Wu <cit.> proved that  radial functions  do not satisfy Strang-Fix conditions for any dimension d≥ 2. Nevertheless, radial functions have been a useful tool for multivariate (high-dimensional) function approximation including interpolation <cit.>, as well as quasi-interpolation  <cit.>. 

Under the framework of approximate approximation <cit.>, Maz'ya and his coauthors studied quasi-interpolation with  scaled radial kernels satisfying  (high-order) moment conditions <cit.>. The resulting quasi-interpolant in the Schoenberg's form can provide approximate convergence order 𝒪(h^ℓ) up to a prescribed saturation error if the radial kernel satisfies moment conditions of order ℓ. They even showed that one can appropriately choose the scale parameter such that the saturation error is negligible to the precision of computer.  But there is no asymptotic results in the classical sense since  approximation errors do not converge to zero.  From the perspective of moving least squares, Fasshauer and Zhang <cit.> constructed  iterated quasi-interpolation with radial kernel that converges to corresponding radial basis function interpolation with respect to the number of iterations. The iterated quasi-interpolation reformulated in matrix form can be interpreted  as laurent expansion of the inversion of the interpolation matrix. Therefore,  the involved radial kernel is required to be positive definite with interpolation matrix whose spectrum is larger than one and less than two to ensure convergence of the asymptotic expansion.  Light and Cheney <cit.> constructed quasi-interpolation with radial kernels satisfying  polynomial reproduction in the convolutional sense <cit.>.  Buhmann, Dyn and Levin <cit.> initiated the idea of taking a sequence of compactly supported radial functions to form quasi-interpolants with local polynomial reproduction. Buhmann and Dai <cit.> derived pointwise error estimates of quasi-interpolation with radial functions in terms of local polynomial reproduction.
Wu and Liu <cit.>   introduced the concept of generalized Strang-Fix condition and constructed  quasi-interpolation with radial kernels for scattered data.   The generalized Strang-Fix condition only requires the first part of Strang-Fix conditions and thus can be easily satisfied by most kernels including radial kernels.  Moreover, Gao and Wu <cit.> even proposed a general approach for constructing radial kernels with high-order generalized Strang-Fix conditions. As a byproduct, they also showed that moment condition, polynomial reproduction in the convolutional sense, and generalized Strang-Fix condition are equivalent to each other.    Gao et al. <cit.> constructed Monte-Carlo approximation based on quasi-interpolation with radial kernels satisfying high-order generalized Strang-Fix condition. 

However,  quasi-interpolation  with radial kernels satisfying generalized Strang-Fix conditions in the Schoneberg's form often provides much lower approximation orders. To improve approximation accuracy, some authors constructed multilevel quasi-interpolation with radial kernels. Usta and Levesley <cit.> constructed multilevel quasi-interpolation using Gaussian kernels on sparse grid. Recently, Franz and Wendland <cit.> proposed a convergent and efficient multilevel quasi-interpolation scheme with compactly supported radial kernels and showed  that it  converges linearly in the number of levels. Later,   Wendland and his coauthors extended the multilevel approach  to high-dimensional function approximation  <cit.> and even manifold-vauled function approximation <cit.>. The multilevel technique can improve approximation accuracy of quasi-interpolation, but it still can not increase the approximation order to the order of  generalized Strang-Fix conditions. 

Motivated from  above discussions, a question arising: can we construct  quasi-interpolation  with a radial kernel in the Schoenberg's form 
that provides the same approximation order as the order of generalized Strang-Fix conditions satisfied by the radial kernel? We shall provide an affirmative answer by constructing a quasi-interpolation scheme  from Gaussian kernels restricted over tori for periodic function approximation. To this end, we first show that when we restrict a two-dimensional generalized Gaussian kernel satisfying certain order of generalized  Strang-Fix condition   to a circle,   the  restricted kernel satisfying periodic Strang-Fix condition of the same order.  Then  we adopt the tensor-product technique to the  restricted generalized Gaussian kernel to get a multivariate  kernel restricted over tori. Finally,  we  construct a periodic quasi-interpolation  with the kernel in the Schoenberg's model from data sampled at regularly spaced  centers over tori. More importantly, we show that our periodic quasi-interpolation provides the  same approximation order as the order of generalized Strang-Fix conditions satisfied by generalized Gaussian kernel. 

We note that in the literature, many researchers restricted the radial basis functions in high dimensions to low dimensional manifolds to obtain important classes of basis function and studied their approximation properties. In particular, spherical basis functions are constructed through the restriction of  radial  functions in ^d+1 to the sphere ^d <cit.>, while   zonal periodic basis functions are considered by restricting Euclidean radial function to a circle <cit.>.

The paper is organized as followings. Section <ref> introduces some concepts related to the paper including  Wiener space and Periodic Strang-Fix conditions. Section <ref> consisting of three subsections is devoted to constructing our main quasi-interpolation scheme and deriving its approximation order. Some numerical simulations are provided in Section  <ref>, while conclusions and remarks are given in Section <ref>.


§ PRELIMINARIES




 §.§ Wiener spaces

We denote  the d-dimensional tori ^d by

    ^d:={=(x_1,…,x_d)∈^d: |x_r|≤π, r=1,…,d}.

Then the $̨-th Fourier coefficient of a functionf∈L_1(^d)is defined via

    ()̨:=∫_^df()e^-ı·̨.

Let(^d)be the space of2π-periodic functions with infinite smoothness, and'(^d)be the dual space. Any functionf∈'(^d)has a Fourier series expansion

    f()=∑_∈̨^d()̨e^ı·̨.

Note that  convergence of above series is related to  decay of  Fourier coefficients, which can be characterized  in Wiener spaces <cit.>. For1≤q≤∞,α≥0, we respectively denote(^d)andA_q,^α(^d)the periodic (isotropic) Wiener space

    (^d):={f()∈ L_1(^d):(1+||̨_2^2)^α/2()̨_ℓ_q(^d)<∞},

and the periodic (mixed) Wiener space

    A_q,^α(^d):={f()∈ L_1(^d):∏_r=1^d(1+|k_r|^2)^α/2()̨_ℓ_q(^d)<∞}.

Particularly, by takingq=1andα=0, the spaceA_1^0(^d)reduces to the well-known Wiener spaceA(^d)containing functions with absolutely summable Fourier series. Moreover, withq=2, the spaceA_2^α(^d)is exactly the Sobolev spaceH_2^α(^d). For otherq,(^d)differs fromW_q^α(^d)but has some embedding relations, which can be found in <cit.>. 

The above Fourier series expansion has a discrete counterpart. More precisely, letN∈ℕ^+be a positive integer and define an index set_Nby_N={=̨(k_1,…,k_d)∈^n: -N/2≤ k_r<N/2, r=1,…,d}.In the sequels, we will work with the equidistant grid set={2π/N, ∈̨_N}on^d. 
Given discrete function values sampled over the set, we useto denote the discrete Fourier series expansion with the$̨-th discrete Fourier coefficient

    _:=1/N^d∑_ℓ∈f(2π/N)e^2πıℓ·/̨N.

In particular, for any f∈(^d), we even have the aliasing formula <cit.>

    _=∑_∈^d(+̨ N).




 §.§ Periodic Strang-Fix conditions


Periodic Strang-Fix conditions <cit.> can be regarded as the periodic counterpart of (complete) Strang-Fix conditions <cit.>. Many functions satisfy periodic Strang-Fix conditions, such as  2π-periodized centered B-splines, linear combination of trigonometric polynomials <cit.>, periodization of Gaussian functions <cit.>,  and so forth. 
(Periodic Strang-Fix conditions)
	Let  {b_}∈ℓ_2(^n)  be some non-negative sequence and _N be the index set defined as above.  Let χ_N be a 2π-periodic continuous function. We say χ_N satisfies  periodic Strang-Fix conditions of order m  if  the following inequalities 
	
    |1-N^d ()̨|   ≤ b_0||̨_2^m N^-m,
    
    		|N^d (+̨ N)|   ≤ b_||̨_2^m N^-m,  ∈^d\{0}

	hold true  for all ∈̨.

Further, if χ_N additionally satisfies the cardinal interpolation property

    χ_N(_)=δ_0,,  _∈, ∈̨,
 then the error of corresponding Lagrange interpolation 

    I_Nf(·)=∑_∈̨f(_)χ_N(·-_)

can be completely characterized by   periodic Strang-Fix conditions <cit.>. 


	Let  f∈ A_q^μ with q≥ 1, μ≥α≥ 0, and μ> d(1-1/q). Suppose that the kernel χ_N  of the above cardinal interpolation I_Nf satisfies  periodic Strang-Fix conditions of order m≥ d/2. Then for σ=min{μ,m+α}, there exists a constant C_σ such that the inequality
	
    f-I_N f_(^d)≤ C_σ(2π/N)^σ-αf_A_q^μ(^d)

	holds true for all f∈ A_q^μ(^d).

The above lemma will be used in the next section to derive error estimate of quasi-interpolation with generalized Gaussian kernels restricted over tori.

	
	
	


	
	
	




§ OUR MAIN RESULTS


We begin with this section by constructing a tensor-product kernel over tori from Gaussian kernel for quasi-interpolation.


 §.§ Generalized Gaussian kernels restricted over tori  ^d 

For a two-dimensional  Gaussian kernel with a positive scale parameter 
    Φ_c( x,  y)=ϕ( x- y;c)=1/√(2π)ce^-s^2/2c^2,
 where s= x- y, x,  y∈ℝ^2. Then by letting 
    x̅= x/ x, y̅= y/ y, s̅=x̅-y̅,
 we can restrict the kernel to a unit circle as

    Φ_c(x̅, y̅)=ϕ(s̅;c)=1/√(2π)ce^-s̅^2/(2c^2)=1/√(2π)ce^-4sin^2(α/2)/2c^2:=ψ(α;c), α∈𝕋,

where α denotes the geodesic distance between x̅ and y̅.   One can verify that the above Gaussian kernel satisfies   generalized Strang-Fix condition of order two <cit.>, that is,

    ϕ(;c)=e^-c^2^2/2=1+𝒪(c^2^2).

Here  we define the Fourier transform pair of a function f∈ L_2(^d) as 

    f(ω)=∫_^df()e^-ıω·,   f()=1/(2π)^d∫_^df(ω)e^ıω·ω.

In particular, for a radial function Φ( x)=ϕ( x)∈ L_1(^d)∩ C(^d),   its Fourier transform is also radial and can be further represented by

    Φ(ω)=ℱ_dϕ(r)=(2π)^d/2r^-(d-2)/2∫_0^∞ϕ(t)t^d/2J_(d-2)/2(rt)dt,  r=ω,

where J_ν(z) is the Bessel function of the first kind.

Furthermore, according to <cit.>, we can construct a generalized Gaussian kernel ϕ_2m+2 satisfying generalized Strang-Fix condition of order 2m+2 as  <cit.>

    ϕ_2m+2(s;c)=1/√(2π)cL_m^(1/2)(s^2/2c^2) e^-s^2/2c^2,

where L_m^(α)(s) is the generalized Laguerre polynomial

    L_m^(α)(s)=∑_j=0^m(-1)^k m+α m-ks^k/k!.

In addition, Fourier transform of the above generalized Gaussian kernel can be expanded in Laguerre polynomials.

	Let ϕ_2m+2(r) be the generalized Gaussian kernel (<ref>), then its Fourier transform is also radial and can be expressed via
	
    ℱ_2ϕ_2m+2(r;c)=√(2π)c∑_j=0^m(-1)^jm+1/2 m-jL_j^(0)(c^2r^2/2)e^-c^2r^2/2, r=ω.



	Note that 
	
    ℱ_2ϕ_2m+2(r;c)    =2π∫_0^∞ϕ_2m+2(t)· t J_0(rt)ṭ
       =∫_0^∞√(2π)/cL_m^(1/2)(t^2/2c^2) e^-t^2/2c^2· tJ_0(rt)ṭ
       =√(2π)/c∑_j=0^m(-1)^j/j!m+1/2 m-j∫_0^∞t^2j+1/(2c^2)^jJ_0(rt) e^-t^2/2c^2ṭ.

	From <cit.>, for any even integer  μ-ν,   we have the general Hankel's formula 
	
    ∫_0^∞J_ν(at)exp(-p^2t^2)· t^μ-1dt=Γ(1/2ν+1/2μ)· (1/2a/p)^ν/2p^μΓ(ν+1)e^-a^2/4p^2_1F_1(1/2ν-1/2μ+1;ν+1;a^2/4p^2),

	where _1F_1 is the confluent hypergeometric function.
	As a special case of  a=r, ν=0, μ=2j+2, p^2=1/2c^2, we get
	
    ∫_0^∞t^2j+1J_0(rt) e^-t^2/2c^2ṭ=j!(2c^2)^j+1/2e^-c^2r^2/2_1F_1(-j;1;c^2r^2/2).

	Moreover, for a positive integer j, the function _1F_1(-j;1;z) reduces to a Laguerre polynomial 
	
    _1F_1(-j;1;z)=L_j^(0)(z)=∑_k=0^j(-1)^k/k!j kz^k.

	Finally, combing above results yields Identity (<ref>).


We go further with providing four lemmas that will be used later. 
(<cit.>)
	Let Φ be a conditionally positive definite  radial function of order q defined on ^d+1 and ϕ(·):=Φ(-)|_,∈^d. If the Fourier transform Φ is measurable on ^d+1, then for ℓ≥ 2q+1, it holds that 
	
    ϕ(ℓ)=∫_0^∞ tΦ(t)J_v^2(t)d t,  v:=ℓ+d-1/2.



	(<cit.>)
	For each fixed ℓ and large z,  I_ℓ(z) has the Hankel’s asymptotic expansion
	
    I_ℓ(z)∼e^z/√(2π z)∑_γ=0^∞(-1)^γa_γ(ℓ)z^-γ,  a_γ(ℓ)=(1/2-ℓ)_γ(1/2+ℓ)_γ/(-2)^γγ!,

	in which (x)_γ=∏_j=0^γ-1(x-j) represents the rising factorial of x. 
	
	Specifically, by taking μ=4ℓ^2, we have
	
    I_ℓ(z)∼e^z/√(2π z)[1-μ-1/8z+(μ-1)(μ-9)/2!(8z)^2-(μ-1)(μ-9)(μ-25)/3!(8z)^3+…].



(<cit.>)
	Let g(t)=∑_l=0^m a_l t^l be a univariate polynomial of degree m, then the polynomial defined by 
	
    G_m(t)=∑_j=0^m(-1)^jt^j/j!g^(j)(t)

	satisfies the condition
	
    G_m(t)=a_0+Ø(t^m+1),  t→ 0.


(A combinatorial identity)
	Let k, m be two nonnegative integers and z∈, then the following identity holds,
	
    ∑_j=k^m(-1)^jz j-k=∑_j=k^m(-1)^jm+1-z  m-jj k.




	Since
	
    ∑_j=0^n(-1)^jz  j   =1+∑_j=1^n(-1)^j[z-1 j+z-1 j-1]
       =∑_j=0^n(-1)^jz-1 j-∑_j=0^n-1z-1 j=(-1)^nz-1 n,

	we have
	
    ∑_j=k^m(-1)^jz j-k=(-1)^k∑_j=0^m-k(-1)^jz j=(-1)^mz-1 m-k.

	
	On the other hand,  with the help of  hypergeometric function, we have
	
    ∑_j=k^m(-1)^jm+1-z m-jj k
    
    		=   ∑_j=0^m-k(-1)^j+km+1-z m-k-jj+k k
    
    		=   (-1)^k1+m-z m-k_2F_1(k-m,1+k,2+k-z;1).

	
	Further, according to the definition of _2F_1 and the Chu-Vandermonde convolution formula <cit.>, we obtain
	
    _2F_1(-n,a,b;1)=∑_k=0^n(-n)_k(a)_k/k!(b)_k=(b-a)_n/(b)_n,

	and thus
	
    _2F_1(k-m,1+k,2+k-z;1)=(1-z)_m-k/(2+k-z)_m-k.

	Moreover, noting that
	
    ∑_j=k^m(-1)^jm+1-z m-jj k
    
    		=   (-1)^k1+m-z k·(1-z)_m-k/(2+k-z)_m-k
    
    		=   (-1)^k(1-z)_m-k/(m-k)!=(-1)^mz-1 m-k,

	we complete the proof.


With  above lemmas, we can derive  the following salient property   of  generalized Gaussian kernels restricted over a circle. More precisely, let ψ_2m+2 be the restriction of ϕ_2m+2 over  𝕋: 

    ψ_2m+2(α;c)=1/√(2π)cL_m^(1/2)(2sin^2(α/2)/c^2) e^-2sin^2(α/2)/c^2, α∈𝕋.

Then we can show that ψ_2m+2 satisfies periodic Strang-Fix conditions of order 2m+2.

	Let ψ_2m+2(α;c) be the restricted Gaussian kernel defined as (<ref>), then we have:
	
    (i)  |_2m+2(ℓ;c)-1|≤ b_0|ℓ|^2m+2c^2m+2,  for fixed ℓ and sufficiently small  c,
       (ii)  |_2m+2(ℓ;c)|  decays exponentially for large ℓ.



	
	
	
	
	
	
	
	By taking d=1 in Lemma <ref> and defining a new variable ρ=c^2, we have
	
    _2(ℓ;ρ)=√(2π)c ∫_0^∞tJ_ℓ^2(t)e^-c^2t^2/2dt=√(2π)ρ^1/2∫_0^∞tJ_ℓ^2(t)e^-ρt^2/2dt.

	
	
	The above integral can be computed exactly in terms of the modified Bessel function of the first kind <cit.> as
	
    _2(ℓ;ρ)=√(2π)ρ^-1/2 e^-1/ρ I_ℓ(1/ρ).

	Using Lemma <ref>, we have the asymptotic expansion
	
    I_ℓ(1/ρ)= e^1/ρρ^1/2/√(2π)(1-(4ℓ^2-1)/8ρ+(4ℓ^2-1)(4ℓ^2-9)/2!8^2ρ^2+Ø(ρ^3)),
  as ρ→ 0.
	Thus for fixed ℓ, we have
	
    _2(ℓ;ρ)= 1-4ℓ^2-1/8ρ+Ø(ρ^2)

	and
	
    |_2(ℓ;ρ)-1|≤1/2ℓ^2ρ=1/2ℓ^2c^2.

	
	To raise the order of expansion in ρ, we can employ   Lemma <ref> to get
	
    ∑_j=0^m(-ρ)^j/j!d^j/dρ^j(_2(ℓ;ρ))=1+𝒪(ℓ^2m+2ρ^m+1)=1+Ø(ℓ^2m+2c^2m+2).

	
	Therefore, the proof of this theorem is equivalent to verify the following identity
	
    R_m:=∑_j=0^m(-ρ)^j/j!d^j/dρ^j(_2(ℓ;ρ)) = _2m+2(ℓ).

	
	
	First, the formula (<ref>) leads to
	
    R_m   =∑_j=0^m(-ρ)^j/j!d^j/dρ^j(_2(ℓ;ρ))
       =∑_j=0^m(-ρ)^j/j!d^j/dρ^j(√(2π)ρ^1/2∫_0^∞tJ_ℓ^2(t)e^-ρt^2/2dt)
       =√(2π)∫_0^∞tJ_ℓ^2(t)∑_j=0^m(-ρ)^j/j!d^j/dρ^j(ρ^1/2e^-ρt^2/2)d t.

	Using the Newton-Leibniz formula
	
    d^j/dρ^j(ρ^1/2e^-ρt^2/2)=∑_k=0^jj k(1/2)_j-kρ^1/2-(j-k)(-t^2/2)^ke^-ρt^2/2,

	we obtain
	
    R_m   =√(2π)ρ^1/2∑_j=0^m(-1)^j/j!∑_k=0^jj k(1/2)_j-k(-ρ)^k∫_0^∞tJ_ℓ^2(t)(t^2/2)^k e^-ρt^2/2dt
       =√(2π)ρ^1/2∑_k=0^m(-1)^k/k!∑_j=k^m(-1)^j1/2 j-k∫_0^∞tJ_ℓ^2(t)(ρ t^2/2)^k e^-ρ t^2/2dt.

	
	
	On the other hand, according to Lemma  <ref> and Lemma <ref>, we have 
	
    _2m+2(ℓ;ρ)   =∫_0^∞tJ_ℓ^2(t) ℱ_2ϕ_2m+2(t)dt
       =√(2π)ρ^1/2∑_j=0^m(-1)^jm+1/2 m-j∫_0^∞tJ_ℓ^2(t)L_j^(0)(ρ t^2/2)e^-ρ t^2/2ṭ
       =√(2π)ρ^1/2∑_j=0^m(-1)^jm+1/2 m-j∑_k=0^j(-1)^k/k!j k∫_0^∞tJ_ℓ^2(t)(ρ t^2/2)^k e^-ρ t^2/2ṭ
       =√(2π)ρ^1/2∑_k=0^m(-1)^k/k!∑_j=k^m(-1)^jm+1/2 m-jj k∫_0^∞tJ_ℓ^2(t)(ρ t^2/2)^k e^-ρ t^2/2ṭ.

	Compare the expressions of _2m+2 and R_m,  proof of identity R_m=_2m+2  reduces to verifying
	
    ∑_j=k^m(-1)^j1/2 j-k=∑_j=k^m(-1)^jm+1/2 m-jj k,

	which is a special case of Lemma <ref> with z=1/2.
	
	(ii) Next, we will show that the kernel ψ_2m+2 satisfies (<ref>). For fixed c>0 and large ℓ, we have from <cit.>,
	
    I_ℓ(1/ρ)∼1/√(2πℓ)(e/2ρℓ)^ℓ,

	and thus
	
    _2(ℓ;ρ)=√(2π)ρ^-1/2 e^-1/ρ I_ℓ(1/ρ)∼e^-1/c^2/2^ℓc^2ℓ+1√(ℓ)(e/ℓ)^ℓ.

	It is straightforward to verify that the coefficients decay exponentially. 
	
	Moreover, since
	
    _2m+2(ℓ;ρ) = ∑_j=0^m(-ρ)^j/j!d^j/dρ^j(_2(ℓ;ρ))

	and
	
    ∂^j I_ℓ(z)/∂ z^j=2^-j∑_k=0^jj kI_2k-j+ℓ(z),

	we have
	
    _2m+2(ℓ;c)=e^-1/c^2·∑_j=0^m ∑_k=0^j p_jk(1/c)I_2k-j+ℓ(1/c^2),

	where p_jk are some polynomials. Consequently,  we can verify  that  coefficient _2m+2(ℓ;ρ) also decays exponentially for sufficiently large  ℓ.


Up to now, we have verified that the restricted Gaussian kernel ψ_2m+2(α;c) satisfies the periodic Strang-Fix conditions of order 2m+2. To approximate functions on tori 𝕋^d, we  employ the tensor-product technique to construct a multivariate  kernel _ h,  s restricted over tori 𝕋^d in the form:

    _ h, s(; c)=∏_j=1^dψ_h_j,s_j(x_j;c_j).

Here h=(h_1,h_2,⋯,h_d), s=(s_1,s_2,⋯, s_d), c=(c_1,c_2,⋯,c_d), ψ_h_j,s_j(x_j;c_j)=ψ_s_j(x_j;c_j)· h_j with parameters h_j=1/N_j, and ψ_s_j(x_j;c_j) is defined in Equation (<ref>) with s_j=2 m_j+2, for some positive integer N_j, m_j. 
We go further with showing that _ h, s satisfies periodic Strang-Fix conditions. To facilitate elaboration, we consider the isotrophic case, that is, N_j=N, m_j=m, c_j=c, h_j =1/N, for j=1,2,⋯, d.  All the results are extendable to the anisotrophic case.


	Let _ h, s  be an isotrophic kernel defined  in Equation (<ref>). Then, the following periodic Strang-Fxi conditions of order s hold by choosing c_j=c=𝒪(1/N), for j=1,2,⋯, d.
	
    |1-N^d_ h, s(;̨ c)|≤ b_0||̨_2^s N^-s,
       |N^d_ h, s(+̨ N; c)|≤ b_||̨_2^s N^-s, ν∈^d\{0},



	We begin with showing the first part  of the above inequality. According to the definition ψ_h,s(x_i;c)=ψ_s(x_i;c)/N,
	we have
	
    |1-N^d_ h, s(;̨ c)|=|1-N^d ∏_r=1^d_h,s(k_r;c)/N^d|=|1-∏_r=1^d(_h,s(k_r;c))|

	Denoting a_r=_h,s(k_r;c), we can obtain
	
    |1-∏_r=1^d a_r|   =|∑_r=1^d(1-a_r)· a_r+1⋯ a_d|
       ≤ C∑_r=1^d|1-a_r|≤ Cb_0N^-s∑_r=1^d|k_r|^s
       ≤ C b_0||̨_2^sN^-s.

	
	
	We are now coming to the second part. Since  ψ(ℓ;c) decays exponentially for large ℓ, we can get 
	
    |_h,s(k_r+ν_r N;c)|≤ b_ν_r|k_r|^sN^-s,
  which is the second part of periodic Strang-Fix conditions (see Equation (<ref>)). 
	This in turn leads to 
	
    |N^d_ h, s(+̨ N;  c)|	=|N^d∏_r=1^d_h,s(k_r+ν_r N;c)/N^d|
    	=    |∏_r=1^d_h,s(k_r+ν_r N;c)|
    		≤∏_r=1^d|_h,s(k_r+ν_r N;c)|
    ≤    ∏_r=1^d (b_ν_r|k_r|^sN^-s)≤ b_(k_1^2+…+k_d^2)^s/2N^-s=b_||̨_2^sN^-s,
 by choosing c_j=c=𝒪(1/N) for j=1,2,⋯,d.




	
	
		
		
		
	
	
	



	
	
		
		
		
	
	
	




 §.§ Quasi-interpolation  with  generalized Gaussian kernels  restricted over tori 𝕋^d

We first consider the data sampled at uniform grid over tori and then extend it to   directly uniform  gird   as well.  

Given the discrete function values {f(_ȷ)}_ȷ∈ sampled from a 2π-periodic continuous function f at the the equidistant grid set :={x_ȷ=2πȷ/N, ȷ∈_N} over ^d, we first  construct an ansatz

    Q_sf() =  ∑_ȷ∈ f(_ȷ)_,(-_ȷ; c).

Then we demonstrate that it is a quasi-interpolant by deriving corresponding error estimates as given in the following theorem.



	
	
		
		
	

	
	
		
		
	
	
		
		
	
	




	
		
		
		
		
		
	



	
		
		
		
	


Let _, be an isotrophic kernel defined in Equation (<ref>) and satisfy the periodic Strang-Fix conditions of order s>d/2. Then, under the assumptions of Lemma <ref> together with μ≥ s+α, there exists a constant C such that

    f-Q_sf_()≤ CN^-s f_A_q^μ()

holds.



Applying triangle inequality to the error f-Q_sf, we have 

    f-Q_sf≤f-I_Nf+I_Nf-Q_sf,
 where I_Nf is  Lagrange interpolation  defined in Equation (<ref>).
In addition, noting that the first part on the right-hand side of the above inequality has been derived in Lemma <ref>, we only need to bound the second part.
To this end, we first use the identity 

    _=∑_∈(+̨ N)

to get

    I_Nf-Q_sf
    
    		=     (_()̨-_()̨)
    
    		=     (+̨ N)(()̨-()̨) 
    
    		=     (+̨ N)(()̨-()̨) 
         +(+̨ N+ N)((+̨ N)-(+̨ N)) e^ı(+̨ N)·.

Moreover, the definition of the norm of  leads to

    I_Nf-Q_sf_^q
    
    		=   (|(+̨ N)(()̨-()̨)|^q
    
    		+   |(+̨ N+ N)((+̨ N)-(+̨ N))|^q
    		).

This together with   periodic Strang-Fix conditions yields

    |()̨-()̨|≤|1-()̨| + |1-()̨|≤ b_0||̨^s N^-s,
       |(+̨ N)-(+̨ N)|≤ b_||̨^s N^-s-α,  ∈.

Thus we have

    I_Nf-Q_sf_^q
    ≤     b_0^q||̨^sq N^-sq |(+̨ N) |^q
         + b_^q||̨^sq N^-sq-α q |(+̨ N+ N)|^q.

We go further with deriving   above two parts respectively. 

The first part can be estimated as

    ||̨^sq N^-sq |(+̨ N) |^q
    ≤      N^-sq|(+̨ N) (1+|+̨ N|_2^2)^(s+α)/2|^q
    
    	=       N^-sq|()̨ (1+||̨_2^2)^(s+α)/2|^q=N^-sq f_A_q^s+α^q.


To derive the second part, we apply the identity 

    |(+̨ N+ N)|^q=∑_∈|(+̨(+) N)|^q=|(+̨ N)|^q, ∈̨, ∈
 to obtain

    b_^q||̨^sq N^-sq-α q |(+̨ N+ N)|^q
    
    	=   |(+̨ N)|^q b_^q||̨^sq N^-sq-α q
    ≤   (|(+̨ N)(1+|+̨ N|_2^2)^μ/2|^q
       × b_^q(1+||̨_2^2)^-μ q/2||̨^sq N^-sq-α q)
    ≤   max_∈̨( b_^q(1+||̨_2^2)^-μ q/2||̨^sq N^-sq-α q) 
       ×(|(+̨ N)(1+|+̨ N|_2^2)^μ/2|^q)
    
    	=   max_∈̨( b_^q(1+||̨_2^2)^-μ q/2||̨^sq N^-sq-α q)
       ×(|(+̨ N)(1+|+̨ N|_2^2)^μ/2|^q)
    
    	=   max_∈̨( b_^q(1+||̨_2^2)^-μ q/2||̨^sq N^-sq-α q)·f_A_q^μ^q.

Furthermore,  the assumption μ> m leads to

    max_∈̨( b_^q(1+||̨_2^2)^-μ q/2||̨^sq N^-sq-α q)
    ≤   max_∈̨( b_^q(1+||̨_2^2)^-(μ-s) q/2 N^-sq-α q)
    ≤    N^-sqmax_∈̨(1+||̨_2^2)^-(μ-s) q/2max_∈̨( b_^q N^-α q)
    
    	=   N^-sqmax_∈̨(1+||̨_2^2)^-(μ-s) q/2max_∈̨( b_^q(1/N^2+|/N+ℓ|_2^2)^α q/2)
    ≤   γ_1^q  N^-sqmax_∈̨(1+||̨_2^2)^-(μ-s) q/2
    	=γ_1^q  N^-sq.


Finally, combining Lemma <ref> with the fact 

    f_A_q^s+α≤f_A_q^μ,  s+α≤μ,

we obtain the error estimate

    f-Q_sf_≤ CN^-(σ-α)f_A_q^μ+N^-sf_A_q^s+α+γ_1  N^-sf_A_q^μ≤ C N^-sf_A_q^μ,

which completes the proof. 



The above results can be easily extended to directly uniform grid by constructing a quasi-interpolant

    Q_,f()=    Q_s_1,h_1Q_s_2,h_2⋯ Q_s_d,h_df() 
    
    		=      ∑_j_1∑_j_2⋯∑_j_d f(x_1^j_1,…,x_d^j_d)∏_r=1^dψ_h_r,s_r(x_r-x_r^j_r;c_r),

with the kernel ψ_h_r,s_r(x_r;c_r)=ψ_s_r(x_r;c_r)· h_r, h_r=1/N_r. Similarly, we can get the following error estimate.


Define the quasi-interpolation Q_,f by (<ref>) with the kernel ψ_h_r,s_r(x_r;c_r) satisfying the periodic Strang-Fix conditions of order s_r, r=1,…,d. Then there exists a constant C such that the inequality

    f-Q_,f_A_q,^α()≤ Cf_A_q,^μ()∑_r=1^dN_r^-s_r
 holds true for any  f∈ A_q,^μ with μ≥α+ max_r{s_r}.



We first rewrite the error as

    f()-Q_,f()
    
    		=   (I-Q_s_1,h_1Q_s_2,h_2⋯ Q_s_d,h_d)f()
    
    		=   ∑_r=1^d(I-Q_s_r,h_r)Q_s_r+1,h_r+1⋯ Q_s_d,h_df().

Then we use the inequality

    Q_s_r,h_rf()_A_q,^α=    Q_s_r,h_rf()-f()_A_q,^α+f()_A_q,^α
    ≤     Cf()_A_q,^s_r+α, r=1,…,d

to get

    f()-Q_,f()_A_q,^α≤   ∑_r=1^d(I-Q_s_r,h_r)Q_s_r+1,h_r+1⋯ Q_s_d,h_df()_A_q,^α
    ≤    C∑_r=1^d N_r^-s_rQ_s_r+1,h_r+1⋯ Q_s_d,h_df()_A_q,^s_r+α
    ≤    Cf_A_q,^μ∑_r=1^d N_r^-s_r.




Note that sparse grid is  a special case of the directly uniform gird that has been widely used for high-dimensional function approximation case to alleviate the curse of dimension <cit.>. Our above constructed quasi-interpolation  Q_, defined in Equation (<ref>) has a sparse grid counterpart.





Let N=(N_1,…,N_d)∈ℕ^d be a d-tupe with length |N|=N_1+…+N_d and _N be a point set consisting of the directionally uniform points with the mesh size 

    h_N:=2^-N=(2^-N_1,…,2^-N_d).

Applying the union operator to point sets _N, we can construct  the sparse grid _ℓ,d at level ℓ in the form 

    _ℓ,d:=⋃_|N|=ℓ+d-1_N.

Besides, we can derive the number of points over the sparse grid _ℓ,d as

    |_ℓ,d|=(-1)^d-1∑_j=0^d-1(-1)^jd-1 j∑_|N|=ℓ+j|_N|,
 which is much less than the one of the full grid or directly uniform grid.
Suppose  we have discrete function values sampled at the above sparse grid at hand, then we can construct a corresponding sparse grid quasi-interpolation


    Q_ℓ,df() = (-1)^d-1∑_j=0^d-1(-1)^jd-1 j∑_|N|=ℓ+jQ_,f()

with

    Q_,f()=∑__N,∈_Nf(_N,)_,_(-_N,; c).



§ NUMERICAL EXAMPLES

This section demonstrates some  numerical simulations of applying our   quasi-interpolation with generalized  Gaussian kernels in approximating periodic functions over tori 𝕋^d.  We consider   finitely smooth functions

    G_p:^d→: ⟼⊗_r=1^d g_p(α_r),

where

    g_p(α_r)=λ_p(2+sgn(α_r-π)·sin^p(α_r))

with λ_p denoting a normalization constant such that g_p_L_2()=1. Note that for ϵ>0, we have g_p∈ H^1/2+p-ϵ() and thus G_p∈ H^1/2+p-ϵ(^d). 

We first explore L_∞-norm approximation errors and corresponding convergence rates of the proposed periodic quasi-interpolation with restricted generalized Gaussian kernels  ψ_2m+2 satisfying different orders  of periodic Strang-Fix conditions to approximate one-dimensional function g_6(α_1). Numerical results under m=0,1, 2 are presented in Table <ref>. It illustrates that  convergence rates are consistent with the orders of periodic Strang-Fix conditions. Moreover, we also approximate a two-dimensional function G_6(^2) on regular tensor grids and present corresponding approximation errors in Figure <ref>. It again validates theoretical analysis. 




Next, we use  quasi-interpolation on sparse grid  with restricted  generalized Gaussian kernels ψ_2m+2(α) (m=0,1,2) to approximate the test function G_p(α) for d=2,3,4,5, respectively. Relative L_∞-norm approximation errors and L_2-norm approximation errors are provided in Figure <ref>-Figure <ref>. We observe a small decrease in convergence rate as the dimension increases. Nevertheless, sparse grid quasi-interpolation with  restricted generalized Gaussian kernels satisfying high-order periodic Strang-Fix conditions do provide better approximation  with  higher convergence rate. More importantly, as demonstrated in Figure <ref>,  there is even an upturn of relative L_∞-norm approximation errors in the case of  restricted generalized Gaussian kernel satisfying periodic Strang-Fix condition of order six. In particular,   convergence order in terms of N exceeds one that surpasses the performance of  well-known and commonly used Monte-Carlo  method for high dimensional function approximation.







	
	
	
	
	
	
	
	
	
	
	
	




	
	
	
	
	




	
	
	
	
	






	
	
	
	
	










	
	
	
	
	















§ CONCLUSIONS AND DISCUSSIONS
 
Quasi-interpolation with radial kernels in the Schoenberg's model  often provides much lower approximation order. To circumvent the pitfall, the paper provides a new perspective of constructing a quasi-interpolation scheme from Gaussian kernel for periodic function approximation such that it provides the highest approximation order as the one of generalized Strang-Fix conditions satisfied by Gaussian kernel. However, the paper is only a starting point, future works will focus on  thorough and comprehensive study of constructing quasi-interpolation with high approximation orders from radial kernels.











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 Paper Title: High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations.







mode=title]High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations
High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations
T. Dzanic and L. Martinelli

1]Tarik Dzanic[orcid=0000-0003-3791-1134]
[1]
[cor1]Corresponding author
dzanic1@llnl.gov


2]Luigi Martinelli[orcid=0000-0001-6319-7646]

[1]Center for Applied Scientific Computing, Lawrence Livermore National Lab, Livermore, CA 94550, USA


[2]Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA


§ ABSTRACT 
The use of limiting methods for high-order numerical approximations of hyperbolic conservation laws generally requires defining an admissible region/bounds for the solution. In this work, we present a novel approach for computing solution bounds and limiting for the Euler equations through the kinetic representation provided by the Boltzmann equation, which allows for extending limiters designed for linear advection directly to the Euler equations. Given an arbitrary set of solution values to compute bounds over (e.g., numerical stencil) and a desired linear advection limiter, the proposed approach yields an analytic expression for the admissible region of particle distribution function values, which may be numerically integrated to yield a set of bounds for the density, momentum, and total energy. These solution bounds are shown to preserve positivity of density/pressure/internal energy and, when paired with a limiting technique, can robustly resolve strong discontinuities while recovering high-order accuracy in smooth regions without any ad hoc corrections (e.g., relaxing the bounds). This approach is demonstrated in the context of an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme for a variety of difficult problems in gas dynamics, including cases with extreme shocks and shock-vortex interactions. Furthermore, this work presents a foundation for limiting techniques for more complex macroscopic governing equations that can be derived from an underlying kinetic representation for which admissible solution bounds are not well-understood. 



Euler equations Boltzmann equation High-order Limiting Positivity-preserving Discontinuous Galerkin







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§ INTRODUCTION


The understanding of the dynamics of inviscid fluid flow governed by the compressible Euler equations plays a critical role in predicting the behavior of fluids in high-speed regimes. These gas dynamics equations are not only fundamental as a governing model for the study of compressible flow phenomena such as shock waves but also serve as a stepping-stone to more complex simulations of viscous compressible flows. Numerical methods for simulating transport-dominated physics have widely relied on the Euler equations as the prototypical model problem, largely due to the challenges posed by the strongly nonlinear interactions between shock waves, discontinuities, and vortical features in the flow. These challenges have driven the development of sophisticated numerical techniques to yield robust and high-fidelity resolution of these complex flow features. 

Various numerical approaches have been utilized for gas dynamics simulations, with the brunt of applications focusing on second-order finite volume and finite difference schemes. Although well-established in the literature and broadly robust for many practical uses, their relatively high numerical dissipation and low accuracy make them prohibitively expensive for high-fidelity (i.e., scale-resolving) simulations of complex unsteady flows. As a result, a particular class of numerical schemes which have more recently grown in popularity are high-order methods, which broadly encompass finite difference, finite volume, and finite element methods that can achieve arbitrarily high-order accuracy. Due to this increased accuracy paired with lower numerical dissipation, these methods can generally offer better resolution of small-scale flow features and transport-dominated flow physics at a lower relative computational cost than low-order schemes <cit.>. However, the high-order approximations which underpin these methods are primarily designed to yield high accuracy in smooth regions of the flow. Around discontinuities or severe gradients, the use of high-order schemes can cause spurious oscillations in the numerical solution, often referred to as Gibbs phenomena. These spurious oscillations result in a numerically ill-behaved solution and, in many cases, the failure of the numerical scheme altogether.

To extend the use of high-order method to problems that exhibit discontinuities and steep gradients, it is often necessary to augment the scheme with an additional numerical stabilization approach to increase its robustness in the vicinity of such features. One such stabilization approach is through a priori or a posteriori limiting on the discrete solution, which relies on using a secondary numerical approximation that is more robust (but generally less accurate). The solutions as predicted by the high-order scheme and the secondary low-order scheme are then combined in a manner which ideally ensures sufficient numerical stabilization around discontinuities without sacrificing the accuracy of the high-order scheme in smooth regions of the flow. Various forms of limiting have been applied to hyperbolic conservation laws, typically falling into variants of the flux-corrected transport method of <cit.> (e.g.,  <cit.>) or through limiting/modification of the underlying solution/operator approximations (e.g., <cit.>). 
We remark here that these examples only cover a small subset of the numerical approaches in the literature and do not by any means encompass all the possible limiting techniques used over the decades. 


One of the primary difficulties in designing performant limiting methods for high-order schemes stems not necessarily from the development of the limiting schemes themselves but instead from the difficulty of defining exactly what a “well-behaved” numerical solution is for nonlinear systems of equations. Limiting between the low-order and high-order solution is typically dictated by some ad hoc heuristic (e.g., indicator functions) or through a set of bounds (i.e., admissible regions/sets) for the solution. While heuristic approaches can be effective for some applications, they often rely on empirical problem-dependent parameters which can misidentify regions that require limiting and degrade the accuracy and robustness of the approach. On the other hand, limiting based on a set of admissible bounds for the solution is typically a highly-robust alternative, often providing guarantees that the limited solution will abide by certain physical constraints <cit.> (although typically at a larger computational expense). However, it is unclear what the optimal bounds are to enforce on the solution, tying in to the problem of defining what a well-behaved numerical solution is. Bounds on the numerical properties of the solution (e.g., total variation diminishing <cit.>, local extremum diminishing methods <cit.>, etc.) have shown to be robust for many applications, but these bounds are often quite restrictive and unnecessarily degrade the accuracy of the high-order scheme in smooth regions without ad hoc corrections, whereas more advanced stencil modification techniques (e.g., ENO/WENO/TENO-type schemes and limiters <cit.>) can retain this accuracy but are typically prohibitively expensive and algorithmically complex for arbitrary mixed unstructured meshes. Bounds based on attributes of the underlying governing equations/systems have shown promise, relying on either physical constraints (e.g., positivity of density/pressure <cit.>) or additional mathematical properties (e.g., minimum entropy principle <cit.>, invariant region <cit.>). Nevertheless, they still exhibit some difficulties, such as sometimes also reducing to first-order accuracy around smooth extrema (requiring ad hoc corrections such as relaxing the bounds), requiring nonlinear limiting procedures which may be computationally expensive, and, most importantly, relying on mathematical properties which may not easily extend to systems which are of more practical interest (e.g., Navier–Stokes equations, multi-physics applications, etc.). 


The methods to be presented in this work are motivated by two primary observations. The first observation is that numerical approximations of the nonlinear Boltzmann equation, a general kinetic description of molecular gas dynamics which underpins the behavior of fluids governed by the Euler and Navier–Stokes equations, are significantly more robust than approximations of the associated continuum conservation laws. In fact, high-order schemes for the Boltzmann equation have been effectively used to directly resolve shock structures without any numerical shock capturing approaches (or with just simple limiting techniques) while accurately resolving smooth features in the flow <cit.>. The second observation is that linear advection, which governs particle transport in the Boltzmann equation, is a significantly easier governing equation to design high-order limiting schemes for. Stabilization methods for high-order approximations of the advection equation have been developed for decades, and due to the linear scalar nature of the equation, simple yet effective bounds can be formed for the solution (e.g., local maximum principle). As such, there exist a variety of high-order limiting schemes for the advection equation that one can choose from, with varying degrees of robustness and accuracy. 


With these observations in mind, the goal of this work is to introduce a novel approach to computing and enforcing bounds for limiting schemes used in high-order approximations of the Euler equations. The basis of this work is the connection between the Euler equations and the underlying kinetic representation provided by the Boltzmann equation, which essentially allows for one to extend limiting techniques designed for linear advection to the Euler equations. The proposed approach relies on the existence of an exact representation of the particle distribution function of the Boltzmann equation in the inviscid limit, for which the desired linear advection limiting technique can analytically be applied to yield a set of admissible bounds for the conserved variables (i.e., density, momentum, energy) in the Euler equations. We demonstrate this approach in the context of an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme <cit.> utilizing an associated local maximum principle preserving limiter <cit.> for the underlying advection equation — however, this approach is broadly applicable in the sense of computing bounds given a set of discrete solution values (e.g., numerical stencil, local neighborhood, etc.), which can be extended to other bounds-preserving limiting methods such as flux-corrected transport and/or slope-limiting techniques. This approach is used to simulate a variety of difficult problems in gas dynamics, including cases with extreme shocks, near-vacuum conditions, and small-scale smooth flow features, showcasing its robustness around strong discontinuities and accuracy in smooth regions of the flow. Furthermore, this work presents a foundation for limiting techniques for more complex macroscopic governing equations that can be derived from an underlying kinetic representation (e.g., Navier–Stokes equations, magnetohydrodynamics, radiation-hydrodynamics, etc.) for which admissible solution bounds are not necessarily well-understood. 


The remainder of this manuscript is organized as follows. In <ref>, some preliminaries on the Euler and Boltzmann equations as well as limiting methods for high-order discontinuous Galerkin schemes are presented. The proposed approach is then introduced in <ref>, followed by numerical analysis in <ref> and implementation details in <ref>. The results of a variety of numerical experiments are then presented in <ref> followed by conclusions in <ref>. 



 


§ PRELIMINARIES





 §.§ Euler equations

The system of interest in this work is represented by the compressible Euler equations for gas dynamics in d dimensions without source terms, written in conservation form as

    𝐰(𝐱, t)t + ∇·𝐅(𝐰) = 0,

where the solution 𝐰 and flux 𝐅(𝐰) are defined as 

    𝐰 = [ ρ; 𝐦; E ]  and  𝐅(𝐰) = [      𝐦^T; 𝐦⊗𝐔 + P𝐈; (E+P)𝐔^T ].

Here, ρ is the density, 𝐦 is the momentum, and E is the total energy. The symbol 𝐈 denotes the identity matrix in ℝ^d × d, 𝐔 = 𝐦/ρ denotes the velocity, and P denotes the pressure, computed through the ideal gas assumption as 

    P = (γ - 1)ρ e = (γ - 1) (E - 1/2𝐦·𝐦/ ρ),

where γ is the specific heat ratio and e is the specific internal energy. It is sometimes convenient to represent the solution in terms of the primitive variables 𝐪 = [ρ, 𝐔, P]^T. The solution of the Euler equations is endowed with a convex invariant set corresponding to the positivity of density (ρ≥ 0) and positivity of pressure/internal energy (P, ρ e ≥ 0). 



 §.§ Boltzmann equation

The Euler equations can be recovered as the limiting behavior of the Boltzmann equation, given as 

    ∂_t f (𝐱, 𝐮, t) + 𝐮·∇ f = 1/Kn𝒞(f, f'),

where f (𝐱, 𝐮, t)∈ℝ^d is a scalar particle distribution function, Kn is the Knudsen number, and 𝒞(f, f') is the collision operator which models particle interactions <cit.>. This distribution function is defined over a higher-dimensional space, which includes not only the spatial position 𝐱 but also the microscopic velocity 𝐮, and represents the probability density of a particle existing at some location 𝐱 with some velocity 𝐮. For brevity, we neglect the internal energy domain here, but the effects of internal energy are later presented. The conserved flow variables 𝐰(𝐱, t) can be recovered through the moments of the distribution function as 

    𝐰(𝐱, t) = [ρ, 𝐦, E ]^T = 
        ∫_ℝ^d f (𝐱, 𝐮, t) ψ (𝐮)  d𝐮,

where ψ (𝐮)  [1, 𝐮, (𝐮·𝐮)/2]^T is the vector of collision invariants. We utilize the notation that the lowercase symbol 𝐮 refers to the microscopic velocity and the uppercase symbol 𝐔 refers to the macroscopic velocity.


In the limit of zero Knudsen number (i.e., zero viscosity), the distribution function f(𝐱, 𝐮, t) tends towards a unique equilibrium distribution function g(𝐱, 𝐮, t) which minimizes (across the velocity domain) the entropy H(z), i.e.,

    g = zarg min H(z),      H(z) = ∫_ℝ^d z log(z)  d𝐮.

and possesses identical primary moments, i.e.,

    ∫_ℝ^d g (𝐱, 𝐮, t) ψ (𝐮)  d𝐮 = ∫_ℝ^d f (𝐱, 𝐮, t) ψ (𝐮)  d𝐮.

From the H-theorem, the equilibrium distribution function (for a monatomic particle) is a Maxwell–Boltzmann distribution of the form

    g(𝐱, 𝐮, t) = ρ (𝐱, t)/[2 πθ(𝐱, t) ]^d/2exp [-𝐮 - 𝐔(𝐱, t) _2^2/2 θ (𝐱, t)],

where θ = P/ρ is a scaled temperature and ·_2^2 denotes the squared norm along the spatial dimension. We sometimes use the notation g(𝐰) to denote the equilibrium distribution corresponding to the macroscopic state 𝐰. The effects of particle collision drive the distribution function f to this equilibrium state g. In particular, the collision operator is known to conserve primary moments, i.e.,

    ∫_ℝ^d  C(f,f') ψ (𝐮)   d𝐮 = 0,

and is identically zero at equilibrium, i.e.,

    C(g,g') = 0.




 §.§ Discontinuous Galerkin methods



   
While the proposed approach to be presented is somewhat agnostic to the numerical method of choice, this work particularly focuses on nodal discontinuous Galerkin (DG)-type <cit.> approximations to <ref>, including strong-form representations such as flux reconstruction <cit.> schemes and collocated discontinuous spectral element methods. In these approaches, the domain Ω is partitioned into N elements Ω_k such that Ω = ⋃_NΩ_k and Ω_i∩Ω_j=∅ for i≠ j. With a slight abuse of notation, we use 𝐰(𝐱)  to denote the discrete solution within each element Ω_k, which is approximated by a set of N_s nodal basis functions as 

    𝐰(𝐱) = ∑_i=1^N_s𝐰_iϕ_i(𝐱)⊂ V_h,

where ϕ_i(𝐱) are the nodal basis functions, 𝐰_i are their associated coefficients, and V_h is the piece-wise polynomial space spanned by the basis functions. The order of the approximation is denoted as ℙ_p, where p is the maximal order of 𝐮(𝐱).



By integrating <ref> with respect to a test function 𝐰_h(𝐱) ⊂ V_h, the DG formulation is obtained as

    ∑_k=1^N{∫_Ω_k∂_t 𝐰·𝐰_h dV  + ∫_∂Ω_k𝐅(𝐰^-, 𝐰^+, 𝐧) ·𝐰_h d𝐬  -  ∫_Ω_k𝐅(𝐰)·∇𝐰_h dV} = 0.

These integrals are typically performed by quadrature over volume and surface quadrature nodes, shown by the blue and red nodes in <ref>, respectively. Similar strong form formulations can be obtained through a correction function-type approach <cit.>, which use corresponding solution and flux nodes akin to volume and surface quadrature nodes. Along each element interface ∂Ω_k, two solutions exist, the solution within the element of interest (denoted by the 𝐰^-) and the solution of its face-adjacent neighbor (denoted by 𝐰^+). The flux function here is replaced by a numerical interface flux 𝐅(𝐰^-, 𝐰^+, 𝐧), which depends on these two solutions as well as the outward facing normal vector 𝐧. The numerical flux is generally computed using exact <cit.> or approximate Riemann solvers (e.g., <cit.>).

While the solution of high-order DG schemes generally does not abide by certain key properties of the underlying hyperbolic systems (e.g., maximum principles, invariant regions, etc.), the scheme possesses a useful feature in that the evolution of the element-wise mean of the solution
𝐰, defined as 

    𝐰 = ∫_Ω_k𝐰 (𝐱) d𝐱/∫_Ω_kd𝐱,

is dependent on only the interface flux, i.e.,

    ∂_t 𝐰 = - ∫_∂Ω_k𝐅(𝐰^-, 𝐰^+, 𝐧)  d𝐬,

which allows it to be represented as a convex combination of temporal updates of first-order Godunov schemes <cit.>. Under some relatively minor assumptions on the numerical scheme (e.g., Courant-–Friedrichs–-Lewy condition, appropriate Riemann solver, explicit strong stability preserving time integration, appropriate quadrature, etc.), this ensures that the element-wise mean preserves convex invariants of the system <cit.>.




 §.§ Maximum principle preserving limiting

This property of the element-wise mean of high-order DG schemes allows it to be used in many numerical stabilization approaches, some examples of which are referenced as follows  <cit.>.
In the simplest example, the element-wise mean can be used to enforce a maximum principle on the advection equation, given as 

    ∂_t u(𝐱,t) + 𝐜·∇ u = 0,

where 𝐜 is the velocity. It is known that the solution of scalar hyperbolic conservation laws abides by a maximum principle in the sense that the extrema are bounded by the initial data u(𝐱, 0). In a discrete setting, this can be formulated as a discrete maximum principle as 

    m ≤ u(𝐱_i,t) ≤ M ∀ i ∈ V, t ≥ 0,

where m and M are the lower and upper bounds of the solution, respectively, and V is some set of nodes over which the maximum principle should be enforced. In general, V corresponds to the discrete locations which the solution needs to be evaluated (e.g., volume/surface quadrature nodes for DG schemes, solution/flux nodes for flux reconstruction schemes, etc.). 

One can apply a global maximum principle, where the bounds are computed over the domain as

    m = 𝐱∈Ωmin u(𝐱, 0),     M = 𝐱∈Ωmax u(𝐱, 0),

which results in a single set of bounds for the entire solution. The discrete solution within each element at some time step t^n can be limited in an a priori (or a posteriori) way using the approach of <cit.> as 

    u^n(𝐱) = u^n + α(u^n(𝐱) -  u^n),

where u^n(𝐱) is the limited solution, u^n is the element-wise mean, and α is the limiting factor. It can be shown that if α is computed as 

    α = min[1, 
         |M - u/i ∈ Vmax u(𝐱_i) - u|, 
         |m - u/i ∈ Vmin u(𝐱_i) - u|],

the limited solution is guaranteed to preserve the maximum principle, i.e., m ≤u (𝐱_i) ≤ M  ∀ i ∈ V.



In practice, this is often not restrictive enough since local extrema can exist in the solution which may not be adequately stabilized by global bounds. It is generally more advisable to enforce a discrete local maximum principle, for which the bounds are computed at each time step over some local stencil S for each element Ω_k as

    m_k^n+1 = i ∈ Smin u(𝐱_i, t^n),     M_k^n+1 = i ∈ Smax u(𝐱_i, t^n).

The local maximum preserving limiting approach requires the user to choose this local stencil S for some corresponding nodal set V to apply the bounds over. Various approaches for this exist, some examples of which are shown in <ref>. A common approach is to apply bounds uniformly across an element, with each volume/surface quadrature node having the same bounds, which may be computed over a local domain of influence such as the face/Voronoi neighbors for that element (<ref>) or the numerical stencil for that element (<ref>). More localized/restrictive approaches exist as well, where unique bounds can be computed over a local subset of nodes (e.g., a first-order stencil) within an element and enforced for each single node (<ref>). These different approaches broadly have their own advantages and disadvantages, with less restrictive bounds (e.g., face neighbors) often resulting in less numerical dissipation around smooth regions of the solution and more restrictive bounds (e.g., nodal neighbors) often resulting in better behaved solutions around discontinuities (particularly for very high-order approximations) and vice versa. The numerical intricacies of these approaches as well as their unique drawbacks are described in more detail in <cit.>, although the examples shown here and in the referenced work for local maximum principle limiters of DG approximations of linear transport are by no means exhaustive. 


   



§ METHODOLOGY

The overarching goal of this work is to use insights from high-order limiting schemes for the Boltzmann equation to guide the development of limiting schemes for the Euler equations. Essentially, this work aims to find a novel approach to computing and enforcing bounds on the solution of Euler equations through its connection with an underlying kinetic transport equation, which allows for the use of limiting techniques developed for linear advection. Due to the scalar linear nature of the advection equation, these limiting techniques have become significantly more established over the decades and are simpler and more robust than methods required for nonlinear hyperbolic systems of equations, with many options of varying degrees of algorithmic simplicity, robustness, and accuracy. Most importantly, through this approach, one instead recovers local bounds for the conserved variables of the solution, i.e.,


    ρ (𝐱_i)    ∈ [ρ_min, ρ_max],
    𝐦_j (𝐱_i)    ∈ [𝐦_j, min, 𝐦_j, max]   for   j ∈{1, …, d},
    
        E (𝐱_i)    ∈ [E_min, E_max],


for some arbitrary point 𝐱_i, which allows for the use of simple scalar limiting techniques and differs from many limiting approaches which enforce bounds on linear/nonlinear functionals of the solution (e.g., density, pressure, entropy, etc.).



 §.§ Boltzmann representation of the Euler equations

The Euler equations can be represented in terms of the zero Knudsen number limit of the Boltzmann equation. Consider an explicit approximation of the Boltzmann equation at some time step t^n (denoted by the superscript n), written as

    ∂_t f^n (𝐱, 𝐮) = - 𝐮·∇ f^n + 1/Kn𝒞(f^n, f^n').

For zero Knudsen number, the distribution function is in equilibrium, such that we obtain

    f^n (𝐱, 𝐮) = g^n (𝐱, 𝐮) = g (𝐰^n (𝐱)),

i.e., the distribution function is the Maxwell–Boltzmann distribution in <ref> for the corresponding macroscopic solution state 𝐰. Furthermore, from <ref>, the collision term in the equilibrium limit for a distribution function in equilibrium is zero, i.e.,

    Kn → 0lim 1/Kn𝒞(g, g') = 0.

Therefore, a single explicit temporal integration step for the Boltzmann equation with a solution initially at equilibrium (f^n = g^n) yields the collisionless Boltzmann equation, written as

    ∂_t f^n (𝐱, 𝐮) = - 𝐮·∇ f^n,

which can be integrated across the velocity domain to yield the Euler equations as 

    ∫_ℝ^d∂_t f^n (𝐱, 𝐮) ψ (𝐮) d𝐮 = ∂_t 𝐰^n (𝐱)   and  ∫_ℝ^d - 𝐮·∇ f^n (𝐱, 𝐮) ψ (𝐮) d𝐮 = - ∇·𝐅(𝐰^n (𝐱)).

Note that this observation is only valid for a single explicit step for an equilibrium state, as any deviations from equilibrium (driven by the transport term) are then immediately reversed by the (infinitely large) collision term at the next time step.  




 §.§ Maximum principle bounds for the Boltzmann equation

The collisionless Boltzmann equation can be considered as a high-dimensional linear transport equation, and for any arbitrary point in the velocity domain 𝐮 = 𝐮_0, f_0 = f(𝐮_0), this simply yields a corresponding linear advection equation,

    ∂_t f_0(𝐱, t) +  𝐮_0 ·∇ f_0 = 0.

As such, we can apply standard stabilization techniques developed for high-order approximations of linear advection such as a local maximum principle preserving limiter. For example, one can apply element-wise forward-in-time maximum principle bounds as presented in <ref>, which, for an arbitrary element Ω_k, enforces bounds on the nodes j ∈ V computed over the stencil i ∈ S as 

    i ∈ Smin f_0^n(𝐱_i) ≤ f_0^n+1(𝐱_j) ≤i ∈ Smax f_0^n(𝐱_i)   ∀ j ∈ V.

We present this in terms a forward Euler approximation, but one can straightforwardly generalize this to higher-order time stepping schemes. Since 𝐮_0 is arbitrary, this local maximum principle can be applied uniformly across the velocity domain, i.e.,

    f_min^n+1(𝐮)≤ f^n+1(𝐱_j, 𝐮) ≤ f_max^n+1(𝐮)  ∀ j ∈ V  ,  ∀ 𝐮∈ℝ^d.

where

    f_min^n+1(𝐮) = i ∈ Smin f^n(𝐱_i, 𝐮) 
          and   
        f_max^n+1(𝐮) = i ∈ Smax f^n(𝐱_i, 𝐮).

We note here that limiting methods of this form are not necessarily just constrained to collisionless Boltzmann approximations, and various techniques that include the effects of collision can also be used (e.g., operator splitting, retaining only positivity bounds <cit.>, etc.). However, this is not necessary in the proposed approach.  




 §.§ From kinetic to macroscopic

Since discrete velocity approximations of the Boltzmann equation are effectively identical to the advection equation, the goal of this approach is to extend limiting techniques (such as the aforementioned maximum principle preserving limiter) from the advection equation to the Euler equations through the Boltzmann equation. The connection between the local maximum principle bounds in <ref> and the Euler equations can be made through the Maxwell–Boltzmann distribution in <ref>, which analytically defines the distribution function as

    f^n(𝐱_i, 𝐮) = g(𝐰^n(𝐱_i)).

An example of this is shown in the schematic in <ref>, which showcases an example Euler solution within a DG element (which aims to local maximum principle bounds on the discrete numerical stencil) and the respective distribution function for the discrete solution nodes across that stencil. 

   

Given the analytic definition of the distribution function for each solution node in the stencil S, the bound for the distribution function at the next time step can be computed as 

    f_min^n+1(𝐱_i, 𝐮) = i ∈ Smin g(𝐰^n(𝐱_i)   and   
         f_max^n+1(𝐱_i, 𝐮) = i ∈ Smax g(𝐰^n(𝐱_i)),

which are piecewise defined over the velocity domain and are effectively the lower and upper bounds of the corresponding distribution functions of each solution node in the stencil S. A schematic of these distribution function bounds for the corresponding solution and stencil in <ref> is shown in <ref>. This essentially yields an expression for the region of “admissible” distribution function values over the velocity domain, bounded from below and above by f_min and f_max, respectively. 

   


Once the admissible distribution function values are defined, it is possible to then recover the corresponding admissible macroscopic states. In particular, we are interested in finding the minimum and maximum values of each component of the macroscopic solution [ρ, 𝐦, E]^T that can be recovered from a distribution function f which resides in the admissible region f_min≤ f ≤ f_max. For the density and energy, where the respective collision invariants are strictly non-negative (i.e., a non-negative measure), we can simply integrate the moments of the minimum/maximum distribution function to yield the minimum/maximum density and energy bounds, i.e.,
    
    ρ_min^n+1 = ∫_ℝ^d f_min^n+1 d𝐮  and  ρ_max^n+1 = ∫_ℝ^d f_max^n+1 d𝐮,

    
    E_min^n+1 = ∫_ℝ^d f_min^n+1 (1/2𝐮·𝐮)  d𝐮  and   E_max^n+1 = ∫_ℝ^d f_max^n+1 (1/2𝐮·𝐮)  d𝐮 d𝐮.

For momentum, where the collision invariants are not strictly non-negative, the process is slightly more involved, requiring splitting the integration over the velocity domain as 
    
    𝐦_i,min^n+1 = ∫_𝐮_i < 0ℝ^d f_max^n+1 𝐮_i  d𝐮 + ∫_𝐮_i ≥ 0ℝ^d f_min^n+1 𝐮_i  d𝐮

and
    
    𝐦_i,max^n+1 = ∫_𝐮_i < 0ℝ^d f_min^n+1 𝐮_i  d𝐮 + ∫_𝐮_i ≥ 0ℝ^d f_max^n+1 𝐮_i  d𝐮,

for each momentum component i. This yields a set of bounds for the conserved variables at the next time step which can be used for the limiting method of choice, which we denote by 𝐰_i,min and 𝐰_i,max, respectively, for each solution component 1 ≤ i ≤ d+2. The technique for approximating these integrals is later presented in <ref>.

    

 §.§ Limiting

As the bounds for the proposed approach are given in terms of the conserved variables instead of functionals thereof (e.g., pressure, entropy, etc.), the limiting can be performed by straightforwardly treating the individual solution components as scalar local maximum principle limiting tasks. In particular, the limiting method in this work is the squeeze limiter of <cit.> presented in <ref>, which uniformly contracts the high-order solution to the element-wise mean. For the proposed approach, the limiting factor α for each solution component i can be computed as

    α_i = min[1, 
         |𝐰_i, max - 𝐰_i/j ∈ Vmax 𝐰_i(𝐱_j) - 𝐰_i|, 
         |𝐰_i, min - 𝐰_i/j ∈ Vmin 𝐰_i(𝐱_j) - 𝐰_i|].

We set the limiting factor explicitly as α_i = 0 if |j ∈ Vmin 𝐰_i(𝐱_j) - 𝐰_i| < ϵ or |j ∈ Vmax 𝐰_i(𝐱_j) - 𝐰_i| < ϵ, where ϵ = 10^-13. This limiting factor is computed separately for each element in the domain. 

As the bounds are decoupled by solution component, the limiting can be performed independently (where each solution component is limited by its respective limiting factor) or coupled (where each solution component is limited by the minimum of all limiting factors). For this work, we use the latter approach, such that the limited solution 𝐰(𝐱) is obtained as

    𝐰(𝐱) = 𝐰 + α_min (𝐰(𝐱) - 𝐰),

where 

    α_min = imin α_i,

although one may obtain a marginally less dissipative approach by using the former. It will later be shown in <ref> that a limiting factor 0 ≤α_min≤ 1 is guaranteed to exist for which the limited solution discretely satisfies the imposed bounds, i.e.,

    j ∈ Vmin 𝐰_i(𝐱_j) ≥𝐰_i, min  and  j ∈ Vmax 𝐰_i(𝐱_j) ≤𝐰_i, max  for   1 ≤ i ≤ d+2.




    

 §.§ Relaxing the bounds

While strictly enforcing a local maximum principle on the high-order scheme is often enough to yield a well-behaved solution around discontinuities (although in some cases more restrictive conditions are needed, see <cit.> for example), it is often too restrictive of a condition in smooth regions of the flow
and will often degrade the accuracy of the scheme around smooth extrema, sometimes referred to as the peak clipping effect. This effect is seen in various limiting approaches for high-order approximations of scalar transport and gas dynamics <cit.>, and one remedy for this effect is to marginally relax the bounds to 
recover high-order accuracy in smooth regions of the flow.

However, as will be shown in the numerical experiments, the proposed approach does not appear to require relaxing the bounds to recover high-order accuracy around smooth extrema whilst simultaneously enforcing sufficient stabilization to ensure convergence around discontinuities. Nevertheless, it can be beneficial to relax the bounds, both to allow for a slight numerical tolerance in terms of limiting and to compensate for any minor numerical integration errors in computing the bounds from the distribution function. As such, we relax the bounds in a relative manner as 

    f_min^n+1(𝐱_i, 𝐮) = i ∈ Smin (1 - r_fac) g(𝐰^n(𝐱_i)   and   
         f_max^n+1(𝐱_i, 𝐮) = i ∈ Smax (1 + r_fac) g(𝐰^n(𝐱_i)),

where the relaxation factor is set as r_fac = 10^-3, which is identical to relaxing f_min^n+1 and f_max^n+1 themselves. Similar relaxation approaches are taken in other limiting methods such as in the works of <cit.>. The effects of enforcing bounds with and without relaxation for smooth and discontinuous solutions will later be shown in the numerical results, but, unless otherwise stated, the relaxed bounds are used in this work.
    
    

 §.§ Modeling internal energy

On its own, the Maxwell–Boltzmann distribution models a monatomic particle which has d translation degrees of freedom. As such, this would constrain the specific heat ratio to γ = 1 + 2/d. To recover any arbitrary value of γ∈ [1, 1 + 2/d] and effectively model polyatomic molecules, it is necessary to include the effects of internal degrees of freedom in the distribution function. This is typically achieved through modeling an additional dimension for the internal energy domain, ζ∈ [0, ∞), with the macroscopic state then computed as

    𝐰(𝐱, t) = 
        ∫_0^∞∫_ℝ^d f (𝐱, 𝐮, ζ, t) ψ (𝐮, ζ)  d𝐮 dζ,

where the updated collision invariants are defined as ψ (𝐮, ζ)  [1, 𝐮, (𝐮·𝐮)/2 + ζ]^T (see <cit.>, for example). However, naively implementing an internal energy model would add another dimension for evaluating/integrating the distribution function, which can drastically increase the computational cost. Since we are only interested in the effects of the internal energy on the macroscopic state, we can instead take a reduced distribution approach <cit.> by analytically integrating over the internal energy domain to compute the macroscopic state as

    𝐰(𝐱, t) = ∫_ℝ^d f_T (𝐱, 𝐮, ζ, t) [1, 𝐮, (𝐮·𝐮)/2]^T  d𝐮 + ∫_ℝ^d f_I (𝐱, 𝐮, ζ, t) [0, 0, 1]^T  d𝐮,

where f_T and f_I represent the translation (monatomic) and internal energy components of the distribution function. This is further simplified through the relation that for an equilibrium distribution function, 

    g_I = θδ/2g_T,

where δ is the number of internal degrees of freedom and θ is the scaled temperature of the macroscopic state. By setting δ = 2/γ -1 - d, one can recover the desired specific heat ratio γ. From a computational perspective, computing the bounds for the distribution function for a particle with internal energy effects (as opposed to the monatomic case) simply requires the addition of the θδ/2 term in the collision invariant for the total energy. 
    
    
    

 §.§ Integrating the distribution function

Computing the bounds requires integrating the distribution functions f_min and f_max. Although it is possible in theory to perform this integration analytically, it is not tractable to do it in any computationally feasible manner outside of simple one-dimensional cases. As such, this integration must be performed numerically via quadrature. The quadrature techniques for this approach broadly follow the techniques used in numerical approximations of the Boltzmann equation, where similar integration over the velocity domain is necessary to recover the moments of the distribution function. However, since the distribution functions to be integrated are no longer the Maxwell–Boltzmann distribution (but a piecewise combination thereof), certain techniques which rely on its properties do not directly apply (e.g., Gauss–Hermite quadrature). 

First, the infinite velocity domain can be truncated to a reduced velocity domain Ω_u ⊂ℝ^d. Since the Maxwell–Boltzmann distribution is a Gaussian distribution centered around 𝐔 with a standard deviation of √(θ), one can quantify the maximum error of truncating the domain to k standard deviations around the mean (i.e., 𝐔± k √(θ)). For example, k =4 and k =8 bound the relative error of

    ∫_ℝ^d g  d𝐮 - ∫_Ω_u g  d𝐮

by approximately 6· 10^-5 and 1 · 10^-15, respectively. We use a similar approach to truncate the velocity domain, where the largest extent over the stencil (along each direction) is used, i.e.,
    
    Ω_u =  [ i ∈ Smin (𝐔(𝐱_𝐢) - k√(θ(𝐱_i))), i ∈ Smax (𝐔(𝐱_𝐢) - k√(θ(𝐱_i))) ]

We assume that k is chosen appropriately to achieve the desired tolerance, which should be related to the relaxation factor used (i.e., k should ostensibly be chosen such that the relative error is ≲ r_fac). For r_fac = 10^-3 in this work, this implies that k ≥ 3.5, whereas without relaxation, one should integrate the bounds to machine precision (i.e, k=8). We note here however that exact integration (even over the truncated domain) isn't actually necessary, and any numerical integration errors simply result in minor under/overprediction in the limiting factors without affecting the positivity-preserving properties of the approach. 

With the truncated domain defined, one can introduce a quadrature operation as
    
    ∑_i = 1^N_u M_i f(𝐮_i) ψ(𝐮_i) ≈∫_ℝ^d f(𝐮) ψ(𝐮) d𝐮,

where M_i is a quadrature weight and 𝐮_i ∈Ω_u correspond to a set of N_u quadrature nodes in the velocity domain. A wide variety of numerical integration techniques can be used here. In this work, we particularly rely on trapezoidal rule integration over an equispaced Cartesian mesh which is commonly employed for discrete velocity approximations of the Boltzman equation. Higher-order (albeit less robust) quadrature techniques can also be used, although the piecewise nature of the underlying distribution function tends to mitigate benefits of many higher-order techniques which have an underlying assumption of the smoothness of the integrand. 

    

 §.§ Bounding density and pressure

As will be shown in <ref>, the proposed bounds guarantee that the density and pressure/internal energy will be positive for any 𝐰∈ [𝐰_min, 𝐰_max]. However, it is often desirable to enforce some minimum threshold on the density/pressure since just strictly enforcing positivity can lead to numerical difficulties for very small values of density/pressure (e.g., ill-behaved Riemann solver, unfeasible time step restrictions). As such, we enforce a floor value of 10^-12 on the density and pressure. For density, this entails simply replacing the minimum density bound ρ_min with max (ρ_min, 10^-12). For pressure, this process is slightly more involved since the proposed bounds are given in terms of the conserved variables. If the discrete limited pressure, computed as 

    P_min = i ∈ Vmin  P(𝐰(𝐱_i)),

was below the threshold, we apply another limiting step as 

    𝐰(𝐱) = 𝐰 + α_P (𝐰(𝐱) - 𝐰),

with the limiting factor computed as

    α_P =
         |10^-12 - P(𝐰)/P_min - P(𝐰)|,

The quasiconcavity of the pressure functional ensures that i ∈ Vmin  P(𝐰(𝐱_i)) ≥ 10^-12.

    

 §.§ Overview


With the details of the proposed approach presented, we summarize here an overview of the method applied to high-order approximations of the Euler equations. This overview is shown in terms of a single temporal update t^n → t^n+1 for an arbitrary element Ω_k, but this can directly be generalized to every element and each substage of higher-order (SSP) time integration schemes. The necessary code for the implementation of this approach is provided in the electronic supplementary material. 


    
  * Collect the macroscopic solution 𝐰^n(𝐱_i) at each node in the chosen stencil S. 
    
  * Evaluate the distribution function bounds f^n+1_min and f^n+1_max (per <ref> and <ref>) and the collision invariants ψ = [1, 𝐮, 𝐮·𝐮/2 + ζ] at each discrete velocity node 𝐮_i. Perform a quadrature summation over the discrete velocity nodes (1 ≤ i ≤ N_u) to recover the macroscopic bounds per <ref>.
    
  * Apply a minimum threshold on the density bound. 
    
  * Advance solution in time 𝐰^n(𝐱) →𝐰^n+1(𝐱).
    
  * Compute the limited solution per <ref>.
    
  * Compute the minimum pressure of the limited solution per <ref>.
    
  * If minimum pressure is below the minimum threshold, apply a second limiting step per <ref>. 
    
  * Replace the high-order DG solution 𝐰^n+1(𝐱) with the limited solution 𝐰^n+1(𝐱)) (or 𝐰^n+1(𝐱)). 


 


§ ANALYSIS


With this formulation, we now move on to state and prove the numerical properties of the proposed approach. The subsequent proofs rely on some (or all) of the following assumptions. 

    
  * Let there be a stencil S over which the bounds are computed for an arbitrary element Ω_k, and let G be the set of solutions with positive density ρ > 0 and positive pressure/internal energy P, ρ e > 0. The initial solution 𝐰_i ∈ G  ∀ i ∈ S at some time t^n.
    
  * The distribution functions are integrated via a discrete integration operator 𝐌 consisting of N_u positive entries corresponding to integration over a subdomain Ω_u ⊂ℝ^d for which we denote its extent as u_max = 𝐮∈Ω_umax 𝐮_2 which is assumed to be at least two thermal/absolute velocity units (i.e., k ≥ 2). It is assumed that the integration operator can integrate moments of an arbitrary distribution function f to some desired numerical tolerance, i.e.,
    
    ∫_ℝ^d f(𝐮) ψ(𝐮) d𝐮≈∑_i = 1^N_u M_i f(𝐮_i) ψ(𝐮_i).

    We denote this operation in shorthand as 𝐌· f ⊗ψ(𝐮). 
    
  * The nodes of the stencil over which to compute bounds (denoted by S) and the nodes over which the limiting is performed (denoted by V ⊂ S) are chosen appropriately such that: 
    
        
  * There exists a set of positive quadrature weights c_i such that 
        
    𝐰 = ∫_Ω_k𝐰(𝐱) d𝐱/∫_Ω_kd𝐱 = ∑_i ∈ V c_i 𝐰(𝐱_i)/∑_i ∈ V c_i.

        
  * There exists a subset of nodes V' ⊂ V (e.g., surface quadrature nodes) with a corresponding set of positive quadrature weights c_i' such that 
        
    ∫_∂Ω_k𝐰(𝐱) d𝐬≈∑_i ∈ V' c_i' 𝐰(𝐱_i).

    
    
  * The interface numerical fluxes are computed using a Rusanov-type approach <cit.> as
    
    𝐅(𝐰^+, 𝐰^-, 𝐧) = 1/2 [𝐅(𝐰^-) + 𝐅(𝐰^+) ]·𝐧 - 1/2λ_max(𝐰^+, 𝐰^-) [𝐰^+ -𝐰^- ],

    where λ_max is an upper bound on the local maximum wavespeed of the system. 





    The proposed bounds preserve constant states in the sense that if 𝐰(𝐱_i, t^n) = 𝐰_0  ∀  i ∈ S for some constant state 𝐰_0, then 𝐰_min^n+1 = 𝐰_max^n+1 = 𝐰_0.


pot_constProof of <ref>

The proof follows directly from the definition of f_min^n+1 and f_max^n+1 in <ref>, which yields 

    f_min^n+1 = f_max^n+1 = g(𝐰_0).

By integrating the moments, one yields

    𝐰_min^n+1 = 𝐰_max^n+1 = 𝐌· g(𝐰_0) ⊗ψ(𝐮) ≈∫_ℝ^d g(𝐰_0)  ψ (𝐮)  d𝐮 = 𝐰_0.




 The proposed bounds preserve positivity of density and pressure/internal energy for all 𝐰∈ [𝐰_min^n+1, 𝐰_max^n+1].


pot_ppProof of <ref>

To show positivity of density, it suffices to show that ρ_min^n+1 > 0, which can be expressed as the condition

    𝐌· f_min^n+1 > 0.

From <ref>, the distribution function g(𝐰) is strictly positive if 𝐰∈ G. Therefore, f_min^n+1(𝐮), f_max^n+1(𝐮) > 0 for all 𝐮∈ℝ^d, which means that ρ_min^n+1 > 0.

To show positivity of pressure/internal energy for all 𝐰∈ [𝐰_min^n+1, 𝐰_max^n+1], we use the relation that any 𝐰_0 ∈ [𝐰_min^n+1, 𝐰_max^n+1] can be expressed as 

    𝐰_0 = 𝐌· f_0^n+1⊗ψ(𝐮)

for some distribution function f_min^n+1(𝐮) ≤ f_0^n+1(𝐮) ≤ f_max^n+1(𝐮), where f_0^n+1(𝐮) > 0. The internal energy can be calculated as 

    ρ e = E - 1/2ρ𝐔·𝐔 = 𝐌· f_0^n+1 (1/2𝐮·𝐮 -  1/2𝐮·𝐔)
        = 𝐌·1/2f_0^n+1𝐜·𝐜,

where 𝐜 = 𝐮 - 𝐔 is the peculiar velocity. From the positivity of the distribution function and measure 𝐜·𝐜, it follows that ρ e > 0 and, by extension, P = (γ-1)ρ e > 0. 




For any arbitrary 𝐰_i, 𝐰_j∈ G and corresponding bounds [f_min, f_max] and [𝐰_min, 𝐰_max], the Riemann-averaged state

    𝐰_ij = 1/2(𝐰_i + 𝐰_j ) - 1/2 λ_max(𝐅(𝐰_j) - 𝐅(𝐰_i) )·𝐧_ij

preserves the bounds in the sense that 𝐰_ij∈ [𝐰_min, 𝐰_max] for any arbitrary normal vector 𝐧_ij if λ_max is an upper bound on the local maximum wavespeed of the system. 

pot_rasProof of <ref>

By using the relation that 

    𝐅(𝐰) = ∫_ℝ^d𝐮 g(𝐰) ψ(𝐮) d𝐮,

the Riemann-averaged state can be expressed in terms of a distribution function g_ij as 

    𝐰_ij = 𝐌·g_ij⊗ψ (𝐮) = 𝐌· [ 1/2(g_i + g_j ) - 1/2 λ_max(g_j - g_i )𝐮·𝐧_ij] ⊗ψ (𝐮).

Let λ_max = u_max≥max (|𝐔_i·𝐧_ij| + √(γθ_i), |𝐔_j·𝐧_ij| + √(γθ_j)). The Riemann-averaged distribution function can be expressed as

    g_ij = g_i  (1/2 + 1/2𝐮·𝐧/λ_max) + g_j  (1/2 - 1/2𝐮·𝐧/λ_max).

Since λ_max≥ |𝐮·𝐧_ij|, g_ij is a convex combination of the states g_i and g_j. Therefore,

    f_min = min (g_i, g_j) ≤g_ij≤max (g_i, g_j) = f_max,

from which it directly follows that 𝐰_ij∈ [𝐰_min, 𝐰_max]. 



    The proof of <ref> is based on an upper bound on the local maximum particle speed u_max (on the truncated domain Ω_u) instead of the local maximum macroscopic wavespeed (e.g., the Davis <cit.> wavespeed estimate λ_max = |𝐔·𝐧| + √(γθ)). At high Mach numbers, where the thermal velocity is small compared to the flow velocity, these estimates do not drastically differ, but at low Mach numbers, the local maximum particle speed may be several times larger than the maximum macroscopic wavespeed — a factor of k/√(γ) — which can affect the maximum admissible time step required to satisfy the bounds. However, we observed in empirical studies that setting λ_max as the local maximum macroscopic wavespeed still always ensured that the Riemann-averaged state satisfied the bounds, but we do not have a rigorous mathematical proof for this behavior.





For any arbitrary 𝐰_0^n, 𝐰_1^n, 𝐰_2^n ∈ G and corresponding bounds [𝐰_min^n+1, 𝐰_max^n+1], the first-order Lax–Friedrichs scheme, 

    𝐰_1^n+1 = 𝐰_1^n - Δ t/Δ x ( 𝐅(𝐰_1^n, 𝐰_2^n) - 𝐅(𝐰_0^n, 𝐰_1^n) ) ,

where 

    𝐅(𝐰_a, 𝐰_b) = 1/2 [𝐅(𝐰_a) + 𝐅(𝐰_b) ] - 1/2λ_max(𝐰_a, 𝐰_b) [𝐰_b -𝐰_a ]

preserves bounds in the sense that 𝐰_1^n+1∈ [𝐰_min^n+1, 𝐰_max^n+1] under the standard Courant-–Friedrichs–-Lewy (CFL) condition λ_maxΔ t/Δ x≤ 1.

pot_foProof of <ref>


The first-order Lax–Friedrichs scheme can be written in terms of the Riemann-averaged states as 

    𝐰_1^n+1 = (1 - λ_max^0,1Δ t/Δ x - λ_max^1,2Δ t/Δ x)𝐰_1^n + λ_max^0,1Δ t/Δ x𝐰_01 + λ_max^1,2Δ t/Δ x𝐰_12,

where λ_max^0,1 = λ_max(𝐰_0, 𝐰_1), λ_max^1,2 = λ_max(𝐰_1, 𝐰_2), and 𝐧_01 = 𝐧_12 = 1. If Δ t is sufficiently small such that λ_maxΔ t/Δ x≤ 1, where λ_max = max (λ_max^0,1, λ_max^1,2), then 𝐰_1^n+1 is a convex combination of states in [𝐰_min^n+1, 𝐰_max^n+1]. As a result, 𝐰_1^n+1∈ [𝐰_min^n+1, 𝐰_max^n+1].








For some arbitrary element Ω_k and its associated mean 𝐰^n = 𝐰(t^n), there exists a sufficiently small Δ t > 0 such that 𝐰(t^n + Δ t) = 𝐰^n+1∈ [𝐰_min^n+1, 𝐰_max^n+1]. 

pot_mbpProof of <ref>


The element-wise mean can be computed as 

    𝐰^n = ∑_i ∈ V c_i  [ 𝐌· g(𝐰_i^n) ⊗ψ (𝐮)]/∑_i ∈ V c_i,

which, by linearity of the quadrature operators, can be expressed as 

    𝐰^n = 𝐌· [ ∑_i ∈ V c_i  g(𝐰_i^n)/∑_i ∈ V c_i ] ⊗ψ (𝐮),

Since, by definition, g(𝐰_i^n) ∈ [f_min^n+1, f_max^n+1]  ∀ 𝐮∈ℝ^d  ∀ i ∈ V and [f_min^n+1, f_max^n+1] is a convex set, the convex combination ∑_i ∈ V c_i  g(𝐰_i^n)/∑_i ∈ V c_i ∈ [f_min^n+1, f_max^n+1]  ∀ 𝐮∈ℝ^d  ∀ i ∈ V. Therefore, 𝐰^n ∈ [𝐰_min^n+1, 𝐰_max^n+1].

It now remains to be shown that 𝐰^n+1∈ [𝐰_min^n+1, 𝐰_max^n+1]. The temporal update of the element-wise mean can be represented as

    𝐰^n+1 = 𝐰^n - Δ t ∑_i ∈ V' c_i' 𝐅(𝐰^n, -_i, 𝐰^n, +_i, 𝐧_i) =  ∑_i ∈ V c_i 𝐰^n_i/∑_i ∈ V c_i - Δ t ∑_i ∈ V' c_i' 𝐅(𝐰^n, -_i, 𝐰^n, +_i, 𝐧_i) ,

where 𝐰^n, -_i, 𝐰^n, +_i ∈ [𝐰_min^n+1, 𝐰_max^n+1] for all i ∈ V'. We use the shorthand 𝐰_i to denote 𝐰(𝐱_i). 𝐰^n+1 can be equivalently expressed as a convex combination as 

    𝐰^n+1 = ∑_i ∈ V∖ V' c_i 𝐰^n_i + ∑_i ∈  V' c_i 𝐰^*_i/∑_i ∈ V c_i ,

where

    𝐰^*_i = 𝐰^n, -_i - Δ t c'_i/c_i𝐅(𝐰^n, -_i, 𝐰^n, +_i, 𝐧_i).

Thus, it is sufficient to show that the intermediate states 𝐰^*_i ∈ [𝐰_min^n+1, 𝐰_max^n+1] for all i ∈ V'. These states can be represented in terms of the Riemann-averaged states as

    𝐰^*_i =   [1 - C ]𝐰^n, -_i + C 𝐰_i^- i^+,

where 

    C = Δ t λ_max(𝐰^n, -_i, 𝐰^n, +_i)c'_i/c_i.

It can be seen that 𝐰^*_i ∈ [𝐰_min^n+1, 𝐰_max^n+1] for any 0 ≤ C ≤ 1. Note here that c_i'/c_i is 𝒪(1/h), where h is some local mesh scale, as it is a surface quadrature weight normalized by a volume quadrature weight. Therefore, the above condition is met under a CFL-like constraint Δ t λ_max(𝐰^n, -_i, 𝐰^n, +_i)c'_i/c_i≤ 1, which then guarantees that 𝐰^n+1∈ [𝐰_min^n+1, 𝐰_max^n+1].





If 𝐰(t^n+1) ∈ [𝐰_min^n+1, 𝐰_max^n+1], there exists a limiting factor α∈ [0,1] such that 𝐰(𝐱_i, t^n+1) ∈ [𝐰_min^n+1, 𝐰_max^n+1] ∀ i ∈ V.




§ IMPLEMENTATION

The proposed limiting approach was implemented using the flux reconstruction (FR) scheme of <cit.>, a strong-form generalization of the nodal DG method, within PyFR <cit.>, a high-order unstructured FR solver that can efficiently target massively-parallel CPU and GPU computing architectures. To showcase generalizability with respect to solution node sets, we utilize open solution/flux node sets (i.e., where the interface flux nodes are not a subset of the interior solution nodes), namely the Gauss–Legendre/Gauss–Legendre solution/flux nodes for line/quadrilateral elements and Williams–Shunn/Gauss–Legendre solution/flux nodes for triangular elements <cit.>. Temporal integration was performed using a three-stage, third-order strong stability preserving Runge–Kutta scheme (SSP-RK3 <cit.>), with limiting performed after each substage of the scheme. Common interface numerical fluxes were computed using the HLLC Riemann solver of <cit.>. We note here that while the numerical analysis in this work only presents some theoretical guarantees for Rusanov-type Riemann solvers, it was found empirically in the numerical experiments that many other positivity-preserving entropy-stable Riemann solvers always preserved the bounds on the element-wise mean (under the standard assumptions), such that less dissipative approaches like HLLC were found to be more performant. 

The bounds were computed over the numerical stencil S, consisting of the interior solution and flux nodes as well as the exterior (face-adjacent) flux nodes, and limiting was performed over the element stencil V, consisting of just the interior solution and flux nodes, as shown in <ref>. The distribution functions were integrated over the velocity domain using uniform trapezoidal rule quadrature with the domain extent set to four thermal/absolute velocity units (i.e., k=4). Similar accuracy was observed with more advanced quadrature techniques such as Romberg integration. For the one- and two-dimensional experiments to be presented, the velocity domain resolution was fixed separately. For one-dimensional problems, we purposely use a large highly-resolved velocity domain (k = 8, N_u  = 2048) which can integrate the bounds to essentially near machine precision levels so that we can effectively compare the accuracy of the approach using strict and relaxed bounds without introducing any implicit relaxation in the bounds due to numerical under-integration of the distribution functions. For two-dimensional problems, we use a more appropriate resolution of k = 4, N_u = 32^2 for every problem, although we note that similar results could be obtained with a lower resolution of N_u = 16^2. In theory, one can use more efficient adaptive quadrature techniques to yield a more performant algorithm since the piecewise nature of the distribution function gives a priori information about the required local resolution, but this is not explored in this work as its primary purpose is to evaluate the accuracy of the proposed approach. 



§ RESULTS




 §.§ Sod shock tube









An initial evaluation of the shock capturing and discontinuity resolving capabilities of the proposed approach was performed with the Sod shock tube <cit.>. This canonical gas dynamics test case consists of the three main features of the Riemann problem: shock waves, rarefaction waves, and contact discontinuities. The domain was set as Ω=[0,1] with Dirichlet boundary conditions, and the initial conditions were set as

    
    𝐪(x,0) = 𝐪_l,     x⩽ 0.5,
    𝐪_r,    ,
              where  𝐪_l = [ 1; 0; 1 ],   𝐪_r = [ 0.125;     0;   0.1 ].


The problem was solved using varying approximation orders and mesh resolution, and the convergence of the density error with respect to the exact solution was computed, calculated as the point-mean L^1 norm on the solution nodes. The errors and rates are shown in <ref> as computed with the relaxed bounds. It can be seen that the optimal rate of unity was nearly achieved at all approximation orders. A marginally higher convergence rate was observed at lower approximation orders (ℙ_2 and ℙ_3) than at higher approximation orders (ℙ_4 and ℙ_5), which is consistent with observations that element-wise local maximum principle limiters are better behaved at lower approximation orders <cit.>. The experiments were also repeated with strict bounds, the results of which are tabulated in <ref>. It can be seen that very similar error magnitudes were obtained with and without relaxing the bounds, with the relaxed bounds even achieving marginally higher convergence rates. 





 §.§ Smooth convecting density pulse


    


    
    
To verify that the proposed approach retains the high-order accuracy of the underlying DG scheme even at smooth extrema, a smooth convecting density pulse was simulated. The problem was evaluated on the periodic domain Ω=[-0.5, 0.5], and the initial conditions are given as
    
        
    ρ    = 1 + exp( - σ x^2)  
    
                u    = 1, 
    
                P    = 1,

    
where σ = 500 is the strength of the Gaussian wave. This problem consists of a density perturbation convecting through the domain at constant velocity. To evaluate the behavior of the limiting approach at smooth extrema, the L^∞ norm of the density error was calculated at the solution nodes after one flow-through of the domain (t = 1). The error and convergence rates at varying approximation orders and mesh resolution are shown in <ref> and <ref> for the limiting with and without relaxing the bounds, respectively. It can be seen that the approach recovers the expected high-order (i.e., approximately p+1) convergence rates for all approximation orders. Most importantly, it can also be seen that strictly enforcing the bounds does not degrade the accuracy of the scheme at smooth extrema, indicating that the proposed approach is not overly restrictive in smooth regions of the flow. This is in contrast to many limiting techniques which require ad hoc corrections to recover high-order accuracy at smooth extrema. 

    
    


 §.§ Double expansion wave


   
    
The positivity-preserving properties of the proposed approach at near-vacuum conditions were then evaluated through the double expansion wave problem. This problem, known as the 123 problem (or Test 2) in <cit.>, was solved on the domain Ω = [0,1] with Dirichlet boundary conditions, and the initial conditions were set as 

    𝐪(x, 0) = 
        𝐪_L,   if  x ≤ 0.5,
    𝐪_R,   else,
          where  𝐪_L = 
        [   1;  -2; 0.4 ],
          𝐪_R = 
        [   1;   2; 0.4 ].

This Riemann problem develops into two outrunning expansion waves, causing a low density and pressure region in the center. The resulting symmetric wave structure and the near-vacuum center region are notably difficult to resolve, with even robust first-order schemes often performing poorly or failing altogether. 


The problem was solved using a ℙ_2 approximation with varying mesh resolution between N = 50 and N = 400 elements. The profiles of density, velocity, pressure, and specific internal energy at t = 0.15 are shown in <ref>. The predicted solution profiles showed clear qualitative convergence to the exact solution with increasing resolution, even in the near-vacuum center region. The specific internal energy profile showed a distinct spike in the center, a very commonly encountered numerical artifact in this problem indicative of spurious physical entropy generation in the center <cit.>. However, as expected, this spike decreased with increasing resolution, also converging toward the exact solution. 



 §.§ Shu–Osher problem

The Shu–Osher problem <cit.> was then simulated, which consists of more complex flow behavior including shock waves and oscillatory small-scale features. The domain was set as Ω = [0,10] with Dirichlet boundary conditions, and the initial conditions were set as 
    
    𝐪(x,0) =  𝐪_l,     x⩽ 1, 
    𝐪_r,    ,
              where  𝐪_l = [  3.857143;  2.629369; 10.333333 ],   𝐪_r = [ 1 + 0.2sin5x;            0;            1 ].

The problem consists of a shock wave interacting with a sinusoidally-perturbed density field which induces instabilities in the flow field that may be erroneously damped by overly dissipative shock capturing schemes. 
    



To explore the efficacy of the proposed approach at both low and high orders, the problem was simulated with both a ℙ_1 and ℙ_3 approximation. A reference solution was computed using a highly-resolved exact Godunov-type solver <cit.>. The density profiles at t=1.8 as computed by a ℙ_1 and ℙ_3 approximation are shown in <ref> in comparison to the reference simulation. It can be seen that the solution was well-behaved around the shock at both approximation orders and various mesh resolutions, with minimal spurious oscillations. Furthermore, good resolution of the oscillatory density field was obtained, with clear qualitative convergence to the exact solution with increasing resolution. As expected, the ℙ_1 was better behaved, with fewer spurious oscillations. However, the ℙ_3 approximation showed better resolution of the oscillatory flow features at lower mesh resolution, indicating that the high-order accuracy of the underlying scheme was sufficiently retained in smooth regions. 



 §.§ Leblanc shock tube

As a final one-dimensional numerical experiment, the Leblanc shock tube was simulated to validate the approach for extreme shocks and flow conditions. This notoriously difficult test case consists of pressure ratios of 𝒪(10^9) and is extremely challenging for numerical schemes to accurately and robustly resolve <cit.>. The problem was solved on the domain Ω = [0,9] with Dirichlet boundary conditions, and the initial conditions were set as

    
    𝐪(x,0) =  𝐪_l,     x⩽ 3, 
    𝐪_r,    ,
              where  𝐪_l = 
            [            1;            0; (γ -1) 10^-1 ],   𝐪_r =  
            [         10^-3;             0; (γ -1) 10^-10 ].

This problem consists of a right-running strong shock and contact discontinuity as well as a left-running expansion wave. Due to the large density jump at the contact discontinuity, which can be smeared by numerical dissipation, it is common for numerical approximations to overpredict the specific internal energy at the contact, which, in turn, overpredicts the shock speed. As such, fine mesh resolution is often required to accurately predict the shock location <cit.>.


   


The problem was solved using a ℙ_3 approximation with various mesh resolutions ranging from N = 100 to N = 1600. The predicted density, velocity, pressure, and specific internal energy fields at t = 6 are shown in <ref>. At all resolutions, the proposed limiting approach effectively stabilized the high-order solution around  very strong discontinuities. At N = 100, the shock speed was noticeably overpredicted, and some minor oscillations were observed around the shock and contact. However, at N=400, a notably accurate solution was obtained despite the relatively coarse resolution, with only a minor overprediction of the shock speed and specific internal energy at the contact. At N=1600, excellent agreement with the exact solution was obtained. 
    
    
    

Since the problem has an exact solution, the convergence behavior of the proposed approach was evaluated for cases with extreme discontinuities. The error of various solution quantities was computed, calculated as the point-mean L^1 norm on the solution nodes normalized by the domain size, including the conserved variables (ρ, m, E) and primitive variables (u, P) as well as non-quasiconcave functionals (e). Furthermore, the physical entropy s = Pρ^-γ was also computed. The error and convergence rates for these quantities are shown in <ref> for the approach with relaxed bounds. It can be seen that the optimal rate of unity was achieved for all quantities. Furthermore, the error and convergence rates for the approach without relaxed bounds are shown in <ref>. Similarly to the Sod shock tube, marginally better error and convergence rates were observed with the relaxed bounds in comparison to the strict bounds. 



 §.§ Sedov blast wave

The extension to two-dimensional problems was first performed with the Sedov blast wave problem <cit.>, which consists of an energy point-source in an ambient gas that drives an outward-running radial explosion. The domain was set as Ω = [-1.2, 1.2]^2 with Dirichlet boundary conditions, and the initial conditions were set to consist of an ambient gas with a constant density ρ_0 = 1 and velocity 𝐔 = 0. We take an identical problem setup to <cit.> by setting the pressure to an ambient value of P_a = 10^-6 everywhere except in the element centered at x = y = 0, where an overpressure value is set in terms of the internal energy e_0 and cell volume V_0 as

    P_0 = 4(γ-1) ρ_0 e_0/V_0.

Here, V_0 = Δ x Δ y = 2.4^2/N is the element volume and ϵ_0 = 0.244816 is the initial specific internal energy. The factor of 4 in the initial pressure is used to account for the fact that the entire domain is being simulated instead of one quadrant.




The problem was solved using a ℙ_2 approximation with N = 257^2 quadrilateral elements. The contours of density as well as the azimuthally-averaged density profile at t=1 are shown in <ref>. We observe the canonical blast wave structure in the solution, with a low density center region and a sharp outward-running shock front. Furthermore, good agreement with the exact solution was obtained, with sharp resolution of the shock and minimal spurious oscillations.



 §.§ Double Mach reflection

The double mach reflection problem of <cit.> was subsequently used to evaluate the proposed approach in the context of strong shocks with small-scale flow features. The problem consists of a Mach 10 shock impinging on a 30 degree ramp, which results in multiple strong shock interactions as well as the onset of small-scale Kelvin–Helmholtz instabilities. The problem was solved on the domain Ω = [0,4] × [0,1], and the initial conditions were set as 

    
    𝐪(𝐱,0) =  𝐪_l,     x < 1/6 + tan (30^∘)y,
    𝐪_r,    ,
              where  𝐪_l = [       8; 7.14471;  -4.125;   116.5 ],   𝐪_r = [ 1.4;   0;   0;   1 ].

Dirichlet boundary conditions were primarily used, with the post-shock state 𝐪_l at the left boundary and the pre-shock state 𝐪_r for the right boundary. At the bottom boundary, the post-shock state 𝐪_l was set for x < 1/6 and a slip adiabatic wall boundary condition was applied for x ⩾1/6. For the top boundary, the exact solution was set, given as
    
    𝐪(𝐱, t)|_y=1 = 𝐪_l,     x⩽ 1/6  + tan (30^∘)y + 10/cos(30^∘)t,
    𝐪_r,    .





The problem was computed using a ℙ_3 approximation with N = 960 × 240 quadrilateral elements. The contours of density at t = 0.2 are shown in <ref>. Excellent resolution of the strong shocks in the flow was observed, with effectively sub-element resolution of discontinuities without noticeable spurious oscillations. Furthermore, the onset of the Kelvin–Helmholtz instabilities rolling up off the contact line was well-resolved by the approach, indicating that the limiter does not erroneously dissipate small-scale flow features.



 §.§ Richtmyer–Meshkov instability




To further evaluate the ability of the approach in resolving small-scale vortical flow features in the vicinity of discontinuities, the Richtmyer–Meshkov instability was simulated. This instability, predicted analytically by <cit.> and shown experimentally by <cit.>, stems from an impulse acting upon a contact discontinuity, generally as a result of a propagating shock wave. The problem was solved on the domain Ω = [0, 0.5] × [0, 4], and the initial conditions were set as
    
    𝐪(𝐱,0) =  𝐪_t,     y ≥ 2 - 1/2cos (2 π x),
    𝐪_m,    1/2≤ y < 2 - 1/2cos (2 π x) ,
    𝐪_b,    ,
              where  𝐪_b = [ 0.25;    0;    0;    1 ],   𝐪_m = [ 1; 0; 0; 1 ],   𝐪_t = [ 8/3;   0;   0; 4.5 ].

These conditions yield an Atwood number of 0.6. Slip adiabatic wall boundary conditions were enforced along the left and right boundaries (mimicking symmetry) and Dirichlet boundary conditions were enforced along the top and bottom boundaries. 

The problem was computed using a ℙ_4 approximation with progressively finer mesh resolution, starting with N = 10 × 80 quadrilateral elements and increasing to N = 160 × 1280 quadrilateral elements. The density contours at t = 2 as computed on the various meshes are shown in <ref>. Good resolution of the discontinuities in the flow was observed even with the higher-order approximation, and the rollup of the vortices was effectively captured by the approach with increasingly smaller-scale vortical structures appearing with higher mesh resolution. These results indicate that the low numerical dissipation and scale-resolving capabilities of the underlying high-order DG scheme are not noticeably degraded with the proposed limiter. 



 §.§ Obstructed radial explosion




The explosion problem of <cit.> was then used to showcase the extension of the approach to unstructured/curved grids. The problem consists of a circular domain of radius 2, with eight circular cylinders of radius 0.3 placed azimuthally equispaced such that their centers were located 1.4 units from the origin. An initial overpressure region in the center drives an outward-running shock and contact discontinuity, which interact with the curved surfaces to form complex shock interactions. The initial conditions were set similarly to a radially-symmetric Sod shock tube problem, with an inner gas region of radius √(0.4) (centered at the origin) set to 𝐪 = [1, 0, 0, 1] and the remaining domain set to 𝐪 = [0.125, 0, 0, 0.1]. Adiabatic slip wall boundary conditions were applied to all surfaces. 



An unstructured mesh was generated with approximately 9.68 × 10^5 second-order triangular elements. The problem was then simulated on this mesh using a ℙ_2 approximation. A Schlieren-type representation of the density gradient norm is shown in <ref> at several time instances. The interactions of the shocks with the contact line, which form Richtmyer–Meshkov instabilities and complex vortical flow, were well captured by the approach. Good resolution of the discontinuities in the flow as well as the small-scale flow features was observed, showcasing the applicability of the approach on unstructured meshes.





 §.§ Mach 800 astrophysical jet

As a final validation study and an evaluation of the approach for very extreme flow conditions, the problem of high-speed astrophysical jets was considered. The test case was introduced  by <cit.> and consists of a Mach 800 jet expanding into an ambient gas. An identical problem setup to <cit.> was used. The half-domain was set as Ω = [0, 0.5] × [0, 1.5], with adiabatic slip wall (symmetry) boundary conditions along the y-axis and Dirichlet boundary conditions on the outer boundaries. The initial conditions were set to 𝐪 = [0.1γ, 0, 0, 1]. At the y=0 boundary, the inlet region was defined on x ⩽ 0.05, with the solution set as 𝐪 = [γ, 0, 800, 1]^T, while the remaining boundary region was set to a Dirichlet boundary conditions.




The problem was solved using a ℙ_3 approximation on a coarse and fine mesh, consisting of N = 200 × 600 and N = 800 × 2400  quadrateral elements, respectively. The contours of density at t = 0.002 as computed on these meshes are shown in <ref>. The strong leading shock wave was robustly resolved without noticeable spurious oscillations, and small-scale flow features around the jet showed excellent resolution. With increasing mesh resolution, smaller-scale flow features were observed around the jet. Given the identical case setup, the results can be compared to the entropy filtering approach in <cit.>. It can be seen that noticeably better resolution of the small-scale flow features was obtained with the proposed approach with similar resolution of the shock structure. 


§ CONCLUSIONS

In this work, we introduced a novel limiting approach for high-order approximations of the Euler equations. Motivated by the observation that high-order approximations of the Boltzmann equation remain robust around discontinuities and that linear particle transport is notably easier to design limiters for, the proposed approach connects the evolution of the macroscopic solution of the Euler equations to the underlying kinetic representation provided by the Boltzmann equation, which allows for extending limiters designed for linear advection directly to the Euler equations. The technique takes an arbitrary stencil of local solution values and, by applying the corresponding local maximum principle to the analytic expression of linear transport of the Boltzmann particle distribution function in the inviscid limit, yields an expression for the admissible region of particle distribution function values, which may be numerically integrated to yield a set of bounds for the density, momentum, and total energy. Several key properties of these bounds are shown, including preservation of the positivity of density/pressure/internal energy. This approach was applied to an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme for a variety of problems in gas dynamics such as extreme shocks, smooth flows, and shock-vortex interactions. It was observed that the approach could robustly resolve strong discontinuities with essentially optimal convergence rates while recovering high-order accuracy in smooth regions without any ad hoc corrections. 

This work presents potential for various extensions and improvements which are in the process of being developed. The approach is not strictly restricted to the limiting method used in this work as it primarily is used to compute solution bounds given a set of discrete solution values. Therefore, it may directly be applied to flux-corrected transport-type techniques which rely on enforcing bounds over a set of Riemann-averaged states <cit.> since it was shown that the Riemann-averaged states preserve these bounds. Furthermore, the kinetic representation provided by the Boltzmann equation underpins many complex macroscopic governing equations, some for which admissible solution bounds are not necessarily well-understood. As such, the extension of this method to these systems, namely the Navier–Stokes equations and multi-physics systems such as magnetohydrodynamics, can result in more physically consistent limiting techniques. Additionally, algorithmic improvements can be made to increase performance, such as applying adaptive quadrature-based integration methods for the distribution functions. 







§ ACKNOWLEDGEMENTS


This work was partially performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE–AC52–07NA27344 and the LLNL-LDRD Program under Project tracking No. 24–ERD–050. Release number LLNL–JRNL–867251–DRAFT.
unsrtnat













 Paper Title: Factorization of a prime matrix in even blocks.




Factorization of a prime matrix]Factorization of a prime matrix in even blocks






Haoming Wang]Haoming Wang
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

wanghm37@mail2.sysu.edu.cn










[2020]Primary 11Axx, 15Axx; Secondary 11A51, 15A69, 15A99.






§ ABSTRACT  In this paper, a matrix is said to be prime if the row and column of this matrix are both prime numbers. We establish various necessary and sufficient conditions for developing matrices into the sum of tensor products of prime matrices. For example, if the diagonal of a matrix blocked evenly are pairwise commutative, it yields such a decomposition. The computational complexity of multiplication of these algorithms is shown to be O(n^5/2). In the section 5, a decomposition is proved to hold if and only if every even natural number greater than 2 is the sum of two prime numbers.


[
    [
    
=====





§ INTRODUCTION

In this note, a prime matrix is a p × q matrix with both p and q prime numbers. Assume m and n are natural numbers with the prime decomposition. The aim of this paper is two-fold:

	
  * Reducing an m× n matrix into the sum of tensor products of prime matrices.
	
  * Reducing an n× n Hermite matrix into the sum of tensor products of prime matrices.



An m × n matrix might be developed into the sum of tensor products of prime matrices, 
In linear algebra, an m × n matrix A has the spectral decomposition in the sense that

    A = ∑_i=1^min{m,n}σ_i u_iv_i^*,

where U = [u_1,u_2,…,u_m] and V = [v_1,v_2, …,v_n] are orthogonal matrices. It's quite hard to compute the σ_i, u_i and v_i when the size of matrix is large. The usual method relies on the computation of eigenvalues and eigenvectors of AA^* and A^*A. If n=m, with the aid of usual elimination algorithm, the computational complexity is O(n^3), which increases the same as a cubic polynomial as the size of matrix gets large. However, this could be remedied provided a tensor form exists by the prime number decomposition theorem. 

If m = p_1^r_1 p_2^r_2… p_k^r_k and n = q_1^s_1 q_2^s_2… q_l^s_l are the prime decomposition of m and n respectively, the m × n matrix A could be blocked according to the prime decomposition.  For example, an m^' p × n^'  q matrix A may be blocked as follows:

    A = [first-row,last-col]
    		q    …    q    
     
    		A_11   …    A_1n^'    p
    ⋮   ⋱   ⋮   ⋮
    
    		A_m^'1   …    A_m^'n^'    p

where each A_ij has p rows and q columns. If the diagonal set of block sub-matrices {A_ii} consists of a pairwise commutative family of matrices of p rows and q columns in the sense that the following equation is satisfied

    AB^* = BA^*,     A^*B = B^*A,

for each A and B in the family of matrices, then there exists a set of orthonormal column vectors {b_i} and {c_j} which diagonalizes each matrix A_ii, say,

    A_ii = σ_ii^k b_kc_k^*,

where the sum is taken over repeated index. Therefore, the m^' p × n^'  q block matrix can be developed into the sum of tensor products

    (1_m^'× m^'⊗ B)^*	 A (1_n^'× n^'⊗ C) =  [first-row,last-col]
    		q    …    q    
    (σ_11^k)    …    A^'_1n^'    p
    ⋮   ⋱   ⋮   ⋮
    
    		A^'_m^'1   …   (σ_m^'n^'^k)     p

where 1_m× n denote the m× n matrix with entries all 1, B = [b_1,b_2,…,b_p], C = [c_1,c_2, …,c_q] and A_ij^'(k,l) = b^*_k A_ij c_l. The symbol ⊗ means the tensor product defined for an m × n matrix A and p × q matrix B: 

    A ⊗ B = 
    	
    		a_11B     a_12B    …    a_1nB 
    
    		a_21B     a_22B    …    a_2nB 
    
    		⋮      ⋱   ⋯
    
    		a_m1B     a_m2B    …    a_mnB

is defined to be a pm× qn block matrix. We will defer the non-square problem to the section 6.

The second aim of this paper is to reduce an n × n Hermite matrix A where A = A^* into the sum of tensor products in the following four type.
  
	
  T_1 The matrix A blocked according to (<ref>) can be developed into the sum
	
    A = ∑_i,j B_ij⊗ C_ij,
 where B_ij and C_ij are n^'× n^' and  q × q matrices respectively.  
	
	
  T_11/2 The matrix A blocked according to (<ref>) can be developed into the sum
	
    A = ∑_i B_i⊗ C_i,
 where B_i and C_i are n^'× n^' and  q × q matrices respectively. 
	
	
  T_2 The matrix A blocked according to (<ref>) can be developed into the sum
	
    A = ∑_i,jα_ij B_i⊗ C_j,
 where α_ij, B_i and C_j are  real numbers, n^'× n^' and  q × q  matrices respectively.
	
	
  T_3 The matrix A blocked according to (<ref>) can be developed into the sum
	
    A =  B ⊗ C,
 where B and C are n^'× n^' and  q × q  matrices respectively.
	

where the required properties for α_ij, B_ij,B_i, B, C_ij,C_i, C will be explained 
in the section 2 explicitly. Here we only illustrate the main idea of this procedure. 

	
  * First, examine whether the diagonal set of block matrix A in (<ref>) is pairwise commutative:
	
    A_iiA_jj = A_jjA_ii,   i,j=1,2,…, n^',

	This requires O(n^5/2) computational complexity.
	
  * Second, examine whether the lower-triangular set of block matrix A in (<ref>) is pairwise commutative. 
	
    A_ijA_i^'j^' = A_i^'j^'A_ij,   i = j,…,n^', i^'=j^',…, n^', j,j^' = 1,…, n^'

	This requires O(n^5/2) computational complexity.
	
  * Third, examine whether the diagonal set of the family of n^'× n^' matrices {B_i} in (<ref>) where B_i(k,l) = b^*_i A_kl c_i is pairwise commutative.
	
    B_iB_j = B_jB_i,   i,j=1,2,…, n^',
 
	This requires O(n^5/2) computational complexity.
	
  * Finally, compute the rank of n^'× n^' real matrix (α_ij)  where α_ij = b_jB_iib_j^* via the similar diagonal form:
	
    P_1P_2… P_n[ 1     O;   1    ;     ⋱  ; O       ]Q_n^-1… Q_2^-1Q_1^-1

	This requires O(n^3/2) computational complexity.

The total computational complexity of these algorithm is therefore O(n^5/2), much less than that of the usual matrix multiplication, which is known to be O(n^3). For this algorithm requires, in the worst case, n^3 multiplications of scalars and n^3-n^2 additions for computing the product of two n × n matrices. This enables us to utilize the above algorithm as a first examination so that if passed, it requires at most 
n^5/2 + n^2 - n^3/2 multiplications of scalars and

n^5/2 - n^3/2 additions for multiplying two n× n matrices. These methods are introduced in section 4 and their algorithms are illustrated in Appendix B.

However, we are only interested in the case n is divisible, that is, n contains at least two prime factors that may not be distinct since if the order n of the square matrix A is a prime number, this problem has only trivial solution,. According to the fundamental theorem of arithmetic, every natural number has a unique decomposition as the product of prime numbers. The proof of arithmetic fundamental theorem is inductive so we use the induction to derive our desired decomposition too. We begin to start with a decomposable number n =a b where a and b are two natural numbers larger than 2. 
 


§ THE CASE N = AB

Here we give some assumptions about the reduction of an square matrix A of order n=ab into the sum of tensor products. Familiar with them is helpful in our coming discussion. Let's look at the Hermite case.
 Let A be an Hermite matrix of order ab. Block A as
	
	
    A = [ A_11 A_12    … A_1a; A_21 A_22    … A_2a;    ⋮         ⋱    ⋮; A_a1 A_a2    … A_aa ]

	where each A_ij is an Hermite matrix of order b. We assume A is decomposable of the type
	
		
  T_1 with respect to some column bases if there exist orthonormal column vectors {c_j} of order b such that A can be developed into the sum
		
    A = ∑_i,j B_ij⊗ C_ij

		where C_ij = c_i c_j^* and B_ij = (c_i^* A_kl c_j). In this case, B_ij(k,k)  = β_i(k)δ_ij. In a word, 
		B_ij vanishing on diagonal if i≠ j.
		
  T_11/2 with respect to some mutually orthogonal row coefficients if it is T_1 decomposable and additionally, B_ij = B_iiδ_ij, i.e. 
		B_ij vanishing if i≠ j.
		
  T_2 with respect to some row-column bases if it is T_1 decomposable and there exist further orthonormal column vectors {b_i} of order a such that A can be developed into the sum
		
    A = ∑_i,jα_ij B_i⊗ C_j,

		where B_i = b_ib_i^* and C_j = c_jc_j^*.
		
  T_3 totally decomposable with respect to some row-column bases if it is T_2 decomposable and additionally, there exist β_i, γ_j such that α_ij = β_iγ_j and
		
    A =  B ⊗ C

		where B = ∑_iβ_i b_i b_i^* and C = ∑_jγ_j c_jc_j^*.
	



As for the non-Hermite case, we will assume A has n distinct eigenvalues. In this setting, replace the word orthogonal with independent and rephrase word for word in the above paragraph. We leave this work to the reader.




§ THE CANONICAL DIAGONAL FORM



We define a binary operation between an m × n matrix A and a k× l matrix B when n and k has common divisor a as follows: block A = [A_1, A_2, …, A_a] and B = [B^*_1, B^*_2, …, B^*_a]^* according to the partition of n = as columns of A and k = at rows of B and m× s sub-matrices A_i and t × l sub-matrices B_i, i=1,2,…,a. 

    A ⊗_a,a B = ∑_i=1^a A_i⊗ B_i

The result A ⊗_a,aB is a tm× ls matrix, dividing the factor of km× ln both by a. 

Besides, we define a unary operation on an m × n matrix A when m and n has both common divisor a: block A according to m=ar and n=as

    [ A_11 A_12    … A_1s; A_21 A_22    … A_2s;    ⋮         ⋱    ⋮; A_r1 A_r2    … A_rs ]

where A_ijs are a × a matrices and then define

    a,a⊕ A = ∑_i,j A_ij


Now return to our topic of decomposing an Hermite matrix of order n =ab. We will denote the block matrix C = (C_ij) according to C_ij as described in section 2 if it appears.
		
	


Since an Hermite matrix may have a canonical diagonal decomposition, we tabulate some equivalent conditions for them to hold explicitly in Table (<ref>):
ccl
	Comparison of T_1,T_11/2,T_2 and T_3.

	
	
		T_1     ⇔      A = ∑_i,j B_ij⊗ C_ij where C_ij=c_ic_j^*.

		     ⇔       C_ij = c_ic_j^* consists of common eigenvectors c_i of A_kk, independent of 

		        k, corresponding to eigenvalues B_ii(k,k).
	
		T_11/2     ⇔     A = ∑_i B_i⊗ C_i where C_i = c_ic_i^*.

		     ⇔       C_i = c_ic_i^* consists of common eigenvectors c_i of A_kl, independent of 

		        k,l, corresponding to eigenvalues B_i(k,l).
	
		T_2     ⇔     A = ∑_i,jα_ij B_i⊗ C_j where B_i = b_ib_i^* and C_j = c_jc_j^*

		    ⇔     B_i = b_ib_i^*, C_j = c_jc_j^* consists of eigenvectors b_i⊗ c_j of A,

		        corresponding to eigenvalues α_ij.

		T_3     ⇔     A= B ⊗ C and B = ∑_iβ_i b_ib_i^*, C = ∑_jγ_j c_j c^*_j.
 
		    ⇔     b_i⊗ c_j are eigenvectors of A, corresponding to eigenvalues β_iγ_j.

		
		


From Table (<ref>), there are some obvious deductions.

 Let A be an n × n Hermite matrix blocked in the form (<ref>).
	
		
  1. T_3⇒ T_2⇒ T_11/2⇒ T_1.
		
  2. The converses T_1⇒ T_11/2 and T_1⇒ T_2 and T_2⇒ T3 are already made clear in the definition.
		
  3. If the diagonal set {A_ii} is pairwise commutative, then T_1 holds.
		
  4. If the lower-triangular set {A_ij} is pairwise commutative, then T_11/2 holds. 
		
  5. T_11/2 + b_j is the eigenvector of B_i, independent of i, corresponding to eigenvalue α_ij ⇒ T_2.
		
  6. T_2 + α_ij=β_iγ_j⇒ T_3;
		
  7. T_11/2 + B_jj = ω_j B where ω_j is constant and B Hermite matrix ⇒ T_3. 
	



	We only prove 3 and 4. This is because any permutation σ_a of {1,2,…,n} can be decomposed into the product of the replacement σ_n = τ_1τ_2…τ_m and commutativity means simultaneous diagonalization.





§ THE COMPUTATIONAL COMPLEXITY


Take the matrix multiplication for example, This algorithm requires, in the worst case, n^3 multiplications of scalars and n^3-n^2 additions for computing the product of two square n × n matrices. Its computational complexity is therefore O(n^3). If the n^2× n^2 matrix blocked evenly in the form (<ref>), we execute the following algorithm:




	
  * First, examine whether the diagonal set of block matrix A in 
	
    A_11    A_12   …    A_1n
    
    		A_21    A_22   …    A_2n
    ⋮      ⋱   ⋮
    
    		A_n1    A_n2   …    A_nn (1-|2) |- (2-|3) |- (3-|4) |- (4-|5);
    		 (2-|1) -| (3-|2) -| (4-|3) -| (5-|4);

	is pairwise commutative:
	
    A_iiA_jj = A_jjA_ii,   i,j=1,2,…, n,

	This step requires, in the worst case 2n  2· n^3 = n^5 - n^4 multiplication of scalars and 2n  2· (n^3 - n^2) = n^5 - 2n^4 +n^3 additions.
	
  * Second, examine whether the lower-triangular set of block matrix A in 
	
    A_11    A_12   …    A_1n
     A_21    A_22   …    A_2n
    ⋮      ⋱   ⋮
     A_n1    A_n2   …    A_nn (1-|2) |- (2-|3) |- (3-|4) |- (4-|5);

	is pairwise commutative. 
	
    A_ijA_i^'j^' = A_i^'j^'A_ij,   i = j,…,n^', i^'=j^',…, n^', j,j^' = 1,…, n

	This step requires, in the worst case 2n^2 2· n = n^5 - n^3 multiplication of scalars and 2n^2 2· (n-1) = n^5 - n^4 - n^3 +n^2 additions.
	
  * Third, examine whether the diagonal set of the family of n × n matrices {B_ii} in 
	
    B_11   …       O
        B_22      ⋮
    ⋮      ⋱   
    
    		O       …    B_nn (1-|2) |- (2-|3) |- (3-|4) |- (4-|5);
    		 (2-|1) -| (3-|2) -| (4-|3) -| (5-|4);

	where B_ii(k,l) = b^*_i A_kl c_i is pairwise commutative.
	
    B_iiB_jj = B_jjB_ii,   i,j=1,2,…, n,
 
	This step requires, in the worst case 2n  2· n^3 = n^5 - n^4 multiplication of scalars and 2n  2· (n^3-1^2) = n^5 - 2n^4 + n^3 additions.
	
  * Finally, examine whether the rank of n  × n real matrix (α_ij) is 1
	
    [  α_11 (= β_1γ_1)  α_12 (= β_1γ_2)                …  α_1n (= β_1γ_n); α_21 (=  β_2γ_1)  α_22 (= β_2γ_2)                …  α_2n (= β_2γ_1);                ⋮                                 ⋱                ⋮;  α_n1 (= β_nγ_1)  α_n2 (= β_nγ_2)                …  α_nn (= β_nγ_n) ]

	where α_ij = b_jB_iib_j^*. This could be done via the similar diagonal form:
	
    P_1P_2… P_n^'[ 1     O;   1    ;     ⋱  ; O       ]Q_n^'^-1… Q_2^-1Q_1^-1

	This step requires, in the worst case 2n  2· n = n^3 - n^2 multiplication of scalars and 2n  2· (n-1) = n^3 - 2n^2 + n additions.





[htbp]ccc
	Computing multiplication of two square n^2× n^2 matrices.

	
	
		    Multiplications     Additions 

		T_1     2A_2^n· n^3 + n^n     n^5 - 2n^4 +n^3

		T_11/2     2A^n^2_2· n + n^4     n^5 - n^4 - n^3 +n^2

		T_2     2A^n^2_2· n + n^3     2n^4 + n^3

		T_3     2A^n_2· n + n^3     n^3 - 2n^2 + n

		
	

The computational complexity is hence not surprisingly O(n^5/2), much less than usual multiplication.







 If A is an Hermite matrix of decomposable order n = p_1^r_1 p_2^r_2…  p_s^r_s, then it has decomposable type 4^r_1 + … + r_s, according to Definition 2.1. The computational complexity of this algorithm is also O(n^5/2).	




§ THE SEMI-PRIME AND QAUSI-PRIME MATRIX



 A semi-prime matrix is an m× n matrix where the minimum of m and n is a prime number.



Recall an m × n matrix A is said to have a left inverse if there exists an m× m matrix B such that BA = c|c
	I_m    O and a right inverse if there exists an n × n matrix C such that AC = c
	I_n

	O. It is said be invertible if both inverses exist and quasi-invertible if there exists only one-side inverse. 

 A matrix is said to be quasi-prime if it is the product of two quasi-invertible semi-prime matrices.


Two n × n matrices A and B are said to be orthogonal if AB^* = A^*B = O.
 

	Every n × n matrix where n is an even natural number greater than 2 is the sum of two orthogonal quasi-prime matrices. 


There are unique orthogonal matrices U and V for any quasi-prime matrix A such that U^*AV is diagonal, and the number of whose non-zero entries is a prime number.

 Suppose there are two A = U_1V_1 = U_2V_2 decomposition of the m × n matrix A given by Definition 5.2, then by the uniqueness of spectral decomposition, the U_1,U_2 and V_1, V_2 are unitarily equivalent matrices.
	If the m × n matrix A has the spectral decomposition, 
	
    A = ∑_i=1^min{m,n}σ^2_i u_iv_i^*

	where U = [u_1,u_2,…,u_m] and V = [v_1,v_2, …,v_n] are orthogonal matrices, it suffices to prove that the number of non-zero σ_i is a prime number. This follows from the unique decomposition of semi-prime matrix.


 Every quasi-invertible semi-prime matrix is quasi-prime.
	

	Suppose m is the minimum of m and n. An m × n matrix that is semi-prime can be expressed as the product A = I_mA. 


	The product of two m × n and n × k quasi-prime matrices is a m × k quasi-prime matrix.





	The conjecture 5.3 holds true if and only if every even natural number greater than 2 is the sum of two prime numbers.

 The necessity is easily seen from the preceding lemma and orthogonal relation.
	It suffices to prove the sufficiency. If an even natural number n larger than 2 can be developed as the sum of two prime numbers p and q, block the n × n matrix A as follows
	
    c|c[first-row,last-col]
    		    p     q    
    
    			A_11       p
        O     	q
    		
    		+  c|c[first-row,last-col]
    			p     q    
        A_12    p
    
    			O        q
    		  + … + c|c[first-row,last-col]
    		 	p     q    
    
    		 	 O        p 
        A_22    q

	and choose two n × n orthogonal matrices U = [u_1, u_2, …, u_n] and V = [v_1, v_2, …, v_n] such that
	
    U^* A V = c|c[first-row,last-col]
    			p     q    
    Λ_1    O     p
    
    			O    Λ_2    q
	
	where Λ_1 = (λ_1,…,λ_p) and Λ_2 = (λ_p+1,…,λ_n). Then block U and V as 
	
    U = c|c[first-row,last-col]
    			p     q    
    
    			U_1    U_2    n 
    		
    
    		V = c|c[first-row,last-col]
    			p     q    
    
    			V_1    V_2    n

	The spectral decomposition yields
	
    A = U_1Λ_1 V^*_1 + U_2Λ_2 V^*_2

	Since U_1, V_1, U_2, V_2 and Λ_1, Λ_2 are n× p, n× p, n× q, n× q semi-prime matrices and p× p, q × q invertible prime matrices, it follows from V_1^*V_2 = U_1^*U_2 = O that the conjecture holds true. 
	

Two m× n matrices A and B are said to be congruent if there are two invertible square matrices U and V of order m and n respectively such that UA = BV.


	For every natural number k, there are infinitely many prime numbers n and (n -2k) × (n) semi-prime matrices that are not congruent to each other. In particular, the semi-prime matrices can be chosen quasi-invertible.


and a weak form of above


	For every natural number k, there are infinitely many prime numbers n and (n-2k) × n quasi-prime matrices that are not congruent to each other.


	The conjecture 5.8 holds true if and only if there are infinitely many twin primes with gaps equal k; The conjecture 5.9 holds true if and only if there are infinitely many twin primes with gaps less or equal than k. 





















	






§ THE NON-SQUARE PROBLEM


The non-square problem arises in real-life applications since in this case the matrix we take into consideration is non-Hermite. It's usually impossible to find a diagonal form but we will try another method to reduce the non-square matrix. For example, if m = p_1^r_1 p_2^r_2… p_k^r_k and n = q_1^s_1 q_2^s_2… q_l^s_l are the prime decomposition of m and n, the m × n matrix A could be blocked as an m^' p_1× n^'  q_1 matrix, say,

    A = [first-row,last-col]
    		q_1   …    q_1   
     
    		A_11   …    A_1n^'    p_1
    ⋮   ⋱   ⋮   ⋮
    
    		A_m^'1   …    A_m^'n^'    p_1

where each A_ij has p_1 rows and q_1 columns. If the diagonal set of block sub-matrices {A_ii} consists of a pairwise commutative family matrices of p_1 rows and q_1 columns in the sense that this relation is fulfilled:

    AB^* = BA^*,     A^*B = B^*A,

for each A and B in the family of matrices, then there exists two p_1× p_1 and q_1× q_1 orthogonal matrices C^L= [c^L_1, c^L_2, …,c^L_p_1] and C^R= [c^R_1, c^R_2, …,c^R_q_1] which diagonalizes each matrix A_ii, say,

    A_ii = σ_ii^k c^L_kc_k^R*,

where the sum is taken over repeated index. Therefore, the m^' p_1× n^'  q_1 block matrix can be developed into the sum of tensor products

    A = ∑_i,j B_ij⊗ C_ij

where C_ij = c^L_i c^R*_j, B_ij = (c^L*_i A_kl c^R_j) and B_ij(k,k)  = β_i(k)δ_ij. In a word, B_ij vanishing on diagonal if i≠ j. Thus the m × n matrix is said to be decomposable of the type


	
  T_1 with respect to some column bases if there exist orthonormal column vectors {c^L_j} and {c^R_j^'} of order p_1 and q_1 respectively such that A can be developed into the sum
	
    A = ∑_i,j B_ij⊗ C_ij

	where C_ij = c^L_i c^R*_j, B_ij = (c_i^L* A_kl c_j^R). In this case, B_ij(k,k)  = β_i(k)δ_ij. In a word, B_ij vanishing on diagonal if i≠ j.
	
  T_11/2 with respect to some mutually orthogonal row coefficients if it is T_1 decomposable and additionally, B_ij = B_iiδ_ij, i.e. B_ij vanishing if i≠ j.
	
  T_2 with respect to some row-column bases if it is T_1 decomposable and there exist further orthonormal column vectors {b^L_i} and {b^R_i^'} of order m^' and n^' respectively such that A can be developed into the sum
	
    A = ∑_i,jα_ij B_i⊗ C_j,

	where B_i = b^L_ib_i^R* and C_j = c_j^Lc_j^R*.
	
  T_3 totally decomposable with respect to some row-column bases if it is T_2 decomposable and additionally, there exist β_i, γ_j such that α_ij = β_iγ_j and
	
    A =  B ⊗ C

	where B = ∑_iβ_i b^L_i b_i^R* and C = ∑_jγ_j c_j^Lc_j^R*.

Equivalent conditions for these assumptions are similar to Table <ref> and Theorem 3.1 provided the commutativity relation replaced by (<ref>).



§ CONCLUSION


There are still possible representations of matrix B_ij(k,l) in T_1. For example, B^'_ij = (c_k^* A_ij c_l). We could develop the matrix A into the sum

    A = ∑_i,j B^'_..(i,j) ⊗ C_ij

and the canonical diagonal form varies according to the representation

		




However, this dosen't indicate explicitly B_ij as the projection of A onto C_ij so we still adopt the former B_ij as our assumption. 7555316FOMATATI2022180Lathauwer2008doi:10.1137/07070111XzbMATH04180669zbMATH07021224





amsplain







§ EXAMPLES OF 4× 4 MATRICES


This problem originates from decomposition of large covariance matrix with hidden structure in spatio-temporal statistics, wireless communication and mathematical physics etc. For example, the white noise problem states that there is a random signal, viewed as a stochastic process on the Minkovski space or a general manifold with 

    ⟨η(𝐱,t)|η(𝐱^',t^') ⟩ = δ(𝐱 - 𝐱^')δ(t - t^')

where δ is the Dirac function. The white noise has the identity matrix as its covariance which can be reduced into the tensor product of two factor matrix

    I_qp = I_p⊗ I_q

where I denotes the identity matrix. This demonstration holds true depends on the hidden structure between 𝐱 and t. In fact, the white noise of form (<ref>) has the matrix normal density with respect to two m × m and n × n Hermite matrices A and B and a m× n matrix M 

    (2π)^-mn/2|A |^-n/2|B |^-m/2etr(-1/2A ^-1(X - M )B ^-1(X - M)^*)∏_i,jx_ij

where etr(A) means exp ( A) and |A| the determinant of A, if and only if, vec(X) viewed as an mn × 1 vector pilling up the columns of the m × n matrix stacked on top of one another, is the multivariate normal distribution N_mn(vec(M), A ⊗ B). The soul problem is to decide whether the covariance of vec(X), an mn× mn Hermite positive definite matrix, is of the form A ⊗ B for some m× m and n × n Hermite matrices A and B. However, if the order n is a product of two divisors a and b, it's still impossible for all n × n matrices to possess a factorization of Kronecker product of two a× a matrix A and b × b matrix B. Let's look at this

 The following square matrix of order 4
	
    [   1 0.5 0.2 0.1; 0.5   1 0.5 0.2; 0.2 0.5   1 0.5; 0.1 0.2 0.5   1 ]

	is not a Kronecker product of two 2 × 2 matrices.
 
Therefore, it sounds absurd to presume such a tensor form since it cannot always describe all situations behind the hidden structure of mathematical models. This initiates our study of weak assumptions for matrices. Although not all square matrices of divisible order n = ab are of the form A × B for some a× a matrix A and b × b matrix B, it's still possible for a square matrix to have a reduced form of the sum of Kronecker products. The following three examples show how this could be done.



[Partial diagonalization]
	
    [   1 0.5 0.2 0.1; 0.5   1 0.5 0.2; 0.2 0.5   1 0.5; 0.1 0.2 0.5   1 ]   =
    		[  0.5 -0.1; -0.1  0.5 ]⊗[  0.5 -0.5; -0.5  0.5 ] + 
    		[ 0.5 0.2; 0.2 1.5 ]⊗[ -0.5 -0.5;  0.5  0.5 ]
        +  
    		[  1.5 -0.2; -0.2  0.5 ]⊗[ -0.5  0.5; -0.5  0.5 ] 
    		+ 
    		[ 1.5 0.5; 0.5 1.5 ]⊗[ 0.5 0.5; 0.5 0.5 ]
    
       = A_11⊗ B_11 + A_12⊗ B_12 + A_21⊗ B_21 + A_22⊗ B_22

	where (B_11) = (B_12) = (B_21) = (B_22) = 1.



[Partially orthogonal diagonalization]
	
    [   1 0.5 0.2 0.3; 0.5   1 0.3 0.2; 0.2 0.3   1 0.5; 0.3 0.2 0.5   1 ]   =
    		[  0.5 -0.1; -0.1  0.5 ]⊗[  0.5 -0.5; -0.5  0.5 ] 
    		+ 
    		[ 1.5 0.5; 0.5 1.5 ]⊗[ 0.5 0.5; 0.5 0.5 ]
       = A_1⊗ B_1 + A_2⊗ B_2

	where (B_1) = (B_2) = 1 and B_1 and B_2 are commutative.



[Totally orthogonal diagonalization]
	
    [    1 0.15 0.15  0.5; 0.15    1  0.5 0.15; 0.15  0.5    1 0.15;  0.5 0.15 0.15    1 ]   =
    		6/5[  0.5 -0.5; -0.5  0.5 ]⊗[  0.5 -0.5; -0.5  0.5 ] 
    		+ 1/2[  0.5 -0.5; -0.5  0.5 ]⊗[ 0.5 0.5; 0.5 0.5 ]
        + 1/2[ 0.5 0.5; 0.5 0.5 ]⊗[  0.5 -0.5; -0.5  0.5 ] 
    		+ 6/5[ 0.5 0.5; 0.5 0.5 ]⊗[ 0.5 0.5; 0.5 0.5 ]
    
       = c_11 A_1⊗ B_1 + c_12 A_1⊗ B_2 + c_21 A_2⊗ B_1 + c_22 A_2⊗ B_2

	where (A_1) = (A_2) = (B_1) = (B_2) = 1, A_1 and A_2 are commutative and B_1 and B_2 are commutative.






§ FACTORIZATION ALGORITHM



















 Paper Title: Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing $B^T φ(A) B$.

 




Monotonicity, bounds and extrapolation of Block-Gauss and Gauss-Radau quadrature for computing B^T ϕ(A) B
    Jörn ZimmerlingUppsala Universitet, Sweden, .
Vladimir DruskinWorcester Polytechnic Institute, USA
Valeria SimonciniDipartimento di Matematica and (AM)^2, Alma Mater Studiorum - Università di Bologna, 40126 Bologna, and IMATI-CNR, Pavia, Italy, 


    August 5, 2024
===========================================================================================================================================================================================================================================================



§ ABSTRACT 
In this paper, we explore quadratures for the evaluation of B^T ϕ(A) B where A is a symmetric nonnegative-definite matrix in ℝ^n × n, B is a tall matrix in ℝ^n × p, and ϕ(·) represents a matrix function that is regular enough in the neighborhood of A's spectrum, e.g., a Stieltjes or exponential function. These formulations, for example, commonly arise in the computation of multiple-input multiple-output (MIMO) transfer functions for diffusion PDEs.

We propose an approximation scheme for B^T ϕ(A) B leveraging the block Lanczos algorithm and its equivalent representation through Stieltjes matrix continued fractions. We extend the notion of Gauss-Radau quadrature to the block case, facilitating the derivation of easily computable error bounds. For problems stemming from the discretization of self-adjoint operators with a continuous spectrum, we obtain sharp estimates grounded in potential theory for Padé approximations and justify extrapolation algorithms at no added computational cost. The obtained results are illustrated on large-scale examples of 2D diffusion and 3D Maxwell's equations as well as a graph from the SNAP repository. We also present promising experimental results on convergence acceleration via random enrichment of the initial block B. 









§ INTRODUCTION



 §.§ Problem statement

Let A=A^T∈^n× n be a symmetric positive definite (s.p.d) matrix and let B be a tall matrix with B∈^n× p, with p≪ n. We refer to p as the block size. We want to approximate 

(s) =B^Tϕ(A,s)B
via approximations of ϕ(A,s)B on a block Krylov subspace 

    𝒦_m(A,B)= blkspan{B, AB, A^2 B , … A^m-1 B }

where blkspan means that the whole space range([B, AB, A^2 B , … A^m-1 B]) is generated.

In the literature various functions ϕ are considered to derive bounds for the scalar case, for instance strictly completely monotonic functions <cit.> or functions with sufficient derivatives on the spectral interval of A <cit.>. Here, we mainly consider the simpler problem ϕ(A,s)=(A+sI)^-1 with s∈_+,
which can be generalized to Stieltjes functions. Our numerical experiments will also 
illustrate the applicability of our results for exponentials and resolvents 
with s∈∖_-. 
To simplify notation, without loss of generality, we assume that the matrix B has orthonormal columns.


In this paper, we extend the notion of Gauss-Radau quadrature to the block case allowing us to obtain two-sided estimates for the solution at the cost of the standard block-Lanczos algorithm on the block-Krylov subspace 𝒦_m(A,B). 


We focus on applications stemming from diffusive electromagnetic problems in an unbounded domain where we show that averaging Gauss-Radau and Gauss quadratures leads to cost-free extrapolation. This
significantly improves the accuracy of the standard Krylov projection approximation,
even though matrix-function approximation by Krylov projection is near optimal in global norms <cit.>.


Monotonicity results for the ordinary Lanczos algorithm were obtained for the exponential function in <cit.> and for more general classes of functions in <cit.>. In this work, we obtain monotonicity results for Gauss and Gauss-Radau quadratures calculated via the block-Lanczos algorithm. Using pairs of Gauss and Gauss-Radau quadratures has for instance been explored for functions of adjacency matrices in <cit.>. For the related conjugate gradient algorithm Gauss-Radau based upper bounds for the scalar case have been developed in <cit.>. For rational functions of matrices error bounds for the classical Lanczos algorithms have been obtained in <cit.>. A connection of block-Lanczos algorithm and matrix continued fraction similar to the one used in this work was described in the context of quantum physics for MIMO transfer function approximations in <cit.>.



The remainder of the article is organized as follows: we review the basic properties of the block-Lanczos algorithm in section <ref>, and we show that the resolvent of the Lanczos block-tridiagonal matrix leads to a Stieltjes-matrix continued fractions in section <ref>. This is used in section <ref> to derive a two-sided error bound and to define the notion of Gauss-Radau quadrature for the block case. An extrapolation formula based on averaged Gauss-Radau and Gauss quadrature is presented in <ref> for the block case. Last, numerical examples are presented in section <ref>.






 §.§ Notation

Given two square symmetric matrices G_1, G_2 we use the notation
G_1<G_2 to mean that
the matrix G_2-G_1 is positive definite. 
A sequence of matrices {G_m}_m≥ 0 is said to be monotonically
increasing (resp. decreasing) if G_m < G_m+1 (resp. G_m+1<G_m)
for all m.
Positive-definite p × p matrices are denoted by Greek 
letters Α,Β, and matrix-valued functions by calligraphic capital letters such as (s) or (s). 
Last, for Α, Β∈ℝ^p× p and Β is nonsingular,
 we use the notation Α/Β:= ΑΒ^-1 (right inversion).
The matrix E_k ∈ℝ^mp× p has zero elements except for the
p× p identity matrix in the k-th block, E_k =[0,…, 0,I, …, 0]^T. 





§ BLOCK GAUSS QUADRATURES VIA THE BLOCK LANCZOS ALGORITHM


For scalar (s) , i.e. p=1, the approximation of (s)=B^T (A+ sI)^-1B by means of
_m(s)=E_1^T (T_m+ sI)^-1E_1, where T_m is a specifically designed s.p.d. tridiagonal
matrix, can be classically interpreted as a quadrature formula for integrals <cit.>. The block setting with p>1 and T_m a specifically designed s.p.d. block-tridiagonal matrix allows for an analogous block quadrature interpretation. 

Consider the eigendecompositions A=ZΛ Z^T with eigenpairs (λ_i,z_i) and T_m=Z̃_m Λ̃_m Z̃_m^T, T_m ∈ℝ^mp× mp with eigenpairs (λ̃_i,z̃_i) . Then we can rewrite (s) as an integral 
B^T (A+ sI)^-1B = ∑_i=1^N (z_i^T B)^T(z_i^T B)/λ_i+s = ∫_0^∞1/λ+s  dμ(λ).
with symmetric, matrix valued spectral measure 
    μ(λ)= ∑_k=1^i(z_k^T B)^T(z_k^T B) H(λ-λ_i),
 with H(·) the Heaviside step function. The approximation _m(s) of this integral can be written as
E_1^T (T_m+ sI)^-1E_1 = ∑_i=1^mp (z̃_i^T E_1)^T(z̃_i^T E_1)/λ̃_i+s 
which can be intepreted as a Gauss-type quadrature rule ∑_i w_i 1/x_i+s with weights w_i=(z̃_i^T E_1)^T(z̃_i^T E_1) and nodes x_i=λ̃_i. 

In the scalar case p=1, the measure μ and quadrature weights are scalar-valued and in Gauss-Radau quadrature one prescribes one of the quadrature nodes at x_1=0. The extension of the Gauss-Radau quadrature to the block case that we propose is to prescribe the first quadrature point at x_1=0 with quadrature weight w_i of rank p.




 

.62
 






To obtain a (block)-symmetric T_m and a quadrature approximation the block Lanczos algorithm (Algoritm <ref>) is used. Let us assume that we can perform m steps, with mp ≤ n, of 
the block Lanczos iteration without breakdowns 
or deflation <cit.>. As a result, 
the orthonormal block vectors Q_i∈ℝ^n × p form 
the matrix Q_m=[Q_1, …, Q_m]∈ℝ^n× mp, whose columns
contain an orthonormal basis for the block Krylov subspace
𝒦_m(A,B)= blkspan{B, AB, A^2 B , … A^m-1 B }.
The Lanczos iteration can then be compactly written as
AQ_m=Q_m T_m + Q_m+1 Β_m+1E_m^T, 
where T_m is the symmetric positive definite block tridiagonal matrix 

T_m=
[     Α_1   Β_2^T                                        ;     Β_2     Α_2   Β_3^T                                ;               ⋱       ⋱       ⋱                        ;                     Β_i     Α_i Β_i+1^T                ;                               ⋱       ⋱       ⋱        ;                                   Β_m-1   Α_m-1   Β_m^T;                                             Β_m     Α_m;         ] ,
and Α_i, Β_i ∈ℝ^p × p are the block coefficients
in Algorithm <ref>. 


Following standard practice,
the matrix Β_i of the orthogonalization coefficients
is obtained by the QR decomposition <cit.> and it is
therefore upper triangular.
For our purposes, since in Algorithm <ref>
the matrix Β_i is only required to satisfy
W^TW=Β_i^TΒ_i>0, in principle Β_i could be taken to
be symmetric and positive definite, e.g., via the polar decomposition of W.
This option is normally used in applications in inverse problems
, e.g., <cit.>. 



 §.§ Block-Gauss quadrature



Using the Lanczos decomposition, (s) can be approximated as

 (s)≈_m(s)= E_1^Tϕ(T_m,s)E_1.
This approximation is known as a block Gauss quadrature rule.
In the following, we show that _m is monotonically increasing with 
m, i.e. _m< _m+1<(s) and thereby converges monotonically. Analogous results for the non-block case (i.e., p=1) for a larger class of functions were obtained in <cit.>. 
We also introduce a computable upper-bound matrix
_m that monotonically decays as iterations increase, that is _m>_m+1>>_m. These two-sided bounds will then provide an error measure and an extrapolation formula. The quantity _m may be interpreted as an extension to p>1 of the approximation via a Gauss-Radau quadrature rule introduced in <cit.>.
 

 
	 The simple Lanczos algorithm using only three-term recursions without re-orthogonalization 
 is known to be unstable due to computer round-offs. The instability is manifested by the loss of orthogonality of the Lanczos vectors and the appearance of spurious copies of the Lanczos eigenvalues. However, this instability only dampens the spectral adaptation of the approximation of actions of regular enough matrix functions and does not affect the converged approximation <cit.>. 
	Likewise, even though the spectral nodes and weights of the Gaussian quadratures computed via the simple Lanczos algorithm can be affected by round-off, the converged quadrature rule is not <cit.>. Moreover, compared to the actions of matrix functions the quadrature convergence is less affected by the loss orthogonality of Lanczos vectors, because these vectors are not explicitly used in the computation of the Gaussian quadratures. We assume that the same reasoning applies to the block extensions as well to the Gauss-Radau quadrature, which we observe in our numerical experiments. 
	






§ GAUSS QUADRATURE AND THE MATRIX S-FRACTION
 


We represent the block-tridiagonal Lanczos matrix T_m (see equation (<ref>)) in a 
block LDL^T decomposition.


	Let T_m be as defined in (<ref>). Then
	there exist 
_i∈^p× p, _1=I_p and _i∈^p× p all full rank such that

    T_m :=( K_m^-1)^T J_mΓ^-1_m J_m^T  K_m^-1
 



where

    J_m^T = 
    [  I_p -I_p          ;         ⋱    ⋱     ;              ⋱ -I_p;                 I_p ]∈^pm× pm,
    	[                  K_m   = blkdiag(_1,…,_m);                  Γ_m = blkdiag(_1,…,_m) , ]

and
Α_1=_1^T_1^-1_1=_1^-1, 
Α_i=_i^T(_i-1^-1+_i^-1)_i and 
Β_i=_i-1^T _i-1^-1_i for i=2,…,m. 


	

The block LDL^T decomposition for positive definite, tridiagonal matrices consists out of a block-bidiagonal L with identity as its diagonal block. and s.p.d. D. The parametrization in equation (<ref>) is a general parameterization of such a LDL^T decomposition with L=( K_m^-1)^T J_m ( K_m)^T and D=( K_m^-1)^T Γ^-1_m  ( K_m^-1).

The p× p matrices must be of full rank, since they are Cholesky factors of the 
s.p.d. Lanczos matrix (<ref>) are of full rank. Further by the same reasoning, the (block)-diagonal factor Γ_m is s.p.d. since D is s.p.d. .





 We can extract the matrices _i's and _i's directly during the block Lanczos recurrences using the coefficients Α_i's and Β_i>0's. This is reported in Algorithm <ref> which in a different parameterization was derived in Appendix C of <cit.>.
From their construction, we obtain that _i, _i are full rank, as long as no deflation occurs
in the block Lanczos recurrence, as Β_i^TΒ_i>0 for i=2,…,m, i.e. nonsingularity of all coefficients Α_i,Β_i ensures that 
Algorithm <ref> runs without breakdowns. This algorithm is also related to the matrix continued fraction construction via block Lanczos algorithm used for computation of quantum scattering problems <cit.>.
 
 

.5
 
 
 

 
 0.1in





To write _m as a matricial continued-S-fractions we first convert T_m to pencil form using 
the factors introduced in (<ref>). To this end, we introduce the matrices _j

_j =_j^T _j,   j=1,…,m.
The matrices _j and _j are known as the Stieltjes parameters <cit.> and they are both s.p.d. if block-Lanczos runs without breakdown.

Let Z_m:= J_m Γ_m^-1J_m^T and Γ_m= blkdiag(_1,…,_m).
Then, due to the initial condition _1=I_p, the function _m can be rewritten using the pencil
form (Z_m,Γ_m), that is

_m(s)=E_1^T(T_m+sI)^-1E_1= E_1^T( Z_m+s Γ_m)^-1E_1.


Next, we show that _m is a matricial Stieltjes continued fraction (S-fraction) that can be written as

	_m(s)= 1s_1+ 1_1 + 1s_2 +1 ⋱1s _m + 1 _m  ,
where the basic building block of these S-fractions is given by the recursion

	_i(s)=1s_i + 1_i+_i+1(s) ,
	    Re{s}>0.





Let the recursion _i(s), i=m-1, …, 1 as in (<ref>) be given.
Then for s∈_+ and _m+1(s) = 0, 

	the function _m is a continued matrix S-fraction defined as

	_m(s):=_1 . 


This result can be formally related to earlier works by Duykarev et al. <cit.> in matricial continued S-fraction.


One can either construct the S-fraction by Gaussian elimination of the block tridiagonal system or use Schur-Complements and the Woodbury identity to relate _i to _i+1. Alternatively, one can show that _j is a discrete Dirichlet-to-Neumann of a circular planar graph and follow the derivation by Ingerman (Section 2.4 <cit.>). A proof will be added in the final manuscript.

 

To prove Proposition (<ref>) we use induction. At every step of the induction, we increase the continued fraction and solve a block tridiagonal pencil problem that increases by one block at each step. It coincides with the full pencil (Z_m+s Γ_m) at the last step. To facilitate this we define ij as a family of p × p real matrices

    {ij∈ℝ^p × p | j=1,…,m   ,    i=j,…,m},

where i corresponds to the blocks coupled through the three-term recurrence relation defined by the pencil (Z_m+s Γ_m) and j corresponds to the steps of the induction starting with j=m. At the j-th induction step we enforce the (block-Neumann boundary) condition 

-^-1_j-1(jj-j-1j)=I_p,
on the last m-j blocks of the three-term recurrence relation defined by the m-j last block rows of (Z_m+s Γ_m) which can be written out as

    -^-1_i(i+1j-ij) +^-1_i-1(ij-i-1j) + s_i ij   =0, 	   i=j,…,m,   j=1,…,m 
         m+1j=0.

Note that for j=1 this together with (<ref>) corresponds to the full pencil problem (Z_m+s Γ_m)[(11)^T, …, (m1)^T ]^T=E_1, which yields _m=11 and will terminate the induction.


Further, we introduce the notation
_i = ii
which due the condition (<ref>) provides the map
-(_j-1)^-1(ij-i-1j) _i↦ ij,   ∀i≥j
since i≥ jj satisfies the same recursion ∀ j. As shown later during the induction these maps are positive definite and thus invertible and the inverse maps

    -(_j-1)^-1(ij-i-1j)   =_i^-1ij  ∀ i≥ j
    i-1j   =(_i+_j-1)_i^-1ij

which for i=j+1 and assuming _j>0 and invertibility of _j+1 gives

_j+1^-1 j+1j=(_j+1+_j)^-1jj=-^-1_j(j+1j-jj).


With this in place consider the recursion (<ref>) for i=j, substitute the condition (<ref>) and replace the term -^-1_j(j+1j-jj) with the expression derived in (<ref>) to relate _j+1 to _j as
(_j+1+_j)^-1_j + s _j s _j =I_p
which gives the basic building block of the matricial S-fraction as

_j(s)=1s_j + 1_j+_j+1(s) ,
	    Re{s}>0.


We are now ready to perform the induction, where we show that indeed _j's are invertible and that the upper recursion gives the desired S-Fraction for _m.


		
  * Base Case: For the case j=m the condition m+1m=0 implies _m+1=0 in (<ref>). Alternatively equation (<ref>) for i=j=m can be written as 
_j^-1_m + s _j s _m =I_p
which starts the S-fraction as

_m(s)=1s_m + 1_m ,     Re{s}>0.
which is positive definite and thus invertible.
	
  * Induction Step: 
	Next for m>j ≥ 1 we assume that _j+1 is positive definite such that

    _j(s)=1s_j + 1_j+_j+1(s)

extends the S-fraction and ensures that _j is positive definite and invertible. The recursion ends with j=1 at which point (<ref>) and (<ref>) coincide with the full pencil and _m=_1(s) defined through recursion (<ref>) and terminated with _m+1=0.
	










§ BLOCK GAUSS-RADAU QUADRATURE

Let us modify the lowest floor of (<ref>) by replacing _m with _m/ϵ, ϵ >0 and keeping the remaining Stieltjes parameters unchanged. In the limit of ϵ→ 0 such a modification yields  
	
	_m(s) =ϵ→0lim
	1s_1+ 1_1 + 1s_2 +1 ⋱1s _m + ϵ _m 
	= 1s_1+ 1_1 + 1s_2 +1 ⋱1s _m  .
	Equivalently, such a limiting transition can be written using the resolvent of a block-tridiagonal matrix as
	
    _m(s)=E_1^T(_m+sI)^-1E_1,

	where 
	
	T̃_m=
	[     Α_1   Β_2^T                                        ;     Β_2     Α_2   Β_3^T                                ;               ⋱       ⋱       ⋱                        ;                     Β_i     Α_i Β_i+1^T                ;                               ⋱       ⋱       ⋱        ;                                   Β_m-1   Α_m-1   Β_m^T;                                             Β_m    Α̃_m;         ] ,
	and Α_i, i=1,…, m-1 and Β_i, i=2,…, m are the same block coefficients
	in Algorithm <ref>,	
	with 
	
	Α̃_m=lim_ϵ→0Α_m(ϵ)=
	 lim_ϵ→0 (_m^-1)^T[ _m-1^-1+ (_m/ϵ)^-1]_m^-1=
	(_m^-1)^T_m-1^-1_m^-1,
	using _m, _m computed via Algorithm <ref>. The following proposition is fundamental for understanding the distinct properties of _m and _m and the relation of _m to the Gauss-Radau quadrature.

0.1in

	The matrix function _m(s) is analytical and of full rank at s=0 as
	
	 
	_m(0)=∑_i=1^m_i. 
	
	Further, _m(s) has a pole of order one at s=0 with a 
	 residue matrix of rank p, specifically

	
	lim_s→0(s_m)=(∑_i=1^m
	_i)^-1. 
	
0.1in

	The basic building block of the considered S-fractions is given in (<ref>)	which allows us to succinctly define the truncated matrix continued S-fraction (<ref>) recursively as
		
		_m(s):=_1,   for  _m+1(s) = +∞,
		in the same fashion we defined _m(s) in equation (<ref>) via (<ref>).
	
	
	We will start with (<ref>) and use the
	recursions (<ref>) and (<ref>) to compute _m(0). First, we notice that _m(0)=_m. Then for i≤ m we get _i-1=_i+_i-1 which proves
	(<ref>) by induction.
	
	Likewise, (<ref>) can be obtained from recursion (<ref>) and (<ref>).
	We will start with computing (lim_s→ 0 s_m)^-1=_m and then get the recursion
	
    (lim_s→ 0s_i-1)^-1=_i-1+(lim_s→ 0s_i)^-1 for  i=m-2,…,2.
 Then (<ref>) will be computed by substituting _1 into the last equality of (<ref>). The inverse of the sum of s.p.d. matrices is of full rank, which concludes the proof.



 
	The poles of _m (as of any matrix S-fraction) are real non-positive and the pole at s=0 has multiplicity p which follows from the second result of Proposition <ref>. Therefore, _m has eigenvalue zero with multiplicity p and we arrive at the following corollary. 
	
	
		The symmetric block tridiagonal matrix _m is semi-definite with p-dimensional null-space.
	
	
	Because it is obtained from T_m by modifying only the last diagonal element Α̃, it can be viewed as a natural extension of the Gauss-Radau quadrature introduced in <cit.> for p=1, thus we will call it block Gauss-Radau quadrature. For p=1 the element Α̃∈ is uniquely defined  by the zero eigenvalue condition. For p>1, Corollary <ref> does not define Α̃_m uniquely, because it has p(p+1)/2 degrees of freedom, however as we shall see, our definition allows _m to retain some useful properties of the classical Gauss-Radau quadrature.

 
 

§ TWO-SIDED ERROR BOUNDS AND CORRECTION


 
 

 §.§ Monotonically convergent lower and upper bounds

 In the absence of deflation, Gaussian quadrature _m(s), as all Krylov approximations eventually do, converges to (s). Gauss-Radau quadrature _m(s) differs from _m(s) only by the absence of the lower floor in the S-fraction representation, thus it also converges to the same limit. We next prove that the two sequences give monotonic two-sided bounds and that their difference defines an upper bound.
 
 
 
 
 0.1in
 
 	For a given real positive s, the following holds,
 	
 	i) The matrix sequence {_m(s)}_m is increasing, i.e. 
 	_m(s)<_m+1(s), ∀ m≥1;
 	
 	ii) The matrix sequence {_m(s)}_m is decreasing, 
 	i.e. _m(s)>_m+1(s), ∀ m≥1;
 	
 	
 	
 	
 
 
 0.1in
   From the recursion 
 		(<ref>) with final condition given by (<ref>) 
 		∀ s>0 and i<m+1 we get _i(s)>0.
 		From (<ref>) it follows that we get 
 		_m-1 if we impose _m=0. 
 		From (<ref>) it also follows
 		that _i is monotonic with respect to _i+1 (_i+1 in the denominator of _i's denominator), so the replacement of the final condition in (<ref>) by _m=0 decreases _m-1, that consequentially decreases _m-2, etc., and by induction we obtain _m-1<_m.  
 	
 	For proving (ii), we recall that
 	one can get _m-1 by terminating the S-fraction with _m=+∞. 
 	Then with the same monotonicity argument as for _m, it follows 
 	that such a substitution recursively increases _i, i=i=m-1,m-2,…, 1.
 
 
 
Now we are ready to formulate a two-sided Gauss-Radau-type estimate.


0.1in

 Let Algorithm <ref> produce 
	Β_j^TΒ_j>0 for j≤ m. Then
	
    0<_m-1(s)<_m(s)< (s)< _m(s)<_m-1(s)   ∀ s∈_+.




We apply Theorem <ref> to _m and _m, yielding 0<_m-1(s)<_m(s) 
and _m(s)<_m-1(s). 
To conclude the proof, we notice that for s satisfying the 
condition of the proposition 
lim_m→∞_m(s)=(s), i.e., _m(s)<(s). 
From T_m- T̃_m≥ 0 we obtain 

	
    _m(s)-_m(s)   =   E_1^T(T̃_m+sI)^-1E_1-E_1^T (T_m+sI)^-1E_1
       =   ∫_0^1d/drE_1^T(T_m+ r(T̃_m-T_m)+sI)^-1E_1 d r
       =   -∫_0^1V(r)^T(T̃_m-T_m)V(r) dr ≥ 0,


with V(r)=[T_m+ r(T̃_m-T_m)+sI]^-1E_1, that concludes the proof.

0.1in
The ordering above provides us with a computable error estimate of the approximation of , that is

    - _m  < _m - _m ,   - _m  < _m - _m .


The two-sided bounds of Proposition <ref> indicate the possibility 
of using their averaging to compute an error correction. 
Such an approach was considered in the works of Reichel and his 
co-authors for Gauss-Radau quadrature with p=1 <cit.> and for 
p>1 for anti-Gauss quadratures <cit.>. Here we identify an important 
class of PDEs operators for which the averaging is particularly efficient. 

 


 §.§ Posterior quadrature error correction for problems with continuous spectrum


Even though we suggest and test extrapolation algorithms for arbitrary 
	p≥ 1, for simplicity in this section we will analyze only the case
	p=1. We also assume that A is obtained via accurate approximations of problems with continuous spectrum (as will be detailed in the following section) and instead of  we consider scalar, single-input single-output (SISO) transfer functions for such problems denoted as . We assume that the support of the continuous part of the spectral measure of (z) consists of the union of intervals of the real positive semiaxis. We also denote _m and _m as the counterparts of respectively of _m and _m.

First, we recall that the Gauss and Gauss-Radau quadratures match m and m-1 matrix moments of 
the spectral measure respectively <cit.>, thus 
 
(s)-_m(s)=O(s^-2m-1),   (s)-_m(s)=O(s^-2m),  s→∞.
Next, we notice, that 

    λ^-1_m(λ^2)
    =1λ_1+ 1λ_1 + 1λ_2 +1⋱1λ_m + 1λ_m,

i.e., λ^-1_m(λ^2) is the 
[2m-1/2m] matrix Padé approximant 
(at λ=∞ ) of
λ^-1(λ^2), where λ=√(s).
Likewise, 	

    λ^-1_m(λ^2)=1λ_1+ 1λ_1 + 1λ_2 +1⋱1λ_m-1 + 1λ_m,
 
i.e., 
λ^-1_m(λ^2) is  [2m-2/2m-1] matrix Padé approximant (at λ=∞ ) of λ^-1(λ^2).
That allows us to  directly adopt Stalh's Padé 
convergence Theorem <cit.> based on 
potential theory, to our case.


	Let g(z) be the potential solution on the complex plane corresponding to the 
	Green function of the unit charge located at z=∞ with Dirichlet 
	boundary conditions on the support of the continuous components of the 
	spectral measure of (z).
	Then for  λ∈ and ∀ϵ>0
		
		e^ -g(λ)-ϵ<lim_m→∞[_m(λ^2) -(λ^2)]^1/4m-3 
		< e^ -g(λ)+ϵ, 
		and
		e^ -g(λ)-ϵ<lim_m→∞[(λ^2)-_m(λ^2)]^1/4m-1 <e^ -g(λ)+ϵ.
	
		


	The potential g(λ) is bounded only if the spectral measure has a nontrivial continuous component. The applicability of Proposition <ref> is thus limited to such problems, e.g., PDE problems in unbounded domains), or its sufficiently accurate approximations. 
	

Proposition <ref> guides us to the error correction formula f̂_m=1/2 (f_m+f̃_m+1) , which in the p>1 setting is

    _m(s):=1/2(_m(s)+_m+1(s)).

We stress that the computation of 
_m+1(s) requires the same m steps of the block 
Lanczos algorithm as _m(s). 
0.1in

	For p=1, small g(√(s)) and thus 1- e^-g(√(s))≈ g(√(s)) the error of the extrapolated f̂_m(s) behaves as
	
    |f(s)-_m(s)|≈ 2g(√(s))((s)- _m(s)).



The potential solution g(√(s)) is small when s 
is close to the boundary of continuous components of the spectra, i.e., in the case of the spectral interval [0, θ_max], for small positive s. Since -4g(√(s)) is the linear rate of convergence of _m, slower convergence of _m leads to better convergence of the extrapolated f̂_m.




	In the context of inverse problem theory, certain PDEs exhibit a significant symmetry structure known as duality, as illustrated in <cit.>. This duality implies that the Neumann-to-Dirichlet map for the primary problem can be linearly scaled to correspond to the Dirichlet-to-Neumann map for the dual problem. The transfer function (s), when B is localized at a (partial) boundary of the computational domain, can be interpreted as a discretization of the partial Neumann-to-Dirichlet map and, similarly, can be scaled to represent the Dirichlet-to-Neumann map of the dual problem, denoted as ^-1(s). Following this logic, it can be demonstrated that _m(s) can be transformed into _m+1(s) for the dual problem. While a rigorous analysis of this phenomenon within the framework of error extrapolation is warranted, we leverage this duality intuitively to refine the extrapolation formula. To adhere to the duality structure, the formula should remain invariant under the substitution of ^-1 for , a property that (<ref>) does not satisfy. Hence, we propose a similar, but dual-invariant, formula that satisfies Corollary <ref>,
	
	
    _m(s):= exp[1/2 [log(_m(s))+log(_m(s))]]

	with 
	
    _m^-1:=1/2 [(_m(s))^-1+(_m+1(s))^-1].





§ NUMERICAL EXAMPLES


 In this section, we consider the application of the developed error bound and the extrapolation formulas for the MIMO transfer function computations for PDE operators with continuous or very dense spectrum when Krylov subspace approximations lose their spectral adaptivity. This phenomenon can only be observed for large-scale problems, thus dimensions of our experiments are of order 10^5-10^6. Though the theory in this manuscript is limited to the resolvent with s∈ℝ^+, we stress that the error bound, monotonicity and extrapolation seem to hold for imaginary s and other functions such as ϕ(A)=exp(-A t). We also explore possible accelerations stemming from adding random vectors to the initial block.
All errors reported are measured in the 2-norm. 






 §.§ Transfer functions of PDE operators on unbounded domains

The computation of transfer functions of self-adjoint PDE operators on unbounded domains is the main target of the proposed approach. For accurate PDE discretization of such domains, we adopted the optimal grid approach (<cit.>) allowing spectral convergence in the exterior part of the computational domain, e.g., see <cit.> with a more detailed explanation of optimal grid implementation. A trade-off of the spectral approximation of the exterior problem is that A's spectrum approximates the continuum spectrum of the exact problem well, and thus the Krylov subspace approximations lose their adaptability. The proposed extrapolation attempts to alleviate this problem by adapting to continuous spectra.



  §.§.§ 2D Diffusion


We discretize the operator

    σ( x)^-1/2Δσ( x)^-1/2

on an unbounded domain using second-order finite differences on a 300× 300 grid. In the grid center where the variations of σ() are supported, we use a grid size of δ_x=1.  This formulation can be associated with both heat transfer and the scalar diffusion of electromagnetic fields. To approximate this, we consider a Dirichlet problem on a bounded domain with N_opt=10 exponentially increasing grid steps in the exterior and a uniform grid in the interior part. Utilizing the optimal geometric factor of exp(π/√(10)) as indicated by <cit.>, we achieve spectral accuracy in the exterior region of the domain, thereby effectively approximating the boundary condition at infinity (exponential decay). The resulting grid and function σ() is shown in Figure <ref>. Only the first two steps of the geometrically increasing grid are shown in the figure since they increase rapidly. In this case, B is a vector of discrete delta functions with entries at grid nodes corresponding to the transducer locations shown in the same figure as Trx. 

The convergence for s=10^-3 is shown in Figure <ref> alongside the convergence curve for purely imaginary shifts s=·10^-3 in Figure <ref>. For imaginary shifts s, block Lanczos converges slower. Despite all derivations relying on s∈ℝ^+, the convergence is monotone, the upper bound holds and the extrapolation formulas reduce the approximation error by an order of magnitude. For the chosen shift both extrapolations perform equally well.

In Figure <ref> the convergence for the matrix exponential ϕ(A)=exp(-At) for t=10^5 is shown. The convergence is superlinear, the error bound becomes tight with increased iterations and the extrapolation from formula (<ref>) performs better for low iteration counts. Numerical results for accelerating convergence further through subspace enrichment by adding random starting vectors to B in 𝒦_m{(A,B)} are presented in the Appendix <ref>.















  §.§.§ 3D Electromagnetic diffusion
 In this example we solve the 3D Maxwell equations in the quasi-stationary (diffusive) approximation. Krylov subspace approximation of action of matrix functions for the solution of that problem was introduced in <cit.> and remains the method of choice for many important geophysical applications. The diffusion Maxwell's system in ^3 can be transformed to the form

(∇×∇×+σ() μ_0 s)ℰ_(r)(,s) = - 𝒥^ext_(r)(,s)   ∈Ωwhere σ is the electrical conductivity, μ_0 the vaccum permeability, 𝒥^ ext_(r) is an external current density indexed by transmitter index (r) causing the electrical field strength ℰ_(r)(,s). For s∉_- the solution of (<ref>) satisfies infinite condition lim_→∞ E()=0.We approximate (<ref>)-(<ref>) by the conservative Yee grid similarly to <cit.>.
 Following the same framework as in the diffusion problem in ^2 described earlier, we truncate the computational domain with Dirichlet boundary conditions for the tangential components of ℰ_(r) on the domain boundary with optimal geometric progression grid, that gives spectral convergence of the infinity condition (<ref>). In our particular realization, we used N_opt=6 gridsteps geometrically increasing with geometric scaling factor exp(π/√(6)) according to <cit.>. The grid is N_x=80, N_y=100 and N_z=120 grid steps large leading to a size of A of N=2  821  100. The configuration has a homogeneous conductivity of σ=1 in the entire domain except for two inclusions shown in Figure <ref> where σ=0.01. As current densities, we use 6 magnetic dipoles (approximated by electric loops), with 3 dipoles located above the inclusion and 3 dipoles located in the middle of the inclusions as highlighted by the arrows in Figure <ref>. Such configuration corresponds to the tri-axial borehole tools used in electromagnetic hydrocarbon exploration <cit.> and the matrix B thus has 6 columns. 

In Figure <ref> the convergence is shown for various real values of s after m=400 block iterations. The error bound is tight across all values for s 
and there is no difference in extrapolation formulas. Further in Figure <ref> the convergence for this test case is shown for a real shift of s=0.05 and in Figure <ref> for an imaginary shift.




 

 §.§ Graph Laplacian
  Efficient computation of SISO and MIMO transfer functions of large graph-Laplacians is important for efficient clustering of big data sets, computation of centrality measures and other applications <cit.>. For our numerical experiments we take the  grap from the SNAP (Standford network analysis project) <cit.> and turn it into an undirected graph by symmetrization. The original graph has 685230 nodes and 7600595 edges and we construct the normalized graph Laplacian A as our test matrix. The nodes of the graph represent pages from berkeley/stanford.edu domains and directed edges (here converted to undirected edges) represent hyperlinks between them. The data were collected in 2002. 
 
 As B matrix we take p=3 delta functions chosen at random nodes[With random seed 1 we obtain  delta functions at nodes 207168, 100562, and 63274.].
 
 In Figure <ref> we evaluate the resolvent ϕ(A,s)=(A+s I)^-1 for s=10^-4 showing that the error bound is tight and the extrapolation formulas give similar results. After m=50 block iterations the error for various values of s is displayed in Figure <ref>.
 
 
 
 
 
  In contrast to PDE operators on unbounded domains, data-related graphs may exhibit spectral distributions with non-vanishing gaps even at very large dimensions. Consequently, they cannot be regarded as approximations of the problems discussed in Proposition <ref>. Nevertheless, as illustrated in Figure <ref>, quadrature approximations for such graphs can demonstrate linear convergence, in which case they satisfy tight block-Gauss-Radau bounds and their extrapolation will be efficient. 
 
 


 


§ CONCLUSIONS


In this paper, we derived a block extension of Gauss-Radau quadrature for the computation of MIMO transfer functions by leveraging the connection between the block-Lanczos algorithm and matrix Stieltjes continued fractions. We proved that the block Gauss and Gauss-Radau quadrature rules provide monotonically convergent two-sided bounds for MIMO transfer functions of linear time-invariant (LTI) dynamical systems with real Laplace frequencies. For problems involving the approximation of operators with a continuous spectrum, we also prove (in the SISO case) that these bounds are tight, thereby justifying extrapolation by averaging Gauss and Gauss-Radau approximations.

Our numerical experiments with large-scale graph-Laplacian, 2D diffusion, and 3D Maxwell systems confirmed the tightness of the derived bounds and the efficiency of the extrapolation formula, not only for the cases analyzed but also for transfer functions of LTI systems with imaginary Laplace frequency and time domain responses. Additionally, we explored the potential of using random enrichment of the initial Lanczos block within the developed algorithm framework. These findings highlight the robustness and broad applicability of our approach targeted for large-scale problems.





§ ACKNOWLEDGEMENT

Part of this work was performed while Valeria Simoncini (VS) was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Numerical PDEs: Analysis, Algorithms, and Data Challenges semester program.


The work of VS was also funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 of 2/2/2022, entitled “Low-rank Structures and Numerical Methods in Matrix and Tensor Computations and their Application”, code 20227PCCKZ – CUP J53D23003620006. VS is a member of the INdAM Research Group GNCS; its continuous support is gladly acknowledged.

The work of Vladimir Druskin was partially supported by AFOSR grants FA 955020-1-0079, FA9550-23-1-0220, and NSF grant DMS-2110773.


The authors like to thank Leonid Knizhnerman for helpful discussions related to the material in section <ref> and Michele Benzi for discussions related to the spectra of large graphs.

plain


 



 

§ RANDOM ENRICHMENT OF INITIAL BLOCK

Adding columns to the initial block B can reduce the number of block-iterations only if 
 the Krylov subspaces generated by the columns have a significant intersection. This is not the case for the problems with localized inputs/outputs considered in section <ref>. Such problems however are important in remote sensing applications. To improve convergence for such a class of problems we experiment with adding random column vectors to B for the problem considered in section <ref>. We used an enriched block B'=[B,R], where R∈^n× p' is a block of random vectors. The Gauss and Gauss-Radau quadratures using 𝒦_i(A,B') are computed and then use the upper right p× p block of the computed approximation to B'^Tϕ(A,s)B' .

In Figure <ref> we illustrate the effect of adding random vectors to B on the convergence of the method for a shift of s=10^-2. The vectors added have independent, identically distributed entries chosen from the normal distribution 𝒩(0,1). Since the block-Lanczos method is a memory-bandwidth limited algorithm on most modern CPU and GPU architectures, adding extra vectors to B comes at a small increase in computational cost per block-Lanczos iteration, yet it greatly affects convergence rates. In the solid black curve only the 3 transducers are used in B, adding one random vector to B increases the convergence rate as shown by the dashed line, and adding two additional random vectors to B accelerated convergence further if expressed in terms of block-iterations. To reach an error of 10^-6 the block-Lanczos iteration with no added vectors needed m=202 iterations (solid black line), with one added random vector in needed m=169 iterations (dashed black line), with 3 added vectors it required m=145 iterations (dotted black line) and with 12 added vectors m=86 iterations (dash-dotted black line) With extrapolation only m=129 iterations were needed for the case of 3 added random vectors. 

 The acceleration of extrapolated results is even greater at higher error levels, which can be qualitatively explained as follows. For problems with an (approximately) continuous spectrum, adding vectors to the subspace does not improve the linear convergence rate, resulting in sub-linear acceleration. This effect is amplified by the extrapolation, which, according to Proposition <ref> is more significant at the low linear convergence rates observed in the final stage of sub-linear convergence. This phenomenon can be practically important for applications to inverse problems that tolerate low-precision computations due to measurement errors.
 
 

It is important to note that adding random starting vectors leads to an increase in the overall size of the matrix T_m due to the enlargement of the block size, despite the acceleration in convergence. Thus any reduction in computation speed comes from memory-bandwidth bound matrix-matrix multiplications AB 





 Paper Title: Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems.









Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

    Lise-Marie Imbert-GérardDepartment of Mathematics, University of Arizona, USA  (mailto:lmig@arizona.edulmig@arizona.edu) ,
Andrea MoiolaDepartment of Mathematics, University of Pavia, Italy       (mailto:andrea.moiola@unipv.itandrea.moiola@unipv.it) ,
Chiara PerinatiDepartment of Mathematics, University of Pavia, Italy (mailto:chiara.perinati01@universitadipavia.itchiara.perinati01@universitadipavia.it) , 
Paul StockerFaculty of Mathematics, University of Vienna, Austria (mailto:paul.stocker@univie.ac.atpaul.stocker@univie.ac.at)
    August 5, 2024
===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
	Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying PDE. 
	Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous, and have piecewise-constant coefficients. 
	However, if the equation has variable coefficients, exact solutions are generally unavailable. 
	Quasi-Trefftz methods overcome this limitation relying on elementwise “approximate solutions” of the PDE, in the sense of Taylor polynomials.
	
	We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term,
	describe their approximation properties and, under a non-degeneracy condition, provide a simple algorithm to compute a basis. 
	We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. 
	The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom.
	For non-homogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. 
	Numerical experiments in 2 and 3 space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.
	
	Keywords: Quasi-Trefftz, 
	Discontinuous Galerkin, 
	Elliptic equation, 
	Diffusion–advection–reaction equation, 
	Smooth coefficients,
	Convergence rates
	
	Mathematics Subject Classification (2020): 
	65N15, 
	65N30, 
	35J25, 
	41A10, 
	41A25 
	
	
	

§ INTRODUCTION

	

 §.§ Motivation for quasi-Trefftz methods

	Classical Galerkin schemes, such as finite element and discontinuous Galerkin (DG) methods, seek an approximation of a boundary value problem (BVP) solution in a piecewise-polynomial discrete space.
	The most common trial and test spaces contain all piecewise polynomials of some given maximal degree, possibly with some inter-element continuity.
	These spaces are not tuned to approximate the solutions of a given partial differential equation (PDE), instead they contain approximations to all sufficiently regular functions.
	To reduce the number of degrees of freedom (DOFs), i.e. the size of the discrete space, and thus the computational cost of the scheme, one can construct more specialized discrete spaces, that are adapted to the PDE to be approximated.
	
	A well-known way to implement this idea is to use a Trefftz method: a scheme where all discrete functions are elementwise solutions of the PDE.
	This is feasible when the PDE has piecewise-constant coefficients.
	For instance, Trefftz methods for the Laplace equation Δ u=0 use harmonic polynomials as basis functions <cit.>, while those for the wave equation ∂_t^2u-Δ u=0 use “polynomial wave” solutions in space–time <cit.>.
	For PDEs with a zero-order term, no polynomial solutions are available, thus Trefftz methods for the Helmholtz equation Δ u+k^2u=0 typically use complex-exponential plane-wave bases <cit.>.

	The common feature of these Trefftz schemes is that they offer the same accuracy as comparable methods based on full polynomial spaces, using much fewer DOFs.
    A sparsity comparison of a Trefftz DG scheme against other polytopal finite element methods, including Hybrid-DG, Hybrid High-Order, and Virtual Element Methods, has been performed in <cit.>. 
    The Trefftz DG scheme is shown to achieve a reduction in complexity comparable to that of the other methods. 
    Furthermore, as the degrees of freedom of the Trefftz DG method are only associated to the mesh elements, the Trefftz DG method generalizes very efficiently to polytopal meshes.
    In <cit.>, the Trefftz DG method is presented for the Stokes problem, and compared to other methods in this context.
	
	However, when the PDE has variable coefficients, the construction of local exact solutions is usually not possible.
	Instead, quasi-Trefftz methods can be applied in this case.
	These rely on discrete spaces of functions that, on each element, are solution of the PDE “up to a small residual”: for the PDE u=f, in each mesh element E with diameter h_E, every function v_h in the discrete trial space satisfies | v_h-f|=(h_E^q) in E for some fixed exponent q∈.
	Both Trefftz and quasi-Trefftz methods are usually formulated as DG schemes.
    Another approach that allows for variable coefficients and non-zero right-hand sides is the embedded Trefftz method <cit.>, where the Trefftz basis functions are not explicitly constructed but embedded in a standard DG method.
	
	So far, the quasi-Trefftz idea has been used for oscillatory problems with smooth coefficients:
    in the time-harmonic regime using both polynomial and complex-exponential basis functions <cit.>,
	in space–time using polynomials for the wave <cit.> and the Schrödinger <cit.> equations.
	Complex-exponential quasi-Trefftz spaces for some homogeneous equations of order m≥ 2 were introduced in <cit.>, while in <cit.> complex-exponential and polynomial quasi-Trefftz spaces were studied for some homogeneous equations of order m=2.
	However, no general treatment of the corresponding quasi-Trefftz methods is available.
	
	

 §.§ The contributions of this paper

	The main goal of this paper is to introduce and analyze the degree-p polynomial quasi-Trefftz space (denoted _f^p(E)) for the general, order-m, linear, partial differential operator =∑_||≤ mα_ D^.
	The main assumption is that the coefficients α_ and the source term f are sufficiently smooth, namely α_,f∈ C^p-m(E) (where E will then be a mesh element).
	We define the affine space _f^p(E) in <Ref>, and prove in <Ref> that it approximates all smooth solutions of u=f with the same convergence rate, with respect to the domain size, compared to the full polynomial space ^p(E) of the same degree.
	Under a simple non-degeneracy condition (<ref>), in <ref> we provide a simple iterative algorithm to compute the monomial expansion of all quasi-Trefftz polynomials.
	These functions are uniquely determined by their “Cauchy data”, i.e. the values of the first m derivatives on a given hyperplane.
	Their computation requires the partial derivatives of the PDE coefficients α_ and right-hand side f at a fixed point ^E.
	<Ref> thus allows to construct simple bases of ^p_0(E) and to verify that the dimension of this space is indeed much smaller than dim(^p(E)), see (<ref>).
	
	In the following sections we study a quasi-Trefftz DG method for elliptic diffusion–advection–reaction problems. 
	We introduce a BVP in <ref>, a polytopal mesh in <ref>, and a DG formulation in <ref>.
	We use the classical symmetric interior penalty for the diffusion term and upwind penalization for the advection term.
	To study the convergence of this method, in <ref> we slightly modify the standard DG analysis of e.g. <cit.> to handle more general polynomial discrete spaces.
	This allows to prove optimal convergence rates for the quasi-Trefftz DG method in <ref>.
	Since the PDE source term f enters the definition of ^p_f(E), the trial space is actually an affine space: to write the Galerkin problem as a linear system, we compute an elementwise approximate PDE solution, using again <Ref>, possibly in parallel, and then solve for the difference, see (<ref>).
	The quasi-Trefftz space could be combined with any other stable and quasi-optimal DG formulation with similar results.
	
	Finally, in <ref> we show some numerical examples in 2 and 3 space dimensions illustrating the capabilities of the method.
	In these examples, the method based on the same DG formulation, discretized with a polynomial quasi-Trefftz discrete space of degree p compared to the full polynomial space of the same degree, achieves the same error and convergence rates, but with considerably fewer DOFs.
	The construction of the quasi-Trefftz basis, following <Ref>, involves a small overhead, but is completely parallelizable: <Ref> shows that the total computational time for the quasi-Trefftz version of the scheme is lower than for the full-polynomial space.
	We also consider two advection-dominated examples and show that the solutions are well captured in both cases.
	
	The quasi-Trefftz DG method for diffusion–advection–reaction equations is implemented in NGSolve <cit.> and the code is freely available.
	This paper is mainly based on the third author's master thesis <cit.>, where some more details can be found.
	
	
	
	
	

§ POLYNOMIAL QUASI-TREFFTZ SPACE
 
	In this section, we first introduce the polynomial quasi-Trefftz space for a general linear PDE with smooth  coefficients and right-hand side, and prove that it contains high-order approximations of all smooth PDE solutions.
	While the definition and the approximation properties of this space only require the governing PDE to have C^p-m-smooth coefficients and right-hand side, p being the polynomial degree of the space and m the order of the PDE, practical aspects of quasi-Trefftz also rely on a simple non-degeneracy assumption on the differential operator.
	In particular, the dimension of the quasi-Trefftz space depends on the differential operator, and we show that under assumption (<ref>) this dimension is much reduced compared to standard polynomial spaces, as expressed in (<ref>).
	In this case, we provide an algorithm for the construction of quasi-Trefftz functions, including for a non-zero-RHS PDE, and more specifically for the construction of a basis for a zero-RHS PDE.
	
	

 §.§ Definitions and notation

	Let d ∈ be the space dimension.
	Multi-indices are denoted :=(i_1,…,i_d)
	∈_0^d, their length ||:=i_1+⋯+i_d, and ≤ denotes the partial order
	defined by ≤ if i_k≤ j_k for all k∈{1,…,d}.
	As a reminder, the multi-index factorial and binomial coefficients are defined as
	
    !:=i_1!⋯ i_d!,     :=!/!(-)!
    	=i_1j_1⋯i_dj_d.

	We use standard multi-index notation D^ f := ∂_x_1^i_1⋯∂_x_d^i_df for derivatives of a function f of ∈^d, and  ^=x_1^i_1⋯ x_d^i_d for monomials.
	
	Let E⊂^d be an open
	set.
	Denote by ^p(E) the space of polynomials of degree at most p∈_0 defined on E.
	The general linear partial differential operator of order m∈, denoted ℳ, is expressed in terms of its variable coefficients α_: E→ for ∈_0^d and ≤ m as
	
    :=∑_∈_0^d, ||≤ mα_ D^.

	The PDE of interest, for the unknown u:E → and source term f:E →, then reads
	u=f in E.
	
	We introduce the polynomial quasi-Trefftz spaces for this PDE, under assumptions of smoothness of the right-hand side and the operator coefficients.
	[Quasi-Trefftz space]
	Let p∈_0, let E ⊂^d be an open set and let ^E∈ E.
	Assume that the coefficients α_∈ C^max{p-m,0}(E) for all ≤ m and that f∈ C^max{p-m,0}(E). 
	We define the polynomial quasi-Trefftz space for the equation u=f in E as
	
    ^p_f(E):={ v∈^p(E) | D^ v (^E)=D^ f (^E)  ∀∈^d_0, ||≤ p-m}.

	
	The choice of the maximal order p-m for the derivatives of ℳv-f that vanish at ^E is optimal in the following sense: for a lower order the space would be larger but it would not have better approximation properties; for a higher order the space would not enjoy the same approximation properties; see <cit.>.
	By definition, ^p_f(E) is a subset of the full polynomial space  ^p(E) and an affine space; when f=0, ^p_0(E) is a vector space.
	For p<m, ^p_f(E) coincides with ^p(E), so we always assume p≥ m.
	
	
	In general,  ^p_f(E)⊄^p+1_f(E), i.e., for increasing polynomial degrees p, the quasi-Trefftz spaces are not nested.
	To see this, consider, for example, 
	the second-order diffusion–advection–reaction operator u:=-Δ u+·∇ u+σ u with ()=(1,…,1)^⊤, σ()=2/x_1^2+1 and f=0.
	Choosing the point ^E= and the function v()=x_1^2+1∈^2(E),  then, v()=-2+2x_1+2, so v(^E)=0.
	Hence v∈^2_0(E), but ∂_x_1 v()=2, implying that v∈^2_0(E)∖^3_0(E).
	
	
	
	Let us consider a constant-coefficient differential operator .
	When all the terms in (<ref>) 
	are derivatives of the same order (i.e. α_=0 for ||<m), such as, for example, in the Laplace and the wave equations, the polynomial Trefftz space 
	^p(E):={v∈^p(E) | v=0  in  E}
	approximates solutions of the homogeneous PDE u=0 with the same orders of h-convergence as the full polynomial space ^p(E) <cit.> (assuming E is star-shaped).
	On the other hand, if the differential operator  includes derivatives of different orders, the convergence rates for the polynomial Trefftz space can be lower.
	For example, for the linear time-dependent Schrödinger equation, in <cit.> the same
	rates are obtained for ^p(E) and ^2p(E), i.e. the Trefftz space requires doubling the polynomial degree.
	In the extreme case when a zero-order term is present, i.e. α_0, such as in the case of the Helmholtz equation, u=0 does not admit polynomial solutions and the polynomial Trefftz space is trivial, ^p(E)={0}.
	The quasi-Trefftz space, instead, is always rich enough to give the same approximation rates as the full ^p(E), as we see below in <Ref>.
	This suggests that quasi-Trefftz methods could be an effective choice also for problems with piecewise-constant coefficients.
	
	
	

 §.§ Approximation properties

	For q∈_0, the standard C^q norms and seminorms are denoted by
	
    v_C^0(E):=sup_∈ E|v()|,    v_C^q(E):=max_∈_0^d,  ||=qD^ v_C^0(E).

	Let p∈_0 and let ^p+1_^E[v]∈^p(E) denote the Taylor polynomial of order p+1 of v∈ C^p(E), centred at ^E∈ E: 
	
    ^p+1_^E[v]():=
    	∑_||≤ p1/!	D^ v(^E)(-^E)^.

	For every multi-index ∈_0^d with ||≤ p
	
    D^^p+1_^E[v]()=   ∑_≤ p
    ≥1/!D^ v(^E)!/(-)!
    	(-^E)^-
    	=∑_≤ p-1/!D^+ v(^E)
    	(-^E)^,
    	
       ⟹   D^^p+1_^E[v]()= ^p+1-_^E[D^v]().

	From the evaluation of this identity at =^E and ^p+1_^E[v]()∈^p(E),
	it follows that
	
    D^^p+1_^E[v](^E)=
    	{[  D^ v(^E) if ||≤ p,;         0  if ||>p. ].

	Recall the Lagrange form of the Taylor remainder <cit.>: if v∈ C^p+1(E) and the segment S with endpoints ^E and  is  contained in E, then exists _*∈ S such that
	
    v()-^p+1_^E[v]()=
    	∑_||=p+11/!D^ v(_*)(-^E)^.

	
	To prove the approximation properties of ^p_f(E) and to construct quasi-Trefftz polynomials, we make the following regularity assumption on the PDE coefficients and right-hand side:
	
    p,m∈,      p≥ m,     α_∈ C^p-m(E)   for all ∈_0^d,    ||≤ m,      
    	f∈ C^p-m(E).

	
	The following theorem provides the key approximation property of quasi-Trefftz spaces:  the orders of h-convergence for quasi-Trefftz spaces ^p_f(E) are the same as those for full polynomial spaces ^p(E) of the same degree.
	We denote the diameter of E as h_E:=sup_,∈ E-.
	
	
	
	Under assumption (<ref>), let
	u∈ C^p+1(E) satisfies u=f in E.
	Then, the Taylor polynomial 𝖳^p+1_^E[u]∈^p_f(E).
	
	Moreover, if E is star-shaped with respect to ^E, then, for all q∈_0 with 
	q≤ p,
	
    inf_v∈^p_f(E)u-v_C^q(E) ≤ u-^p+1_^E[u]_C^q(E) ≤ d^p+1-q/(p+1-q)! h_E^p+1-qu_C^p+1(E).

	
	
	First we prove that ^p+1_^E[u]∈^p_f(E).
	By definition, ^p+1_^E[u]∈^p(E).
	Moreover, from the definition (<ref>) of  and the Leibniz product rule,
	for all v∈ C^p(E) and ||≤ p-m 
	
    D^ v(^E)=∑_≤ m D^(α_ (^E)D^ v(^E))
    	=∑_≤ m∑_≤ D^α_ (^E)D^-+v(^E).

	Hence, with v= 𝖳^p+1_^E[u], we have for all ||≤ p-m
	
    D^𝖳^p+1_^E[u](^E)   =∑_≤ m∑_≤ D^α_(^E) D^-+𝖳^p+1_^E[u](^E)
       =∑_≤ m∑_≤ D^α_(^E) D^-+ u(^E)
    	=D^ u (^E) = D^ f (^E) .

	The second equality
	follows from the property (<ref>) with partial derivatives of order at most equal to +m≤ p, while the third one is (<ref>) again  with v=u∈ C^p+1(E).
	In the last step we use that u is solution of u=f in E.
	This shows that the Taylor polynomial 𝖳^p+1_^E[u] belongs to the quasi-Trefftz space ^p_f(E).
	
	This immediately implies the first inequality in the best-approximation bound (<ref>).
	To prove the second inequality, fix q with 0≤ q≤ p.
	Using the ·_C^q-seminorm definition, 
	the identity D^^p+1_^E[u]=^p+1-||_^E[D^ u] for ||=q≤ p from (<ref>), 
	estimating the Lagrange form of the Taylor remainder (<ref>), which is applicable because u∈ C^p+1(E) and E is star-shaped with respect to ^E,
	we obtain the assertion:
	
    u-^p+1_^E[u]_C^q(E)   =max_∈_0^d,  ||=qD^ (u-𝖳^p+1_^E[u])_C^0(E)
       =max_∈_0^d,  ||=qD^ u-𝖳^p+1-q_^E[D^ u]_C^0(E)
       ≤max_∈_0^d,  ||=q∑_||=p+1-q1/!sup_,_*∈ E	D^+u(_*)(-^E)^
       ≤d^p+1-q/(p+1-q)! h_E^p+1-qu_C^p+1(E).
 	
	We have used the formula 
	∑_||=k1/!= d^k/k! with k=p+1-q, 
	obtained from the multinomial theorem 
	(w_1+…+w_d)^k=∑_∈_0^d, =kk!/!^
	by choosing =(1,…,1).
	
	
	Bound (<ref>) is an h-approximation estimate: it ensures convergence of the approximation error to zero when the size of the domain E decreases.
	For analytic functions whose seminorm sequence p↦|u|_C^p(E) increases at most exponentially, it ensures also p-convergence, namely convergence on a fixed E when p→∞.
	
	

 §.§ Construction of quasi-Trefftz functions

	Under assumption (<ref>), this section proposes an explicit procedure to construct quasi-Trefftz functions under a further non-degeneracy assumption on the differential operator , namely that 
	
    α_^*(^E)0   for ^*=(m,0,…,0)=m_1.

	Here we denote by _k∈^d the elements of the canonical basis of ^d, defined by (_k)_l=δ_kl, 1≤ k,l≤ d.
	Assuming instead that ^*=m_k for any k between 2 and d would allow for the same reasoning.
	Condition (<ref>) might be circumvented with a more algebraic approach to the construction of quasi-Trefftz functions, which is currently under development.
	
	Constructing a polynomial v∈^p_f(E) boils down to computing 
	the coefficients {a_,∈_0^d,≤ p} of its expansion as a linear combination of scaled monomials centered at ^E∈ E:
	
    v()=∑_∈_0^d,||≤ p a_(-^E/h_E)^,  from which D^ v(^E)=! /h_E^a_.

	In order to state
	the conditions D^ℳv(^E)=D^f(^E) for ≤ p-m
	in terms of the coefficients a_, we note from (<ref>) that 
	
    D^ v(^E)
    	   =∑_≤ m∑_≤ D^α_ (^E) 
    	(-+)! /h_E^-+a_-+
       
    	=	∑_≤ m∑_≤- D^-α_ (^E) 
    	(+)! /h_E^+a_+         ||≤ p-m.

	Under assumption (<ref>), each of these conditions for ≤ p-m can be equivalently stated as
	
    a_+m_1=h_E^+m/α_m_1(^E)(+m_1)!( D^	f(^E)-∑_≤ m
    ≤
     (,)≠(m_1,) -  D^-α_ (^E) 
    	(+)! /h_E^+ a_+),

	dividing by α_m_1(^E)0.
	Imposing (<ref>) following an order such that, at each step, all the a_+ appearing at the right-hand side are known, would provide an iterative formula to compute all the a_ such that k_1≥ m. 
	
	Accordingly, we propose to start by fixing all the coefficients  a_=h_E^/!D^ v(^E) such that k_1< m. 
	This is  equivalent to choosing m polynomials ψ_r∈^p-r() for r=0,…,m-1 such that ∂_x_1^r v(x_1^E,·)= ψ_r, where v(x^E_1,·) denotes the restriction of v to the hyperplane {x_1=x^E_1}. 
	We call this set of functions {ψ_r,0≤ r<m} the “Cauchy data” of v  in analogy to the case of the wave equation <cit.> (with m=2 and x_1 corresponding to the time variable). This step is referred to as the initialization.
	
	Next, given the Cauchy data of v, we propose the following precise ordering of the multi-indices  via three nested loops to compute iteratively the coefficients a_+m_1 in (<ref>): 
	
	
  * we start by looping over the length q= increasingly from q=0 to q=p-m;
	
  * at fixed q, we loop over the first component i_1 of , from i_1=0 to i_1=q;
	
  * at fixed q and i_1, we compute a_+m_1 for the indices (i_2,…,i_d) such that 
	i_2+⋯+i_d=q-i_1, 
	in any arbitrary order.
	 
	
	<Ref> summarizes the procedure comprised of the initialization and the iterative step. Does it fulfill the goal of constructing a quasi-Trefftz function? It does, as for fixed q and i_1 all the  coefficients a_+ appearing in the right-hand side of (<ref>) are already known: 
	(1) for ℓ_1+j_1<m they are fixed from the initialization;
	(2) for ℓ_1+j_1≥ m and |+|< ||+m they are computed at a previous iteration of the outer loop for q'<q;
	(3)  for ℓ_1+j_1≥ m, |+|=||+m  and (,)≠(m_1,) they are computed at the same iteration of the outer loop, 
	but at a previous iteration of the second loop for i'_1 =ℓ_1+j_1<i_1+m. 
	
	
	
	
	
	All the information about the PDE required by <Ref> is encoded in the values at ^E of the partial derivatives of order up to p-m of the coefficients α_, and of the right-hand side f.
	
	
	
	We will see in <ref> that, in order to treat non-homogeneous BVPs, we need to construct an elementwise approximate particular solution, i.e. an element of _f^p(E) for each mesh element E.
	To this purpose, it is sufficient to choose any Cauchy data (ψ_r)_r=0,…,m-1 and apply <Ref>.
	In practice we will choose ψ_r=0 for all r.
	
 	Next we turn to the question of the uniqueness of the quasi-Trefftz polynomial with given Cauchy data. 
	
	
	
	
	Assume that the regularity and the non-degeneracy conditions (<ref>) and (<ref>) are satisfied for an open, connected set E⊂^d,
	and let ^E∈ E. 
	Given any set of m polynomials ψ_r∈^p-r(ℝ^d-1) for r=0,…,m-1, there exists a unique v∈^p_f(E) such that ∂_x_1^r v(x_1^E,·)= ψ_r for r=0,…,m-1.
	
	
	Any polynomial v∈^p_f(E) is uniquely determined by the sets of its coefficients  
	{a_=h_E^/!D^ v(^E), ∈ℕ_0^d, ||≤ p} as in (<ref>). 
	The corresponding index set can be split  as
	
    {∈ℕ_0^d,  ||≤ p}  
    	= {∈ℕ_0^d,  ||≤ p,  k_1<m}∪{∈ℕ_0^d,  ||≤ p,  k_1≥ m} .

	On the one hand, the first set of coefficients a_ is uniquely defined by imposing ∂_x_1^r v(x_1^E,·)= ψ_r for r=0,…,m-1 since then D^ v(^E)=D^(k_2,…,k_d)ψ_k_1(x_2^E,…,x_d^E) for all  with k_1<m.
	On the other hand, Algorithm <ref> shows that it is possible to compute the coefficients a_ for all the indices  in the second set, with k_1≥ m, thus there exists a v∈^p_f(E) as desired.
	This is unique because the coefficients of any quasi-Trefftz v must satisfy equation (<ref>), thus each of those in the form a_+m_1 are determined by the a_ that appear earlier in the ordering given by the nested loops of <Ref>.
	In particular, if v_1,v_2∈_f^p(E) share the same first set of coefficients, then they coincide.
	
	The lowest-dimensional cases are ideal for a visual representation of the iterated loops in <Ref>.
	In the 1D case (d=1), only the outermost loop over q=i is present, and the algorithm reduces to the sequential computation of a_i+m from i=0 to i=p-m.
	In the 2D case (d=2), the innermost loop degenerates to the computation of the single coefficient a_i_1+m,q-i_1.
	<Ref> illustrates the dependence between the coefficients a_ of the monomial expansion of v∈^p_f(E) and their ordering as they are computed in <Ref> for d=2, m=2 and p=6.
	The dots in the quarter-plane of multi-indices =(k_1,k_2)∈ℕ_0^2  represent the coefficients a_k_1,k_2.
	Under the constraint that k_1+ k_2≤ p, these dots form a triangular shape in the plane.
	To initialize the algorithm we choose the Cauchy data, which consists of two functions ψ_0∈^p(ℝ^d-1) and ψ_1∈^p-1(ℝ^d-1) such that  v(x^E_1,·)= ψ_0 and  ∂_x_1 v(x^E_1,·)= ψ_1.
	This choice determines the coefficients a_0, k _2 with  0≤ k _2≤ p and a_1, k _2 with  0≤  k _2≤ p-1, represented by the shaded yellow area in the figure.
	All the other coefficients are then uniquely determined and can be computed in the iterative part of the algorithm using relation (<ref>).
	See <cit.> for a similar figure with d=3.
	
	
	
	
	
	
	
	
	

 §.§ Construction of a basis for the homogeneous equation

	In this section, we define, for any p∈, a basis for the quasi-Trefftz space ^p_0(E) for the homogeneous equation u=0 and use <Ref> to explicitly construct it.
	We denote
	
    S_d,p:=dim(^p(ℝ^d))=p+dd   and    
    	I_d,p,m:={(r,s)∈_0^2 0≤ r≤ m-1,
    1≤	s≤ S_d-1,p-r}.

	In order to define a set of quasi-Trefftz functions, we first choose m polynomial bases:
	
    {ψ_(r,s)}_(r,s)∈ I_d,p,msuch that, ∀ r∈{0,…,m-1},  {ψ_(r,s)}_s=1,…,S_d-1,p-r is a basis for ^p-r(ℝ^d-1).

	Their total cardinality is 
	
    N_d,p:=card(I_d,p,m)=S_d-1,p+…+S_d-1,p-m+1=p+d-1d-1+…+p+d-md-1.

	We then define the following set of N_d,p elements of ^p_0(E):
	
    ℬ_E^p:={b_(r,s)∈^p_0(E)  ∂_x_1^rb_(r,s)(x^E_1,·)=	ψ_(r,s),
    ∂_x_1^r'b_(r,s)(x^E_1,·)=0   for  r'=0,…, m-1, r'≠ r}_(r,s)∈ I_d,p,m.

	Equivalently, for each (r,s)∈ I_d,p,m, the element b_(r,s) is a polynomial of degree at most p satisfying the quasi-Trefftz property and with prescribed Cauchy data:
	
    D^ b_(r,s)(^E)=0 
    	     ∈_0^d,  ||≤ p-m,
    ∂_x_1^rb_(r,s)(x^E_1,·)=	ψ_(r,s)
    ∂_x_1^r'b_(r,s)(x^E_1,·)=0       r'=0,…, m-1,  r'≠ r.

	Next we show that the set ℬ_E^p of the elements b_(r,s) for all (r,s)∈ I_d,p,m forms a basis of ^p_0(E).
	
	Assume that the regularity and non-degeneracy conditions (<ref>) and (<ref>) are satisfied for an open, connected set E⊂^d,
	and let ^E∈ E. 
	Let {ψ_(r,s)}_s=1,…,S_d-1,p-r be a basis of ^p-r(ℝ^d-1) for each r∈{0,…,m-1}.
	Then the set ℬ^p_E in (<ref>) is a basis of the space ^p_0(E).
	
	
	Each b_(r,s)∈^p_E in (<ref>) is uniquely defined by
	<Ref>. 
	We need to verify that  ℬ_E^p is a spanning set of linearly independent functions.
	
	For any v∈^p_0(E) and r=0,…,m-1, since the restriction to {x_1=x_1^E} of the derivative ∂^r_x_1 v is a polynomial of degree p-r, there exist some coefficients {λ_(r,s)}_(r,s)∈ I_d,p,m⊂ℝ such that
	
    ∂^r_x_1 v(x^E_1,·)
    	=∑_s=1^S_d-1,p-rλ_(r,s)ψ_(r,s)
    	=∑_s=1^S_d-1,p-rλ_(r,s)∂^r_x_1b_(r,s)(x^E_1,·)
    	=∂^r_x_1(∑_s=1^S_d-1,p-rλ_(r,s)b_(r,s)_=:w_r)
    	(x^E_1,·).
 
	Set w:=∑_r=0^m-1w_r=∑_(r,s)∈ I_d,p,mλ_(r,s)b_(r,s).
	By (<ref>), ∂^r'_x_1 w_r(x_1^E,·)=0 for all r' r, 
	thus ∂^r_x_1 w(x_1^E,·)=∂^r_x_1 w_r(x_1^E,·)=∂^r_x_1 v(x_1^E,·) for all r=0,…,m-1.
	Hence, v and w are both elements of ^p_0(E) and they coincide by <Ref>, so that v
	is indeed a linear combination of b_(r,s). 
	This proves that ℬ_E^p is a spanning set for ^p_0(E).
	
	Next we show that the  polynomials {b_(r,s)}_(r,s)∈ I_d,p,m are linearly independent.
	Assume that ∑_(r,s)∈ I_d,p,mc_(r,s) b_(r,s)=0 for some coefficients {c_(r,s)}_(r,s)∈ I_d,p,m⊂ℝ.
	Then, fixing any r̃∈{0,…,m-1} and restricting to {x_1=x^E_1}, we obtain
	
    0=∑_(r,s)∈ I_d,p,mc_(r,s)∂_x_1^r̃ b_(r,s)(x^E_1,·)
    	=∑_s=1^S_d-1,p-r̃
    	c_(r̃,s)∂_x_1^r̃ b_(r̃,s)(x^E_1,·)
    	=∑_s=1^S_d-1,p-r̃c_(r̃,s)ψ_(r̃,s).

	This implies that c_(r̃,s)=0 for each (r̃,s)∈ I_d,p,m, since 
	{ψ_(r̃,s)}_s=1,…, S_d-1,p-r̃ are linearly independent. It concludes the proof.
	
	
	<Ref> implies
	that the conditions in the definition of ^p_0(E) are linearly independent:
	
    (^p(E))-card{∈_0^d| | |≤ p-m}=
    	p+dd-p+d-md
    	=N_d,p=(^p_0(E)).

	The equality between p+dd-p+d-md and the sum in (<ref>) follows from manipulations of the binomials and the formula ∑_k=0^nr+kk=r+n+1n for n,r∈_0, under the assumption that p≥ m.
	In particular, we have 
	
    (^p_0(E))=N_d,p
    	=
    	m     d=1,
     
    	m(p-m/2+3/2)     d=2,
     
    	m(1/2p^2+2p+11/6-1/2mp-m+m^2/6)     d=3.
 
	For m=2, this expression simplifies to N_2,p=2p+1 and N_3,p=(p+1)^2.
	This means that, for second-order PDEs, ^p_0(E) has the same dimension of the space of harmonic polynomials in ^d of degree at most p, see <Ref>.
	In the one-dimensional case, when increasing the polynomial degree p the dimension of the quasi-Trefftz space remains the same, but the space changes; see <cit.> for an example.
	
	
	
	
	
	Comparing against the dimension of the full polynomial space ^p(E), we observe that
	
    (^p_0(E))
    	=_p→∞(p^d-1)  ≪  (^p(E))
    	=p+dd=_p→∞(p^d).

	Thus, for large polynomial degrees p, the dimension of the quasi-Trefftz space is much smaller than the dimension of the full polynomial space of the same degree.
	
	Combined with <Ref>, this implies that smooth solutions of PDEs with smooth coefficients are approximated by ^p_0(E) and by ^p(E) with the same convergence rates with respect to the meshsize h, but with significantly less degrees of freedom in the quasi-Trefftz case.
	
	For f0, the space _f^p(E) is not a linear space but an affine one.
	Given any v_f∈_f^p(E), which can be constructed using <Ref> with any choice of Cauchy data, we have _f^p(E)=v_f+^p_0(E), therefore (_f^p(E))=(^p_0(E))=N_d,p.
	
	
	
	
	
	

§ DIFFUSION–ADVECTION–REACTION EQUATION

	Let Ω be an open, bounded, Lipschitz subset of ℝ^d and denote by Γ:=∂Ω its boundary. 
	We define the second-order, linear diffusion–advection–reaction operator , applied to  v:Ω→ℝ, as
	
    ℒv:=	div(-∇ v  +  v ) +σ v,

	with coefficients
	:Ω→ℝ^d× d, :Ω→ℝ^d and σ:Ω→ℝ.
	
	Let Γ_D and Γ_N be sufficiently regular subsets of the boundary such that Γ_D≠∅, Γ=Γ_D∪Γ_N and  Γ_D∩Γ_N=∅. Dirichlet and Neumann boundary conditions are imposed on Γ_D and Γ_N, respectively.
	Let () be the outward unit normal vector to the boundary at ∈Γ.
	
	Let f∈ L^2(Ω), g_D∈ H^1/2(Γ_D) and g_N∈ L^2(Γ_N).
	We consider the following boundary value problem for the diffusion–advection–reaction equation:
	
	
	div(-∇u +u) +σu    = f           in  Ω,

	u   =g_D          on Γ_D, 

	- ∇u  ·   =g_N          on Γ_N.
	
	
	We make the following assumptions on the data:
	
    =^⊤∈[L^∞(Ω)]^d× d,  ∈[W^1,∞(Ω)]^d,  σ∈ L^∞(Ω).

	In particular, this implies 
	∈ H(div;Ω).
	We will write ^2_L^∞(Ω) for the L^∞(Ω) norm of the 2-norm of the matrix , i.e. its spectral radius.
	We also assume that the ellipticity condition is satisfied, i.e. there exists a constant >0 such that
	
    ^⊤() ≥^2    ∀∈ℝ^d, 
    	 a.e. ∈Ω,

	where · denotes the Euclidean norm in ℝ^d.
	Choosing =(1,0,…,0)^⊤ in (<ref>) implies
	
    _11()≥ >0   a.e. ∈Ω.

	Under the ellipticity condition,  is a non-degenerate second-order partial differential operator; in particular, (<ref>) implies (<ref>) with m=2 and ^*=2_1 if the PDE coefficients are sufficiently smooth. 
	Moreover, we make the following assumption: if at least one among  and σ is not null, then there exists a constant σ_0>0 such that 
	
    σ()+1/2div(())≥σ_0   a.e. ∈Ω.

	
	When the advection term  is non-zero, we distinguish between the inflow and outflow parts of the boundary Γ, defined as 
	
    Γ_-:={∈Γ|() ·()<0 },     Γ_+:={∈Γ|() ·()≥0},

	respectively.
	Following e.g. <cit.> and <cit.>, we assume that 
	·≥ 0 on Γ_N when Γ_N is nonempty:
	
    Γ_N⊂Γ_+,     equivalently,    Γ_- ⊂Γ_D.

	This is done for simplicity but is also physically reasonable, for example, to model the movement of a substance knowing its concentration at the flow entrance but not at the exit.
	
	The classical variational formulation of problem (<ref>) is described in <cit.>. 
	In particular, <cit.> proves that, under these assumptions, (<ref>) admits a unique weak solution u∈ H^1(Ω). 
	
	
	

§ DISCONTINUOUS GALERKIN DISCRETIZATION

	

 §.§ Mesh assumptions and notation

	We assume that the domain Ω is a polytope of ℝ^d. 	
	We define polytopes by induction: a 0-dimensional polytope is a subset of ℝ^d containing a single point.
	For n∈, 1≤ n≤ d, a n-dimensional polytope of ℝ^d is a relatively open, bounded, connected and Lipschitz subset of a n-dimensional affine subspace of ℝ^d, such that its relative boundary is a finite union of (n-1)-facets, i.e., closures of (n-1)-dimensional polytopes.
	For n=1,2,3, polytopes are simply segments, polygons and polyhedra, respectively.
	
	We discretize the domain Ω using a polytopal mesh _h, where each mesh element E∈_h is a d-dimensional polytope  with  diameter h_E := sup_,∈ E- and the meshsize is h:= sup_E∈_hh_E.
	To analyze the DG method h-convergence, we consider a mesh sequence _ℋ:={_h}_h∈ℋ where ℋ is a countable subset of {h∈| h>0} having only 0 as accumulation point.
	
	For E∈_h, we denote by ρ_E the radius of the largest ball inscribed in E,
	and by E its d-dimensional measure.
	The boundary of E is indicated by ∂ E and  its (d-1)-dimensional measure by ∂ E. We define _E on  ∂ E as the unit outward normal vector to the element E.
	
	We consider conforming meshes: for all E, E'∈_h, E≠ E', the intersection ∂ E∩∂ E' is either empty or a common n-dimensional facet with n≤ d-1.
	Distinct facets of E may be co-planar.
	
	A mesh facet is a (d-1)-facet of a polytopal mesh element E∈_h, i.e. the closure of a (d-1)-dimensional polytope that is part of the boundary ∂ E.
	We denote by _h the set of all facets of _h.
	We assume that each F∈_h is either an interior facet for which there exist two distinct elements E_1, E_2∈_h such that F=∂ E_1∩∂ E_2, or a boundary facet for which there exists an element E∈_h such that F⊂∂ E∩∂Ω.
	The sets of interior and boundary facets are denoted by _h^I and _h^B, respectively.
	We assume that it is possible to collect the boundary facets where Dirichlet conditions are assigned in a set, denoted _h^D, and the boundary facets where Neumann conditions are assigned in another set, denoted  _h^N.
	Similarly, _h^- and  _h^+ denote the sets of inflow and outflow boundary facets.
	Thus _h=_h^I∪_h^D∪_h^N
	=_h^I∪_h^-∪_h^+, where all unions are disjoint.
	For F∈_h, we denote by	h_F the diameter of the facet F, by F its (d-1)-dimensional measure and we associate to it a unit normal vector _F. 
	If F∈_h^B then _F is chosen equal to , i.e. pointing outward from Ω.
	For each element E∈_h we define the set of all its facets as 
	_E:={F∈_h| F⊂∂ E}.
	The maximum number of mesh facets composing the boundary of a mesh element is denoted by
	
    N_∂:=max_E∈_hcard(_E).

	
	
	We assume to work with mesh sequences that satisfy the following properties:
	
	
	
  *  Star-shaped property:
	there exists 0< r_⋆≤1/2 such that,  for all h∈ℋ, each E∈_h is star-shaped with respect to a ball centred at some ∈ E and with radius r_⋆ h_E.
	
  * 
	Graded mesh(<cit.>):
	there exists >0 such that,  for all h∈ℋ, for all E∈_h and for all F∈_E,
	
    h_E≤ h_F.

	
	The star-shaped property <ref> implies the classical shape-regularity property (e.g. <cit.>):
	
    h_E≤ρ_E,     with  =r_⋆^-1.

	
	The star-shaped property <ref> is used in the DG stability analysis of <ref>, while the graded-mesh condition <ref> is only used to prove quasi-Trefftz convergence rates in <Ref>.
	The star-shaped property <ref> implies also 
	the “chunkiness” of the mesh sequence, which will be used
	in the proof of Theorem <ref>.
	
	Let E⊂^d be a polytope with diameter h_E that is star-shaped with respect to an open ball B of radius ρ_⋆ h_E, for 0<ρ_⋆≤1/2.
	Then, 
	
    h_E|∂ E|≤d/ρ_⋆|E|.

	
	
	Assume without loss of generality that B is centered at the origin .
	For each (d-1)-dimensional facet F∈_E of E, define Y_F:={=t|∈ F, 0≤ t<1}, the d-dimensional pyramid with basis F and apex at the origin.
	By the star-shapedness with respect to the origin of E, we have that E=⋃_F∈_EY_F and that Y_F_1∩ Y_F_2 has zero d-dimensional measure for different facets F_1,F_2∈_E.
	The d-dimensional measure of Y_F is |Y_F|=1/d H_F|F|, where the pyramid height H_F is the distance between the hyperplane Π_F containing F and the origin (special cases are the usual triangle area formula “half base times height”, and the 3D pyramid volume “one third base area times height”).
	Since Π_F contains a boundary facet and E is star-shaped with respect to B, Π_F cannot intersect B, thus H_F≥ρ_⋆ h_E.
	Then the assertion follows:
	
    h_E|∂ E|/|E|=h_E∑_F∈_E|F|/∑_F∈_E|Y_F|
    	=dh_E∑_F∈_E|F|/∑_F∈_EH_F|F|≤dh_E/inf_F∈_EH_F≤d/ρ_⋆.

	
	Inequality (<ref>) is an equality when each facet of E belongs to a hyperplane tangential to the ball B; this is the case, e.g., for all simplices, hypercubes, regular polygons and regular polyhedra.
	
	To apply the quasi-Trefftz approximation result of <Ref>, E has to be star-shaped with respect to the point ^E used to define the local discrete space ^p_f(E): this point need not be the center of the ball in <Ref>.
	
	<Ref> ensures that, under assumption <ref>, inequality (<ref>) holds for all E∈_h with ρ_⋆=r_⋆.
	
	
	We recall the definition of the broken Sobolev spaces:
	
    H^m(_h):=   {φ∈ L^2(Ω) |φ_|_E∈ H^m(E)   ∀ E ∈_h },    m∈_0,
    
    	H(div;_h):=   {∈[ L^2(Ω)] ^d|_|_E∈ H(div;E)    ∀ E ∈_h }.

	
	We use the standard DG notation <cit.> for averages · and jumps · of any scalar function φ∈ H^1(_h) and any vector-valued function  ∈ [H^1(_h)]^d across the mesh facets: 
	
    φ:=    φ_|_E_1+φ_|_E_2/2,        :=    _|_E_1+_|_E_2/2,
    φ:=    φ_|_E_1_E_1+φ_|_E_2_E_2,       :=    _|_E_1·_E_1+_|_E_2·_E_2,
    	  on   F=∂ E_1 ∩∂ E_2,
    	
       φ :=    φ_|_E,         :=    _|_E,
    φ:=    φ_|_E_E,       :=    _|_E·_E,
    	    on   F⊂∂ E ∩∂Ω.

	We will use the “DG magic formula” <cit.>:
	for all φ∈ H^1(_h) and for all ∈[H^1(_h)]^d,
	
    ∑_E∈_h∫_∂  E·_ Eφ =
    	∑_ F∈_h^I∫_ F(·φ +  φ)
    	+∫_∂Ω·φ.

	
	For p∈_0, we define the broken polynomial space of degree at most p on the mesh, as
	^p(_h):={v∈ L^2(Ω) | 
	v_|_E∈^p(E)    ∀ E ∈_h }.
	Finally, we recall the following discrete inverse trace inequality for mesh sequences _ℋ enjoying the star-shaped property <ref>: given p∈_0, there exists >0, depending only on d, p, and r_⋆ (defined in <ref>) such that
	
    v^2_L^2(∂ E)≤ h_E^-1v^2_L^2(E)    ∀ h∈ℋ,    E∈_h,    v∈^p(E).

	Inequality (<ref>) follows combining <cit.> and <cit.>, and recalling that shape-regularity (<ref>) follows from property <ref>.
	Since in the DG scheme we will use the quasi-Trefftz space, which is a subset of the full polynomial space, this inequality can be applied.
	
	
	
	
	
	

 §.§ Discontinuous Galerkin formulation

	We describe the DG variational formulation of the dif­fu­sion–ad­vec­tion–reaction problem (<ref>).
	We consider the Symmetric Interior Penalty Galerkin (SIPG) method <cit.> to handle the diffusion term, and the upwind DG method to handle the advection–reaction terms, following mostly 
	<cit.>.
	
	We define the DG scheme and carry out the abstract error analysis for a general discrete sub­space V_h of the broken polynomial space ^p(_h). 
	We will choose a global quasi-Trefftz space in <ref> and prove convergence rates for it.
	Following the non-conforming analysis of <cit.> we define 
	
    V_*:= H^1(Ω) ∩ H^2(_h),      V_*h := V_* + V_h.

	
	Let u be the weak solution of problem (<ref>).
	We assume u∈ V_*, which is guaranteed e.g. if Γ_N=∅, Ω is convex, and the PDE data are sufficiently smooth, by e.g. <cit.>.
	[The choice of requiring H^2 elementwise regularity is made only for simplicity: what we really need is that the trace of ∇ u is in L^2(∂ E)^d for all elements, which is ensured by u∈ V_*:=H^1(Ω)∩ H^3/2+ϵ(_h) for some ϵ>0.]
	
	We consider the following discretization of problem (<ref>):  
	
    Find  u_h ∈ V_h  such that _h^dar(u_h,v_h)=L_h(v_h)   ∀ v_h ∈ V_h,

	with the DG bilinear form 
	_h^dar: V_*h× V_h →ℝ,
	
    _h^dar(w,v_h):=    _h^d(w,v_h)+_h^ar(w,v_h),
    	
    _h^d(w,v_h):=   ∑_E∈_h∫_E ∇ w ·∇  v_h
       
    	+
    	∑_F∈_h^I∫_F (-∇ w ·v_h- w·∇ v_h  
    	+ γK_F/h_Fw·v_h)
       
    	+
    	∑_F∈_h^D∫_F (-∇ w · v_h- w ∇ v_h ·+ γK_F/h_F w v_h),
    	
    _h^ar(w,v_h):=   ∑_E∈_h∫_E  (-( w) ·∇  v_h + σ w  v_h)
       +
    	∑_F∈ℱ_h^I∫_F (   w·v_h+1/2|·_F|w·v_h)+
    	∑_F∈ℱ_h^+∫_F (   w) · v_h,

	and the linear form L_h:V_h→,
	
    L_h(v_h):=∑_E∈_h∫_E  f v_h - ∑_F∈ℱ_h^N∫_F  g_N v_h +
    	∑_F∈ℱ_h^D∫_F g_D(- ∇ v_h  ·  + γK_F/h_F v_h)
    	-∑_F∈ℱ_h^-∫_F g_D· v_h.

	The bilinear form _h^dar depends on the penalty parameters γ,K_F>0 that penalize the jumps of the function values.
	The quantity γ>0 is a dimensionless constant independent of the diffusion coefficient , while K_F is a diffusion-dependent penalty parameter defined on each facet such that ≤ K_F≤_L^∞(E_1∪ E_2) for all F∈_h^I with F=∂ E_1 ∩∂ E_2, and ≤ K_F≤_L^∞(E) for all F∈_h^B with F⊂∂ E ∩Γ.
	
	Problem (<ref>) is independent of the choice of the normal _F on the internal facets, since its only occurrence in _h^dar is inside the absolute value.
	
	The term on the interior facets in _h^ar is the penalization form of the classical upwind flux, <cit.>.
	Indeed, if ↦() ·_F() does not change sign in any given F∈_h^I, then, for F=∂ E_1 ∩∂ E_2 and φ∈ H^1(_h),
	
    φ·_F +1/2·_Fφ·_F
    	=φ_upw·_F
    	  where φ_upw: = φ_|_E_2   if ·_E_1<0,
    φ_|_E_1   if ·_E_1>0,
    φ   if ·_E_1=0.

	
	
	
	The diffusion part of the DG formulation (<ref>) corresponds to the formulation in <cit.> with α=0 (no reaction), ϵ=-1 (SIPG), and σ_e^1=0 for all facets (no gradient jump stabilization term).
	The penalty term is slightly different: on each facet, <cit.> uses a number divided by a power of the (d-1)-dimensional measure of the facet, while we use a constant γ (independent of the facet) times a diffusion-dependent penalty parameter K_F, divided by the facet's diameter h_F, following <cit.>.
	In turn, <cit.> assumes piecewise-constant diffusion, uses diffusion-dependent weights for the average and K_F is chosen as the harmonic mean of (scalar)  across F. 
	This penalty strategy is particularly important in the advection-dominated/reaction-dominated regimes to tune automatically the penalty parameter and reduce spurious oscillations, see <cit.> and section <ref> below.
	
	For what concerns the advection–reaction terms, (<ref>) follows <cit.> with η=1 and with the right-hand side as in  <cit.>.
	
	
	
	
	

 §.§ Mesh-dependent norms
	
	For all v∈ V_*h we define four mesh-dependent norms and the seminorm |·|_J:
	
    v^2_d:=   ∑_E∈_h∫_E∇ v·∇ v+v_J^2,     v_J^2:=∑_F∈_h^IγK_F/h_F∫_F v^2+∑_F∈_h^DγK_F/h_F∫_Fv^2,
    v^2_ar:=   σ_0v^2_L^2(Ω)+ 1/2∑_F∈_h∫_F·_Fv^2,
    v^2_dar:=   v^2_d+	v^2_ar, 
    v^2_dar,*:=   v^2_dar+∑_E∈_hh_E^1/2∇ v·_E^2_L^2(∂ E)+∑_E∈_h_L^∞(E)v^2_L^2(∂ E).

	We write ^1/2 for the unique positive-definite matrix field such that ^1/2^1/2= in Ω.
	Note that ·_d is a norm because we have assumed that Γ_D is not empty.
	
	
	
	

 §.§ Well-posedness, stability, quasi-optimality

	
	The aim of this section is to prove the well-posedness of the discrete DG problem (<ref>) and the quasi-optimality error estimates of the DG method.
	The proof of the next theorem relies on Lax–Milgram theorem and consists of verifying the three assumptions of the abstract result in <cit.>:
	consistency, discrete coercivity and boundedness.
	
	<Ref> holds for arbitrary polynomial spaces V_h⊂^p(_h) (more generally, for any discrete space for which an inverse trace inequality such as (<ref>) holds).
	With this generality, we cannot immediately apply standard results such as those in <cit.>: their analysis of the advection–reaction bilinear form relies on the “boundedness on orthogonal subscales” <cit.>, whose proof requires that (piecewise) partial derivatives of elements of V_h belong to V_h, a property satisfied by ^p(_h) but not all its subspaces.
	Similar assumptions are common in the literature, e.g. <cit.>. Many works also assume piecewise-constant diffusion, e.g. <cit.>, <cit.>, <cit.>, while we are interested in the general case ∈ [L^∞(Ω)]^d× d.
	These hypothesis are not necessary and are often made for simplicity of presentation, however we can not directly rely on their analysis.
	We refer to <cit.> for a more general analysis of an inconsistent variant of the SIP-upwind DG method for second-order PDEs with nonnegative characteristic form.
	
	<Ref> gives an explicit estimate, which in <ref> will be combined with
	the local approximation bound (<ref>) of the quasi-Trefftz space.
	In particular, our analysis for the discrete coercivity of the diffusion bilinear form follows <cit.>, while the continuity is similar to <cit.>. 
	Concerning the advection–reaction bilinear form, for the coercivity we follow <cit.>, while, 
	to prove continuity avoiding
	conditions like <cit.>, we estimate the quantity
	∑_E∈_h∫_E  ( v) ·∇  w_h  
	using the diffusion norm w_h_d for the second term. 
	
	
	
	Under the assumptions on the BVP and the mesh made in <ref>, let γ_0:=^2_L^∞(Ω)/ ^2N_∂ > 0 with  defined in (<ref>), N_∂ in (<ref>) and  in (<ref>), and recall σ_0 from (<ref>). 
	Assume that the penalty parameter satisfies γ>γ_0, and set 
	
    α:=1-√(γ_0/γ),    
    	M:=5+ _L^∞(Ω)/√(σ_0)+	σ_L^∞(Ω)/σ_0
    	+(_L^∞(Ω)/γ)^1/2.

	Then the bilinear form _h^dar is coercive on V_h in ·_dar norm:
	
    _h^dar(v_h,v_h)≥αv_h^2_dar    ∀ v_h∈ V_h.

	The DG variational problem (<ref>) admits a unique solution u_h∈ V_h, for any subspace 
	V_h⊂^p(_h).
	The bilinear form _h^dar is bounded on V_*h× V_h in ·_dar,*–·_dar norms: 
	
    _h^dar(v,w_h)≤ Mv_dar,*w_h_dar    ∀ (v,w_h)∈ V_*h× V_h.

	The weak solution u of the BVP (<ref>)
	solves the variational problem (<ref>), i.e. (<ref>) is consistent.
	Moreover, the following quasi-optimality error estimate holds true:
	
    u-u_h_dar≤(1+M/α) inf_v_h∈ V_hu-v_h_dar,*.

	
	
	
	Discrete Coercivity: First we establish the coercivity of the diffusion bilinear form _h^d on V_h with respect to the ·_d-norm, then we show that the advection–reaction bilinear form _h^ar is coercive on V_h with respect to the ·_ar-norm.  Combining these two results, we deduce the discrete coercivity of the diffusion–advection–reaction bilinear form a^dar_h with respect to the ·_dar-norm.
	
	Let v_h∈ V_h.
	Applying Young's inequality to the bound (<ref>) proved in the appendix we deduce 
	
    |∑_F∈_h^I∪_h^D∫_F  ∇ v_h·v_h|
    	≤_L^∞(Ω)/( N_∂/γ)^1/2(   1/2∑_E∈_h^1/2∇ v_h_L^2( E)^2
    	+  1/2v_h_J^2).

	Using this bound we achieve
	_h^d(v_h,v_h)≥(1-	_L^∞(Ω)/
	(N_∂/γ)^1/2)v_h_d^2.
	Choosing γ large enough, γ > γ_0=  ^2_L^∞(Ω)/k_min^2
	N_∂, we obtain the discrete coercivity 
	_h^d(v_h,v_h)≥ (1-√(γ_0/γ))v_h_d^2
	of the diffusion bilinear form.
	
	On the other hand, integration by parts yields
	
    ∑_E∈_h∫_E  ( v_h) ·∇  v_h=∑_E∈_h∫_E ·∇(v_h^2/2)
    	=-∑_E∈_h∫_E  div() v_h^2/2 +∑_E∈_h∫_∂ E·_E v_h^2/2.

	Applying the DG magic formula (<ref>) on the last term with = and φ=v_h^2, using the formula 
	1/2·v_h^2= v_h·v_h-1/4|v_h|^2
	on each interior facet, and observing that =0 on F∈_h^I by the regularity assumption (<ref>), we get
	
    ∑_E∈_h∫_∂ E·_E v_h^2/2=
    	∑_F∈_h^I∫_F v_h·v_h+1/2∑_F∈_h^B∫_F· v_h^2.

	Combining the previous steps, the bilinear form _h^ar(v_h,v_h) can be rewritten as follows:
	
    ∑_E∈_h∫_E  (σ+div/2) v_h^2 
    	-1/2∑_F∈_h^-∫_F· v_h^2
    	+1/2∑_F∈_h^+∫_F· v_h^2+
    	1/2∑_F∈_h^I∫_F  |·_F|v_h^2.

	Recalling the definition (<ref>) of Γ_± and using the lower bound (<ref>) on σ+1/2div,
	we deduce that
	
    _h^ar(v_h,v_h)≥σ_0v_h_L^2(Ω)^2+
    	1/2∑_F∈_h∫_F  |·_F|v_h^2
    	=v_h_ar^2,

	hence the coercivity constant for the advection–reaction bilinear form is 1.  
	Since ^dar_h=^d_h+^ar_h and 
	·_dar^2=·_d^2+·_ar^2, we obtain the discrete coercivity (<ref>).
	The discrete coercivity implies the well-posedness of the discrete DG problem  (<ref>) since it is a sufficient condition for discrete stability <cit.>.
	
	
	Consistency:
	Let u∈ V^* be the weak solution of problem (<ref>). 
	We show that u satisfies the variational problem (<ref>), i.e.
	_h^dar(u,v_h)=L_h(v_h) for all v_h ∈ V_h.
	We multiply (<ref>) by v_h∈ V_h, integrate by parts on each element E and sum over all the elements:
	
    -∑_E∈_h∫_E (-∇ u+ u) ·∇  v_h +
    	∑_E∈_h∫_∂ E(-∇ u+ u) ·_E v_h +
    	∑_E∈_h∫_E  σ u  v_h= 	∑_E∈_h∫_E f  v_h.

	Using the DG magic formula (<ref>)  with =-∇ u+ u and φ=v_h on the second term and observing that -∇ u+ u =0 on each interior facet since -∇ u + u belongs to H(div;Ω), and using the Dirichlet and Neumann boundary conditions (<ref>)–(<ref>), we find
	
    -∑_E∈_h∫_E (-∇ u+ u) ·∇  v_h -
    	∑_F∈_h^I∫_F  ∇ u ·v_h +
    	∑_F∈_h^I∪_h^+∫_F    u·v_h
       
    	-	∑_F∈_h^D∫_F  ∇ u · v_h+∑_E∈_h∫_E  σ u  v_h= ∫_Ω f  v_h - ∑_F∈_h^N∫_F g_N v_h   
    	- ∑_F∈_h^-∫_F  (   g_D)· v_h .

	Using the fact that u=0 on each interior facet since u∈ H^1(Ω), and that u satisfies the Dirichlet boundary condition (<ref>), the variational formulation (<ref>) evaluated in u coincides with the above equality, 
	implying the consistency of the DG scheme.
	
	Boundedness:
	Let (v,w_h)∈ V_*h× V_h. We decompose the bilinear form _h^dar in eight terms:
	
    _h^dar(v,w_h)=   ∑_E∈_h∫_E ∇ v ·∇  w_h +∑_F∈_h^I∪_h^DγK_F/h_F∫_F  v·w_h-
    	∑_F∈_h^I∪_h^D∫_F  ∇ v ·w_h
       
    	-∑_F∈_h^I∪_h^D∫_F  v·∇ w_h 
    	+∑_E∈_h∫_E  (-( v) ·∇  w_h + σ  v w_h)
       +∑_F∈_h^I∫_F    v·w_h+
    	1/2∑_F∈_h^I∫_F  |·_F|v·w_h  +
    	∑_F∈_h^+∫_F (   v) ·_F w_h
    
    	=:   𝔗_1+𝔗_2+𝔗_3+𝔗_4+𝔗_5+𝔗_6+𝔗_7+𝔗_8.

	The Cauchy–Schwarz inequality and the ellipticity condition (<ref>) yield
	
    𝔗_1 + 𝔗_2≤    v_dw_h_d,
    𝔗_7 + 𝔗_8≤    2 v_ar w_h_ar,
    𝔗_5≤    _L^∞(Ω)/√(σ_0)v_arw_h_d+
    	σ_L^∞(Ω)/σ_0v_ar w_h_ar.

	Moreover, using the continuity of  (<ref>) and the Cauchy–Schwarz inequality, we infer
	
    𝔗_6   ≤(2∑_F∈_h^I∫_F ·_Fv^2)^1/2 (1/2∑_F∈_h^I∫_F ·_Fw_h^2)^1/2
       ≤(  ∑_E∈_h_L^∞(E)v^2_L^2(∂ E))^1/2  w_h_ar≤v_dar,* w_h_ar,

	where in the second step we use the formula 2v^2=1/2(v_1+v_2)^2≤ v_1^2+v_2^2. 
	
	Since h_F≤ h_E for all F∈_E, E∈_h, and ≤ K_F for all F∈_h, from to the bound (<ref>) we get
	𝔗_3≤(_L^∞(Ω)/γ)^1/2v_dar,*w_h_d.
	Finally, we control the remaining term using bound (<ref>):
	𝔗_4≤(γ_0/γ)^1/2v_dw_h_d≤v_dw_h_d.
	By combining all these bounds we infer the boundedness of _h^dar with 
	M as in the statement.
	
	Since discrete stability, consistency and boundedness hold, we conclude applying 
	<cit.>.
	
	
	
	Given the quasi-optimality inequality (<ref>), the convergence of the DG method follows studying
	the approximation properties of the particular discrete space V_h chosen.
	In <Ref> we do this for the h-convergence of the quasi-Trefftz version of the DG scheme.
	
	Among all the constants and the parameters appearing in <Ref>, only the maximal number of facets per element N_∂ (<ref>) and the inverse trace constant  (<ref>) depend on the mesh _h and on the discrete space V_h.
	The first one is easily computed, the second one is well-studied for some classes of meshes (e.g. 
	<cit.>).
	
	
	
	
	
	
	
	

§ QUASI-TREFFTZ DG DISCRETIZATION

	We fix a point ^E∈ E for each mesh element E∈_h.
	Since the diffusion–advection–reaction operator , defined in (<ref>), is a linear partial differential operator of order m=2, the quasi-Trefftz space (<ref>) for the equation u = f on a mesh element E∈_h is 
	
    ^p_f(E)={ v∈^p(E) | D^ℒv (^E)=D^ f (^E)   ∀∈_0^d,  ||≤ p-2},     p∈.

	For p=1 we have ^1_f(E)=^1(E), so we fix p≥2.
	Recall (<ref>): the non-degeneracy condition (<ref>) is ensured by ellipticity (<ref>).
	The space ^p_f(E) is well-defined if the PDE coefficients ,  and σ and the source term f are sufficiently smooth. 
	We expand the operator v=div(-∇ v+ v)+σ v in the form (<ref>) using the Leibniz product rule:
	
    ℒv =∑_j=1^d[∑_m=1^d(-_jm  D^_j+_m v-D^_j_jmD^_m v ) +	
    	_j 	 D^_j v+(D^_j_j) v ]+σ v.

	Recalling the regularity hypothesis (<ref>) made for general differential operators, assume
	
    ∈ C^p-2 (E)^d× d,     div, ∈ C^p-2(E)^d,    div, σ, f∈ C^p-2(E).

	where the matrix divergence div is taken column-wise.
	Then the quasi-Trefftz space (<ref>) for the diffusion–advection–reaction equation is well-defined and all the results in <ref> apply.
	The detailed description of <Ref> for the homogeneous diffusion–advection–reaction equation, for the case d=1, d=2, and for the general d-dimensional case, can be found in <cit.>.
	
	We discretize the DG formulation (<ref>) choosing as trial space the global quasi-Trefftz space 
	^p_f(_h):={v∈ L^2(Ω) | v_|_T∈^p_f(E)   ∀ E ∈_h } and as test space the global quasi-Trefftz space 
	^p_0(_h):={v∈ L^2(Ω) | v_|_T∈^p_0(E)   ∀ E ∈_h }.
	The quasi-Trefftz DG method is then: 
	
    Find  u_h ∈^p_f(_h)  such that   _h^dar(u_h,v_h)=L_h(v_h) 
    	    ∀ v_h ∈^p_0(_h).

	If the source term f vanishes, existence and uniqueness of u_h in (<ref>) follow from 
	<Ref>.
	However, in the general case with f≠ 0, 
	<Ref> does not apply directly,
	since the trial and test spaces are different.
	In this case, we choose a lifting u_h,f∈^p_f(_h).
	This can be computed by applying <Ref> in each element E∈_h, with any choice of Cauchy data (ψ_0,ψ_1)∈^p(^d-1)×^p-1(^d-1).
	For simplicity, in the 
	experiments of <ref> we take ψ_0=ψ_1=0.
	Then we consider the problem
	
    Find  u_h,0∈^p_0(_h)  s.t.   _h^dar(u_h,0,v_h)=L_h(v_h)-_h^dar(u_h,f,v_h)     ∀ v_h ∈^p_0(_h).

	Here trial and test spaces coincide and <Ref> applies, so problem (<ref>) admits a unique solution u_h,0.
	Then u_h=u_h,0+u_h,f is the solution to (<ref>).
	
	
	

 §.§ h-convergence of the quasi-Trefftz DG method

	
	
	The aim of this section is to infer the convergence rate in h for the quasi-Trefftz Galerkin error u-u_h measured in the ·_dar-norm.	
	We first adapt the DG stability analysis of <ref> to problem (<ref>), which is posed on an affine trial space.
	
	
	Under the assumptions of <Ref>
	and (<ref>), problem (<ref>) is well-posed and 
	the following error estimate holds true:
	
    u-u_h_dar≤(1+M/α) inf_v_h∈^p_f(𝒯_h)u-v_h_dar,*.

	
	
	Under the assumptions made, <Ref> ensures the existence of u_h,f∈_f^p(_h).
	<Ref> implies the existence of u_h,0 solving (<ref>), and so of u_h=u_h,0+u_h,f solving (<ref>).
	The uniqueness of u_h follows because (<ref>) is a square discrete linear problem as 
	(^p_f(_h))=(^p_0(_h)).
	
	
	To show (<ref>), we adapt Céa lemma to the affine space in (<ref>).
	For any v_h∈^p_f(𝒯_h), u_h-v_h∈^p_0(𝒯_h) and therefore 
	_h^dar(u-u_h, u_h-v_h)=0 by (<ref>) and the consistency of the scheme.
	Coercivity and continuity of the DG formulation yield the estimate
	
    αu_h-v_h_dar^2    ≤_h^dar(u_h-v_h,u_h-v_h)
       =^dar_h(u-v_h,u_h-v_h)
    	≤ Mu-v_h_dar,*u_h-v_h_dar    ∀ v_h∈^p_f(_h).

	Estimate (<ref>) follows by applying the triangle inequality and recalling that 
	·_dar≤·_dar,*.
	
	
	
	From this quasi-optimality result we deduce the optimal convergence rate for the quasi-Trefftz DG method, using the approximation estimate (<ref>).
	We define the broken space C^q(_h):={v ∈ L^2(Ω) | v_|_E∈ C^q(E)    ∀ E ∈_h } for q∈_0 and recall that V_* :=H^1(Ω) ∩ H^2(_h) 
	
	
	Let p∈ and let u∈ V_*∩ C^p+1(_h) solve the BVP (<ref>) under the assumptions made in <ref> and (<ref>).
	Let u_h solve (<ref>), with a mesh _h as in <ref>, and
	penalty parameter γ as in <Ref>.
	Assume that each mesh element E is star-shaped with respect to ^E.
	Then, the following error bound holds:
	
    u-u_h_dar≤    (1+M/α)d^p/p!(∑_E∈_h G_E Eh_E^2pu^2_C^p+1(E))^1/2
    ≤    (1+M/α)d^p/p!|Ω|^1/2  h^p  max_E∈_h(G_E^1/2u_C^p+1(E))
    	,    where
    
    	G_E:=[(1+   d/r_⋆)_L^∞(E) 
    	+d^2/(p+1)^2(2γ_L^∞(_E)d/r_⋆
    	+2_L^∞(E)d/r_⋆ h_E+σ_0 h_E^2
    	)],

	with α and M as in <Ref>, r_⋆, in <ref>–<ref>, σ_0 in (<ref>) and 
	_E:=E∪⋃_F=∂ E∩∂ E'∈_h^IE'  
	the patch of mesh elements adjacents to the element E.
	
	     
	We estimate the quantity
	inf_v_h∈^p_f(_h)u-v_h_dar,* on the right-hand side of the quasi-optimality inequality (<ref>), with the ·_dar,*-norm defined in <ref> for  v∈ V_*h=V_*+^p_f(_h).
	We use v^2=(v_1-v_2)^2≤ 2(v_1^2+v_2^2) on internal facets F=∂ E_1∩∂ E_2,
	and v^2=v^2 on boundary facets F. 
	We recall h_E≤ h_F for F∈_E by the graded-mesh assumption (<ref>), and that K_F≤_L^∞(_E) for all facets F∈_E. 
	Using these facts, we rearrange the sums over parts of the mesh skeleton as sums over elements and obtain the bound:
	
    v^2_dar,*≤∑_E∈_h(   _L^∞(E)∇ v^2_L^2(E) +2 γ/h_E_L^∞(_E)v^2_L^2(∂ E)
    	+σ_0v^2_L^2(E)
       
    	+_L^∞(E)h_E∇ v^2_L^2(∂ E)
    	+2_L^∞(E)v^2_L^2(∂ E)).

	Next, we use the definition of the ·_C^m-norms and obtain
	
    v^2_dar,*≤∑_E∈_h(   _L^∞(E)E∇ v^2_C^0(E) +
    	2γ/h_E_L^∞(_E)∂ Ev^2_C^0( E)
    	+σ_0Ev^2_C^0(E)
       
    	+_L^∞(E)∂ Eh_E∇ v^2_C^0( E)
    	+2_L^∞(E)∂ Ev^2_C^0(E)).

	Considering the quantity of interest and using the quasi-Trefftz approximation estimate (<ref>) we get 
	
    inf_v_h∈^p_f(_h)u-v_h^2_dar,*
    ≤   ∑_E∈_hinf_v_h∈^p_f(E)[(E+∂ Eh_E)_L^∞(E)∇ (u-v_h)^2_C^0(E)
       +(2( γ/h_E_L^∞(_E)
    	+_L^∞(E))∂ E
    	+σ_0E)u-v_h^2_C^0(E)]
    	
    ≤   ∑_E∈_h[(E+∂ Eh_E)_L^∞(E)d^2p/(p!)^2 h_E^2pu^2_C^p+1(E). 
       .
    	+(2( γ/h_E_L^∞(_E) +_L^∞(E)) ∂ E+σ_0E)
    	d^2(p+1)/((p+1)!)^2 h_E^2(p+1)u^2_C^p+1(E)].

	By the chunkiness property h_E|∂ E|≤d/r_⋆|E| (<ref>) on the mesh,
	the last expression is bounded by
	
    inf_v_h∈^p_f(_h)u-v_h^2_dar,*≤    d^2p/(p!)^2∑_E∈_h[(1+d/r_⋆)E_L^∞(E) 
    	+d^2/(p+1)^2Eh_E^2×
       (2	γ/h_E_L^∞(_E)d/r_⋆ h_E^-1
    	+2_L^∞(E)d/r_⋆ h_E^-1+σ_0)] h_E^2pu^2_C^p+1(E).

	Combining this bound  with the quasi-optimality inequality (<ref>) yields the assertion.
	
	
	The estimate (<ref>) can immediately be adapted to the case 
	where a different polynomial degree p_E∈ is used in each element.
	
	For the quasi-Trefftz DG error estimate (<ref>) to hold, the solution u needs to belong to C^p+1(_h), which is a stronger regularity assumption than the usual u∈ H^p+1(_h).
	This is a consequence of the approximation estimate (<ref>), which is based on a Taylor argument.
	For the Trefftz space the analysis has been extended to the case of solutions in H^p+1(_h) using the fact that the “averaged Taylor polynomials” of exact solutions are Trefftz functions <cit.>. 
	However, we cannot use this argument since, in general, averaged Taylor polynomials are not quasi-Trefftz functions and, to our knowledge, a quasi-Trefftz convergence analysis using Sobolev norms is still missing <cit.>.
	Apart from this difference, (<ref>) shows optimal h-convergence rates in the ·_dar-norm.
	
	
	
	

§ NUMERICAL EXPERIMENTS

	
	The quasi-Trefftz DG method has been implemented using  <cit.> and  <cit.>[Reproduction material is available in <cit.>.].
	The computations were performed with parallelization limited to 16 threads on a server with two Intel(R) Xeon(R) CPU E5-2687W v4, with 12 cores each.
	The derivatives required for the computation of the quasi-Trefftz functions are computed using the symbolic differentiation capabilities of , with evaluation of <Ref> performed in parallel elementwise. 
	To initialize the algorithm, we choose the Cauchy data ψ_0=ψ_1=0 for constructing the lifting u_h,f∈^p_f(_h), and centered monomial bases of ^p(^d-1) and of ^p-1(^d-1) as Cauchy data for the quasi-Trefftz basis (<ref>) of ^p_0(_h).
	We use a direct solver based on the  library.
	The diffusion-dependent penalty parameter K_F is chosen equal to k_min on each facet F∈_h^I. 
	Additional experiments and details on a 2D Matlab implementation for the homogeneous case can be found in <cit.>.
	
	

 §.§ Non-homogeneous Dirichlet problem

	We consider a non-homogeneous diffusion-dominated problem in the unit cube Ω=(0,1)^3.
	The PDE coefficients and the solution are chosen as
	
    =(1+x_1+x_2+x_3)_3, =[ sin x_1; sin x_2; sin x_3 ],σ=4/1+x_1+x_2+x_3, 
    	u_ex = sin(π(x_1+x_2+x_3)).

	Here _3 is the 3×3 identity matrix.
	The right-hand side f is constructed in order to manufacture the solution u_ex in (<ref>).
	Dirichlet boundary conditions are imposed on the entire boundary of the domain matching the exact solution.
	We consider a sequence of tetrahedral meshes obtained by refinement of an unstructured quasi-uniform tetrahedral initial mesh.
	The penalization parameters are chosen as γ =50p^2 and K_F==1.
	
	In <Ref> we show the absolute errors of the quasi-Trefftz and the standard (full-polynomial space) DG methods, for the same polynomial degrees p∈{2,3,4} and under mesh refinement. 
	We observe that the quasi-Trefftz DG method converges with the expected orders h^p+1 in the L^2(Ω) norm and h^p in the ·_dar-norm, the latter in agreement with <Ref> and both norms matching the convergence rates of the standard DG method.
	The errors of the two methods are similar, but the quasi-Trefftz DG error is slightly larger by a constant factor (within a factor 1.65 for the ·_dar-error for h<0.5).
	
	The assembly of the quasi-Trefftz DG linear system has an overhead given by the computation of the basis functions and the particular approximate solution u_h,f.
	To assess this, in <Ref> we compare the computing time of the quasi-Trefftz and the full-polynomial version of the DG method. 
	We observe that, as soon as h is sufficiently small or p large, the quasi-Trefftz version requires considerably less time: the basis computation time is offset by the reduced number of degrees of freedom. 
	
	
	
	
	
	
	The left panel in Figure <ref> shows how the advantage provided by the quasi-Trefftz approach improves with higher polynomial degrees p.
	This seems to confirm the exp(-bp^1/(d-1)) behavior of Trefftz and quasi-Trefftz errors, as opposed to the 
	exp(-bp^1/d) dependence for methods based on classical polynomial spaces, see <cit.> and <cit.>.
	Note however that we are not aware of any rigorous quasi-Trefftz p-convergence result.

    

  §.§.§ Conditioning

	
	We study the condition number for the quasi-Trefftz DG method and the standard DG method. 
	We consider a 2D Dirichlet problem in the unit square Ω=(0,1)^2 with coefficients
	=(1+x_1+x_2)_2, =(1,0)^⊤, and σ=3/1+x_1+x_2, with _2 the identity matrix in ℝ^2× 2.
	The right panel of Figure <ref> shows the condition numbers of the matrices for the quasi-Trefftz DG method and the standard DG methods.
    We see that for the quasi-Trefftz DG the condition number grows asymptotically as 𝒪(h^-0.5) or less for all p∈, and for the standard DG it grows asymptotically as 𝒪(h^-1.5) for all p∈.
	However, for increasing values of p, the condition number of the quasi-Trefftz DG appears to grow exponentially.
	This is to be expected since we initialize the quasi-Trefftz Cauchy data by monomials.
	The selection of Cauchy data that ensure better-conditioned quasi-Trefftz bases is currently under investigation.
	
	
	
	

 §.§ Advection-dominated problems

	
	We investigate advection-dominated problems to assess the capabilities of the method also in such more challenging setting.
	In section <ref> we consider a solution that presents an internal layer while in section <ref> a solution with boundary layers and corner singularities. 
	In both examples the advection field  is divergence-free and the reaction σ=0, hence assumption (<ref>) is violated.
	Even if the stability theory does not apply, the method performs well.
	
	

  §.§.§ Internal layer

	We consider the homogeneous problem f≡ 0 with coefficients
	
    =ν_2,    =(x_2 e^x_1-1/2x_2^2, e^x_1-1/2x_2^2)^⊤,     σ=0,

	in Ω=(0,1)^2. The streamlines of the advection field  are the parabolas x_1=x_2^2/2+c.
	We consider different values of the parameter ν to investigate the influence of the advection term: ν=10^-j for j=1,2,3,4. 
	We set the Dirichlet boundary Γ_D={(x_1,x_2)∈∂Ω| x_1=0  or  x_2=0} with the data g_D=1 if x_1≤1/3 and g_D=0 otherwise,
	and Neumann boundary Γ_N=∂Ω∖Γ_D with the data g_N=0.
	The choice of the penalization parameter for this kind of problems is particularly delicate, as for small values of γ coercivity fails, but large values of γ often introduce spurious oscillations in the solution.
	Here we choose γ = 100 and K_F==ν.
	
	The results with mesh size h=2^-6 are shown in <Ref>, where we compare the results of the quasi-Trefftz method (lower row) to those of the full-polynomial DG (upper row).
	Both methods show similar results: the flat part of the solution is well approximated, and the discontinuity at the boundary and the internal layer are well captured with small oscillations.
	
	
	
	
	
	

  §.§.§ L-shaped domain

	We apply the method to a strongly advection-dominated BVP from <cit.>. 
	The coefficients are 
	
    =ν_2,    =(-x_2,x_1)^⊤,    σ=0,

	with ν=5× 10^-3.
	The source term is f=0, the problem is posed on the L-shaped domain Ω=(0,1)^2∖[0,0.5]^2,
	and the Dirichlet boundary condition g_D=1 on x_2=0 and g_D=0 elsewhere is imposed on ∂Ω.
	The solution u exhibits boundary layers and corner singularities.
	
	We fix K_F==ν, γ = 50 and choose a mesh 4246 triangular elements and polynomial degree p=3.
	<Ref> shows the quasi-Trefftz DG solution, in perfect visual agreement with <cit.>, and the difference against the full-polynomial space DG solution.
	We observe that this difference is concentrated at the singular corners and at the outflow layer.
	
	
	
	

§ CONCLUSIONS AND FUTURE DEVELOPMENTS
 
	We have examined the polynomial quasi-Trefftz space for linear PDEs with smooth coefficients and right-hand side. We have shown it can approximate smooth solutions to the PDE with the same accuracy as the full polynomial space but requiring fewer degrees of freedom, and described an algorithm for the construction of a quasi-Trefftz basis.
	Then we have analyzed a quasi-Trefftz DG method for elliptic diffusion–advection–reaction BVPs with piecewise-smooth data,
	proving optimal-rate h-convergence, as confirmed by numerical results.
	
	Further investigation into the properties of the quasi-Trefftz basis functions is needed to optimize the choice of the Cauchy data, i.e. of the m polynomial bases in the initialization step of the algorithm for the construction of the quasi-Trefftz basis functions, aiming at further improving accuracy, conditioning and computing time.
	
	Analysis of non-polynomial quasi-Trefftz functions could be useful for efficiently approximating solutions with boundary layers or less regular solutions, such as those with corner singularities.
	
	Further research is required to obtain approximation estimates in Sobolev norms and to establish optimal DG error bounds in L^2(Ω)-norm, as suggested by the numerics. 
	A challenging extension, which has not yet been achieved for quasi-Trefftz methods, is the analysis of the approximation properties for increasing polynomial degrees (p-convergence).
	
	Another interesting extension is the application of this method to PDEs whose nature changes in the domain, such as the Euler-Tricomi equation (∂^2_xu +x ∂^2_yu= 0), modeling transonic flow. 	
	
	

§ ACKNOWLEDGEMENTS

	LMIG, AM and PS gratefully acknowledge the Centro Internazionale per la Ricerca Matematica (CIRM, Trento) for hosting them in the Research-in-Pairs program.
	AM and CP acknowledge support from PRIN projects “ASTICE” (202292JW3F) and “NA-FROM-PDEs” (201752HKH8), GNCS–INDAM, and PNRR-M4C2-I1.4-NC-HPC-Spoke6, which are partly funded by the European Union – NextGenerationEU.
This research was funded in part by the Austrian Science Fund (FWF) 
https://doi.org/10.55776/F6510.55776/F65 and 
https://doi.org/10.55776/ESP438982410.55776/ESP4389824.
For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.
	LMIG acknowledges support from the US National Science Foundation (NSF): this material is based upon work supported by the NSF under Grant No. https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110407   HistoricalAwards=falseDMS-2110407.
	
	[heading=bibintoc,title=Bibliography]
	
	

§ ESTIMATES ON JUMP–AVERAGE TERMS

	
	For all v,w∈ H^2(_h), 
	
    |∑_F∈_h^I∪_h^D∫_F  ∇ v· w| ≤_L^∞(Ω)^1/2(∑_E∈_h∑_F∈_Eh_F/γ K_F(^1/2∇ v)_|_E·_F_L^2(F)^2
    	)^1/2w_J.

	
	
	Using the Cauchy–Schwarz inequality and recalling the definition (<ref>) of |·|_J, we have
	
    |∑_F∈_h^I∪_h^D∫_F  ∇ v· w|
    	≤   (∑_F∈_h^I∪_h^Dh_F/γ K_F∫_F (∇ v·_F)^2)^1/2
    	|w|_J.

	In the first term, for all F=∂ E_1 ∩∂ E_2∈_h^I, Young's inequality yields
	
    ∫_F ( ∇ v·_F)^2   
    	=∫_F[1/2(_|_E_1^1/2 (^1/2∇ v)_|_E_1+_|_E_2^1/2 (^1/2∇ v)_|_E_2)  ·_F]^2
       ≤1/2_L^∞(Ω)((^1/2∇ v)_|_E_1·_F^2_L^2(F)+(^1/2∇ v)_|_E_2·_F^2_L^2(F)).

	For all F∈_h^D with F⊂∂ E, we obtain
	∫_F ( ∇ v·_F)^2
	≤_L^∞(Ω)(^1/2∇ v)_|_E·_F^2_L^2(F).
	Combining these two bounds, the thesis is derived by collecting the facet contributions of each mesh element.
	
	
	
	
	For all (v,w_h)∈ H^2(_h) ×  V_h,
	
    |∑_F∈_h^I∪_h^D∫_F  v·∇ w_h| 
    	≤v_J_L^∞(Ω)/( N_∂/γ)^1/2(∑_E∈_h^1/2∇ w_h_L^2( E)^2)^1/2.

	
	
	As in the previous proof, Cauchy–Schwarz inequality leads to
	
    |∑_F∈_h^I∪_h^D∫_F  v·∇ w_h|
    	≤(∑_F∈_h^I∪_h^Dh_F/γ K_F∫_F (∇ w_h·_F)^2)^1/2
    	|v|_J.

	For all F∈_h^I with F=∂ E_1 ∩∂ E_2, Young's inequality yields
	
    ∫_F ( ∇ w_h·_F)^2   
    	=∫_F[1/2(	(∇ w_h)_|_E_1+ (∇ w_h)_|_E_2)  ·_F]^2
       ≤1/2_L^∞(Ω)^2 ((∇ w_h)_|_E_1·_F^2_L^2(F)+(∇ w_h)_|_E_2·_F^2_L^2(F)),

	and for F∈_h^D with F⊂∂ E, we have 
	∫_F ( ∇ w_h·_F)^2
	≤_L^∞(Ω)^2 (∇ w_h)_|_E·_F^2_L^2(F).
	Aggregating the contributions of each element, owing to the facts that h_F≤ h_E for all F∈_E, E∈_h, that ≤ K_F for all F∈_h, and to the discrete trace inequality (<ref>), we deduce 
	
    |∑_F∈_h^I∪_h^D∫_F  v·∇ w_h| 
    	   ≤v_J_L^∞(Ω)(∑_E∈_h∑_F∈_Eh_F/γ K_F(∇ w_h)_|_E·_F_L^2(F)^2
    	)^1/2
       ≤v_J_L^∞(Ω)( 	∑_E∈_h∑_F∈_Eh_E/γ  h_E^-1∇ w_h_L^2(E)^2
    	)^1/2 .

	The assertion is obtained recalling the definition (<ref>) of N_∂ and using the ellipticity condition (<ref>) as ∇ w_h≤ ^-1/2^1/2∇ w_h.
	
	
	In the proof of <Ref> we could have used the bound (<ref>) where  is already inside the L^2-norm, instead of using (<ref>) and paying the factor 1/. 
	However, in general ^1/2∇ v_h is not a polynomial, so the classical discrete trace inequality (<ref>) would not be applicable.


