# Two and three dimensional $H^{2}$-conforming finite element approximations
without $C^{1}$-elements 111The authors acknowledge that this material is
based upon work supported by the National Science Foundation under Award No.
DMS-2324364 for the first author and Award No. DMS-2201487 for the second
author.

Mark Ainsworth<EMAIL_ADDRESS>Charles Parker
<EMAIL_ADDRESS>
###### Abstract

We develop a method to compute $H^{2}$-conforming finite element
approximations in both two and three space dimensions using readily available
finite element spaces. This is accomplished by deriving a novel, equivalent
mixed variational formulation involving spaces with at most
$H^{1}$-smoothness, so that conforming discretizations require at most
$C^{0}$-continuity. The method is demonstrated on arbitrary order
$C^{1}$-splines.

###### keywords:

$H^{2}$-conforming finite elements , $C^{1}$ finite elements

###### MSC:

[2020] 65N30 , 65N12

††journal: Computer Methods in Applied Mechanics and Engineering

[1]organization=Division of Applied Mathematics, Brown University,
addressline=Box F, 182 George Street, city=Providence, postcode=02912,
state=RI, country=USA

[2]organization=Mathematical Institute, University of Oxford,
addressline=Andrew Wiles Building, Woodstock Road, city=Oxford, postcode=OX2
6GG, country=UK

##  1 Introduction

Fourth-order partial differential equations arise in a variety of applications
including thin plates and shells [1], the separation of binary alloys [2],
condensed matter physics [3], and liquid crystals [4], among many others; see
e.g. [5] and references therein. The variational formulation of such problems,
or their linearizations, are well-posed in subspaces of $H^{2}$ so that a
standard Galerkin scheme would require $H^{2}$-conforming finite element
methods. Conforming schemes are attractive in that they retain the stability
of the variational formulation. While the $H^{2}$-conforming 2D Argyris
element appears in Firedrake [6, 7, 8] and scikit-fem [9] and the 2D Hsieh-
Clough-Tocher (HCT) element appears in FreeFEM [10], the choice is otherwise
rather limited. For instance, 3D $H^{2}$-conforming elements seem to be absent
from all major packages.

Why are $H^{2}$-conforming elements not more widely available? One possible
answer is that most $H^{2}$-conforming elements are $C^{1}$-continuous [11]
and imposing $C^{1}$-continuity is less straightforward then imposing
$C^{0}$-continuity. Consider, for instance, the simple setting where the
domain is the unit cube and the finite element space is the space of $C^{1}$
piecewise polynomials of degree $p\in\mathbb{N}$ in the special case of
tetrahedral Freudenthal partitions [12] of the cube. Even counting the
dimension of the associated $H^{2}$-conforming space is a nontrivial matter.
In [13], a count of the dimension of the space is given explicitly for all
$p\in\mathbb{N}$ along with a choice of basis. However, the basis cannot be
constructed in terms of local basis functions defined on a reference element.
Instead, the basis is global in nature and thus not amenable to standard
finite element procedures such as subassembly. On a general tetrahedral
partition, even counting the dimension of the associated $C^{1}$ piecewise
polynomials is an open problem. Standard $H^{2}$-conforming finite elements
based on locally defined basis functions are, of course, possible but only if
the degree is sufficiently high ($p\geq 5$ in 2D [14] and $p\geq 9$ [15,
Chapter 18.2] in 3D).

In this work, we develop a method to compute $H^{2}$-conforming finite element
approximations in both two and three space dimensions using finite element
spaces that impose lower continuity than $C^{1}$, and which are more readily
available. One benefit of such a scheme is that $H^{2}$-conforming
approximations can be obtained using existing software packages even when
$H^{2}$-spaces are not offered. This is accomplished by deriving a novel,
equivalent mixed variational formulation involving spaces with at most
$H^{1}$-smoothness, so that conforming discretizations require at most
$C^{0}$-continuity. Particular choices of conforming spaces recover
$H^{2}$-conforming approximations of the original $H^{2}$-variational problem,
including the space in [13]. The method generalizes our previous work [16] to
$H^{2}$-problems with lower order terms (e.g. time dependent problems) and to
three space dimensions.

There are many alternative mixed methods for $H^{2}$-variational problems
currently available – most of which specifically focus on the biharmonic
equation or Kirchhoff plate problem in 2D; see e.g. [17, 18, 19, 20, 21, 22].
Methods that are applicable to more general $H^{2}$-variational problems in
two and three dimensions are less common; see e.g. [23, 24, 25]. In any case,
none of the above methods will, in general, recover the $H^{2}$-conforming
approximation. In contrast, our method uses standard spaces to recover the
$H^{2}$-conforming approximation in two and three dimensions for problems with
general boundary conditions.

The remainder of the paper is organized as follows. In section 2, we describe
the problem setting that naturally leads to the equivalent mixed variational
formulation analyzed in section 3. Section 4 details well-posed
discretizations of the mixed formulation and their relationship to
$H^{2}$-conforming approximations. The implementation of the method and
several numerical examples are given in sections 5 and 6, while section 7
discusses the extension to time-dependent problems. For the 2D Morgan-Scott
element [26], we show in section 8 that the inf-sup condition associated with
the discrete mixed problem is uniformly bounded away from zero with respect to
both the polynomial degree and mesh size. Finally, in section 9, we show
precisely how the proposed method generalizes our previous work [16].

##  2 Problem setting

Let $\Omega\subset\mathbb{R}^{d}$, $d\in\\{2,3\\}$, be a connected Lipschitz
polyhedral domain. The boundary of $\Omega$, denoted $\Gamma:=\partial\Omega$,
is partitioned into disjoint sets $\Gamma_{c}$, $\Gamma_{s}$, and
$\Gamma_{f}$. For $d=2$, we assume that there exists simply-connected
Lipschitz polygons
$\Omega_{0},\Omega_{1},\ldots,\Omega_{N_{H}}\subset\mathbb{R}^{2}$ such that
$\bar{\Omega}_{j}\subsetneq\Omega_{0}$ for $1\leq j\leq N_{H}$,
$\bar{\Omega}_{i}\cap\bar{\Omega}_{j}=\emptyset$ for $1\leq i,j\leq N_{H}$
with $i\neq j$, and
$\Omega=\Omega_{0}\setminus\cup_{j=1}^{N_{H}}\bar{\Omega}_{j}$. For $d=3$, we
assume that the geometry of $\Gamma_{cs}:=\Gamma_{c}\cup\Gamma_{s}$ satisfies
the following conditions [27]: For each of the $N$ connected components
$\Gamma_{cs}^{(i)}$, $1\leq i\leq N$, of $\Gamma_{cs}$, there exists an open
Lipschitz domain $\Omega_{i}\subset\mathbb{R}^{3}$ such that
$\bar{\Omega}_{i}\cap\bar{\Omega}=\bar{\Gamma}_{cs}^{(i)}$,
$\Omega_{i}\cap\Omega=\emptyset$, the distance between $\Omega_{i}$ and
$\Omega_{j}$ is strictly positive for $i\neq j$, and
$\bar{\Omega}\cup\bar{\Omega}_{1}\cup\ldots\cup\bar{\Omega}_{N}$ is a
Lipschitz domain.

We consider $H^{2}(\Omega)$ variational problems of the form

$\displaystyle w\in H^{2}_{\Gamma}(\Omega):\qquad B(w,v)=F(v)\qquad\forall
v\in H^{2}_{\Gamma}(\Omega),$ (2.1)

where

$\displaystyle H^{2}_{\Gamma}(\Omega):=\\{v\in
H^{2}(\Omega):v|_{\Gamma_{cs}}=0\text{ and }\partial_{n}v|_{\Gamma_{c}}=0\\},$

$F$ is a linear functional on $H^{2}_{\Gamma}(\Omega)$, and
$B:H^{2}(\Omega)\times H^{2}(\Omega)\to\mathbb{R}$ is a bilinear form with the
following properties.

  * 1.

(Boundedness) There exists $M_{B}>0$ such that

$\displaystyle|B(u,v)|\leq M_{B}\|u\|_{2}\|v\|_{2}\qquad\forall u,v\in
H^{2}(\Omega).$ (2.2)

  * 2.

(Inf-sup condition) There exists $\alpha>0$ such that

$\displaystyle\inf_{0\neq u\in H^{2}_{\Gamma}(\Omega)}\sup_{0\neq v\in
H^{2}_{\Gamma}(\Omega)}\frac{B(u,v)}{\|u\|_{2}\|v\|_{2}}\geq\alpha.$ (2.3)

  * 3.

(Uniqueness condition) For every $v\in H^{2}_{\Gamma}(\Omega)$, there holds

$\displaystyle\sup_{u\in H^{2}_{\Gamma}(\Omega)}B(u,v)>0.$ (2.4)

Then, problem 2.1 is well-posed [28, Theorem 2.6].

Let $\mathcal{T}$ be a shape regular [29, Definition (4.4.13)] partitioning of
$\Omega$ into simplices such that the nonempty intersection of any two
distinct elements from $\mathcal{T}$ is a single common sub-simplex of both
elements. Given an $H^{2}$-conforming finite element space $\mathbb{W}\subset
H^{2}(\Omega)$ on $\mathcal{T}$, typically consisting of $C^{1}$-continuous
piecewise polynomials of an arbitrary but fixed degree $p$, the corresponding
$H^{2}$-conforming Galerkin finite element scheme for 2.1 reads

$\displaystyle w_{\mathcal{T}}\in\mathbb{W}_{\Gamma}:\qquad
B(w_{\mathcal{T}},v)=F(v)\qquad\forall v\in\mathbb{W}_{\Gamma},$ (2.5)

where $\mathbb{W}_{\Gamma}:=\mathbb{W}\cap H^{2}_{\Gamma}(\Omega)$. If there
exists a positive constant $\alpha_{\mathcal{T}}>0$, possibly depending on the
dimension of $\mathbb{W}_{\Gamma}$, such that the following discrete analogue
of the inf-sup condition 2.3 holds:

$\displaystyle\alpha_{\mathcal{T}}$ $\displaystyle:=\inf_{0\neq
u\in\mathbb{W}_{\Gamma}}\sup_{0\neq
v\in\mathbb{W}_{\Gamma}}\frac{B(u,v)}{\|u\|_{2}\|v\|_{2}}>0,$

then 2.5 has a unique solution. Moreover, standard error estimates (e.g. [28,
Lemma 2.28]) show that $w_{\mathcal{T}}$ is, up to a constant, the best
approximation to $w$ in $\mathbb{W}_{\Gamma}$:

$\displaystyle\|w-w_{\mathcal{T}}\|_{2}\leq\left(1+\frac{M_{B}}{\alpha_{\mathcal{T}}}\right)\inf_{v\in\mathbb{W}_{\Gamma}}\|w-v\|_{2}.$

###  2.1 Typical Problems

In practice, the bilinear form $B(\cdot,\cdot)$ often has more structure
beyond that indicated by the abstract assumptions 2.2, 2.3, and 2.4. For
example, in the case of the biharmonic equation $B(\cdot,\cdot)$ depends only
on the gradient of its arguments i.e.

$\displaystyle
B(w,v)=a(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v),$ (2.6)

where the bilinear form
$a(\bm{\theta},\bm{\psi}):=(\operatorname{div}\bm{\theta},\operatorname{div}\bm{\psi})$
is bounded on $\bm{H}^{1}(\Omega)$. In a similar vein, the bilinear form
$B(\cdot,\cdot)$ for a Kirchhoff plate in equilibrium (see e.g. Chapter 4 of
[1]) again has the form 2.6, this time with

$\displaystyle
a(\bm{\theta},\bm{\psi}):=D\left[(1-\nu)(\bm{\varepsilon}(\bm{\theta}),\bm{\varepsilon}(\bm{\psi}))+\nu(\operatorname{div}\bm{\theta},\operatorname{div}\bm{\psi})\right],$
(2.7)

where
$\bm{\varepsilon}(\bm{\theta})=(\operatorname{\mathbf{grad}}\bm{\theta}+(\operatorname{\mathbf{grad}}\bm{\theta})^{T})/2$
and $D$ and $\nu$ are positive physical parameters so that the corresponding
bilinear form $a(\cdot,\cdot)$ is again bounded on $\bm{H}^{1}(\Omega)$. Here,
$\Gamma_{c}$, $\Gamma_{s}$, and $\Gamma_{f}$ correspond to the parts of the
boundary where the plate is clamped, simply-supported, and left free.

Discretizations of time dependent problems offer another useful example.
Consider, for instance, the time dependent biharmonic equation

$\displaystyle\partial_{t}w+\Delta^{2}w=G\quad\text{with}\quad w|_{t=0}=g$

in variational form: find $w(t)\in H^{2}_{\Gamma}(\Omega)$ such that

$\displaystyle\frac{d}{dt}(w(t),v)+(\Delta w(t),\Delta v)=G(v)\qquad\forall
v\in H^{2}_{\Gamma}(\Omega)$

with $w(0)=g$. This problem can be accommodated in the foregoing framework
using Rothe’s method [30] to first discretize in time (using e.g. Crank-
Nicolson with a step size $\Delta t>0$) and thereby obtain the semi-discrete
scheme: For $n\in\mathbb{N}_{0}$, find $w_{n}\in H^{2}_{\Gamma}(\Omega)$ such
that

$\displaystyle(w_{n},v)+\frac{\Delta t}{2}(\Delta w_{n},\Delta
v)=(w_{n-1},v)-\frac{\Delta t}{2}(\Delta w_{n-1},\Delta v)+\Delta
tG(v)\qquad\forall v\in H^{2}_{\Gamma}(\Omega)$ (2.8)

with $w_{0}=g$. This problem is of the form 2.1 where is given by

$\displaystyle B(w,v)=\frac{\Delta t}{2}(\Delta w,\Delta v)+(w,v)=\frac{\Delta
t}{2}a(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v)+(w,v),$

where $a(\cdot,\cdot)$ is the bilinear form that appears in the equilibrium
case 2.6. Other fourth-order problems, such as the Cahn-Hilliard equations
[2], give rise to problems of the form 2.1 with the same structure as in 2.8.
The same structure also arises for problems with lower order terms. For
example, the weak formulation of the PDE $\Delta^{2}u-\Delta
u+\bm{b}\cdot\operatorname{\mathbf{grad}}u+u=f$ with $\Gamma_{c}=\Gamma$
reads: Find $u\in H^{2}_{\Gamma}(\Omega)$ such that

$\displaystyle(\Delta u,\Delta
v)+(\operatorname{\mathbf{grad}}u,\operatorname{\mathbf{grad}}v)+(\bm{b}\cdot\operatorname{\mathbf{grad}}u,v)+(u,v)=(f,v)\qquad\forall
v\in H^{2}_{\Gamma}(\Omega).$ (2.9)

###  2.2 Structural assumption

In view of the above examples, we assume that the bilinear form
$B(\cdot,\cdot)$ can be written in the form

$\displaystyle
B(w,v)=a(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v)+c(w,v)\qquad\forall
w,v\in H^{2}(\Omega),$ (2.10)

where $a:\bm{H}^{1}(\Omega)\times\bm{H}^{1}(\Omega)\to\mathbb{R}$ and
$c:H^{1}(\Omega)\times H^{1}(\Omega)\to\mathbb{R}$ are bilinear forms for
which there exists a constant $M>0$ such that

$\displaystyle|a(\bm{\theta},\bm{\psi})|$ $\displaystyle\leq
M\|\bm{\theta}\|_{1}\|\bm{\psi}\|_{1}\qquad$
$\displaystyle\forall\bm{\theta},\bm{\psi}\in\bm{H}^{1}(\Omega),$ (2.11a)
$\displaystyle|c(w,v)|$ $\displaystyle\leq M\|w\|_{1}\|v\|_{1}\qquad$
$\displaystyle\forall w,v\in H^{1}(\Omega),$ (2.11b)

where the $H^{1}(\Omega)$-norm of a vector-valued function is the sum of the
$H^{1}(\Omega)$-norm of its components. Similarly, we assume that the right
hand side $F$ can be written as the sum of two linear functionals
$F_{1}\in\bm{H}^{1}(\Omega)^{*}$ and $F_{2}\in H^{1}(\Omega)^{*}$ satisfying

$\displaystyle F(v)=F_{1}(\operatorname{\mathbf{grad}}v)+F_{2}(v)\qquad\forall
v\in H^{2}(\Omega).$ (2.12)

The decompositions 2.10 and 2.12 need not be unique. For instance, the
bilinear form appearing in the example in 2.9 with $\bm{b}=\bm{0}$ satisfies
the decomposition condition with either
$a(\cdot,\cdot)=(\operatorname{div}\cdot,\operatorname{div}\cdot)$ and
$c(\cdot,\cdot)=(\operatorname{\mathbf{grad}}\cdot,\operatorname{\mathbf{grad}}\cdot)+(\cdot,\cdot)$
or alternatively
$a(\cdot,\cdot)=(\operatorname{div}\cdot,\operatorname{div}\cdot)+(\cdot,\cdot)$
and $c(\cdot,\cdot)=(\cdot,\cdot)$. In both cases, $a(\cdot,\cdot)$ and
$c(\cdot,\cdot)$ are bounded on $\bm{H}^{1}(\Omega)$ and $H^{1}(\Omega)$
respectively. The ensuing analysis makes use of the existence of
decompositions 2.10 and 2.12, but does not require their uniqueness.

##  3 A novel mixed formulation for the continuous problem

The original primal variational problem 2.1 can be written as a mixed
formulation involving spaces that have weaker smoothness requirements than
$H^{2}(\Omega)$ as follows. Let $\bm{\gamma}=\operatorname{\mathbf{grad}}w$
and observe that, since $w\in H^{2}_{\Gamma}(\Omega)$, the gradient
$\bm{\gamma}$ belongs to the space

$\displaystyle\bm{G}_{\Gamma}(\Omega):=\\{\bm{\psi}\in
H^{1}(\Omega)^{d}:\bm{\psi}|_{\Gamma_{c}}=\bm{0}\text{ and
}\bm{\pi}_{T}\bm{\psi}|_{\Gamma_{s}}=\bm{0}\\},$

where $\bm{\pi}_{T}$ is the tangential projection operator defined by

$\displaystyle\bm{\pi}_{T}\bm{\psi}:=\bm{\psi}-(\bm{\psi}\cdot\bm{n})\bm{n}\qquad\text{on
}\Gamma.$

Observing that $w\in H^{2}_{\Gamma}(\Omega)\subset H^{1}_{\Gamma}(\Omega)$,
where

$\displaystyle H^{1}_{\Gamma}(\Omega):=\\{\tilde{v}\in
H^{1}(\Omega):\tilde{v}|_{\Gamma_{cs}}=0\\},$

it follows that $(w,\bm{\gamma})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ satisfies

$\displaystyle a(\bm{\gamma},\operatorname{\mathbf{grad}}v)+c(w,v)$
$\displaystyle=F_{1}(\operatorname{\mathbf{grad}}v)+F_{2}(v)\qquad\forall v\in
H^{2}_{\Gamma}(\Omega).$ (3.1)

Equation 3.1 holds for functions belonging to $H^{2}_{\Gamma}(\Omega)$. It is
not difficult to see that, with the above notation, $H^{2}_{\Gamma}(\Omega)$
can be precisely characterized as the following subspace of
$H^{1}_{\Gamma}(\Omega)$:

$\displaystyle H^{2}_{\Gamma}(\Omega)=\\{\tilde{v}\in
H^{1}_{\Gamma}(\Omega):\operatorname{\mathbf{grad}}\tilde{v}\in\bm{G}_{\Gamma}(\Omega)\\}.$
(3.2)

Hence, viewing the relation $\bm{\gamma}=\operatorname{\mathbf{grad}}w$ as a
side constraint for 3.1 to be imposed using a Lagrange multiplier $\bm{\phi}$,
we arrive at the following mixed problem: Find
$(\tilde{w},\bm{\gamma},\bm{\phi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\times\bm{\Phi}_{\Gamma}(\Omega)$
such that

$\displaystyle
a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})+\langle\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\phi}\rangle$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega),$ (3.3a)
$\displaystyle\langle\bm{\Xi}(\tilde{w},\bm{\gamma}),\bm{\eta}\rangle$
$\displaystyle=0\qquad$
$\displaystyle\forall\bm{\eta}\in\bm{\Phi}_{\Gamma}(\Omega),$ (3.3b)

where the operator $\bm{\Xi}$ is defined by the rule

$\displaystyle\bm{\Xi}(\tilde{v},\bm{\psi}):=\operatorname{\mathbf{grad}}\tilde{v}-\bm{\psi}\qquad\forall(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega).$ (3.4)

Observe that we write $\tilde{w}$ rather than $w$ in 3.3 to reflect the fact
that, starting with 3.3, it is not obvious (at this stage) that $\tilde{w}$
satisfies the original variational problem 2.1. Later, we shall see that if
the function space $\bm{\Phi}_{\Gamma}(\Omega)$ and duality pairing
$\langle\cdot,\cdot\rangle$ are chosen appropriately, then $\tilde{w}$ does
indeed satisfy 2.1 and, as a consequence $w=\tilde{w}$.

It remains to choose an appropriate function space
$\bm{\Phi}_{\Gamma}(\Omega)$ and duality pairing $\langle\cdot,\cdot\rangle$
with which to impose the side constraint. We make the natural choice and take
$\bm{\Phi}_{\Gamma}(\Omega):=(\operatorname{Im}\bm{\Xi})^{*}$, where

$\displaystyle\operatorname{Im}\bm{\Xi}:=\\{\bm{\Xi}(\tilde{v},\bm{\psi}):(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\\},$

so that $\operatorname{Im}\bm{\Xi}\subset{L}^{2}(\Omega)^{d}$ and the duality
pairing $\langle\cdot,\cdot\rangle$ is taken to be the extension of the
$L^{2}(\Omega)^{d}$ inner-product to
$\operatorname{Im}\bm{\Xi}\times(\operatorname{Im}\bm{\Xi})^{*}$.

###  3.1 Characterizing $(\operatorname{Im}\bm{\Xi})^{*}$

The mixed formulation 3.3 achieves the primary goal of reducing the maximum
smoothness of the involved spaces:
$H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\subset
H^{1}(\Omega)\times H^{1}(\Omega)^{d}$ and
$L^{2}(\Omega)^{d}\subset(\operatorname{Im}\bm{\Xi})^{*}$. However, the
presence of the nonstandard space $(\operatorname{Im}\bm{\Xi})^{*}$ means that
neither the well-posedness of 3.3 nor its discretization are clear-cut. For
instance, while it is clear from definition 3.4 that
$\operatorname{Im}\bm{\Xi}\subseteq{L}^{2}(\Omega)^{d}$, we will show in this
section that $\operatorname{Im}\bm{\Xi}\neq{L}^{2}(\Omega)^{d}$. That is,
$(\operatorname{Im}\bm{\Xi})^{*}$ contains objects that are less smooth than
${L}^{2}(\Omega)^{d}$ which has repercussions both in the analysis of 3.3 and
the development of a Galerkin scheme for 3.3.

The first step in analyzing the mixed problem 3.3 is to develop a concrete
characterization for the image space $\operatorname{Im}\bm{\Xi}$. For
$(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$, there holds

$\displaystyle\bm{\Xi}(\tilde{v},\bm{\psi})=\operatorname{\mathbf{grad}}\tilde{v}-\bm{\psi}\in{L}^{2}(\Omega)^{d}\quad\text{and}\quad\operatorname{curl}\bm{\Xi}(\tilde{v},\bm{\psi})=-\operatorname{curl}\bm{\psi}\in
L^{2}(\Omega)^{d^{*}},$

where $d^{*}=1$ if $d=2$ and $d^{*}=3$ if $d=3$. Here, we use
$\operatorname{curl}$ to denote either the scalar curl of a 2D vector (i.e.
$\operatorname{curl}\bm{\theta}:=\partial_{x}\theta_{2}-\partial_{y}\theta_{1}$)
or the usual curl operator of a 3D vector depending on the context.
Consequently,
$\operatorname{Im}\bm{\Xi}\subseteq\bm{H}(\mathrm{curl};\Omega)$, where

$\displaystyle\bm{H}(\mathrm{curl};\Omega):=\\{\bm{\theta}\in{L}^{2}(\Omega)^{d}:\operatorname{curl}\bm{\theta}\in
L^{2}(\Omega)^{d^{*}}\\}\quad\text{with}\quad\|\bm{\theta}\|_{\operatorname{curl}}^{2}:=\|\bm{\theta}\|^{2}+\|\operatorname{curl}\bm{\theta}\|^{2}.$

We shall also need to characterize the traces of functions in
$\operatorname{Im}\bm{\Xi}$. The tangential trace operator $\operatorname{Tr}$
given by

$\displaystyle\bm{H}(\mathrm{curl};\Omega)\ni\bm{\theta}\mapsto\operatorname{Tr}\bm{\theta}:=\begin{cases}\bm{t}\cdot\bm{\theta}|_{\Gamma}&\text{if
}d=2\\\ \bm{\pi}_{T}\bm{\theta}|_{\Gamma}&\text{if }d=3\end{cases}\in
H^{-1/2}(\Gamma)^{d^{*}}$

is well-defined on $\bm{H}(\mathrm{curl};\Omega)$ (see e.g. [31, p. 27 Theorem
2.5] for the case $d=2$ and [32, 33] for the case $d=3$). For any $\hat{w}\in
C^{\infty}(\bar{\Omega})^{d^{*}}$ with $\hat{w}|_{\Gamma_{f}}=0$, there holds

$\displaystyle\langle\operatorname{Tr}\bm{\Xi}(\tilde{v},\bm{\psi}),\hat{w}\rangle_{\Gamma}=\langle\operatorname{Tr}\operatorname{\mathbf{grad}}\tilde{v},\hat{w}\rangle_{\Gamma}+(\operatorname{Tr}\bm{\psi},\hat{w})_{\Gamma}=\begin{cases}-(\tilde{v},\bm{t}\cdot\operatorname{\mathbf{grad}}\hat{w})_{\Gamma}=0&d=2,\\\
-(\tilde{v},\operatorname{div}\bm{\pi}_{T}\hat{w})_{\Gamma}=0&d=3,\end{cases}$

where $\langle\cdot,\cdot\rangle_{\Gamma}$ denotes the extension of the
$L^{2}(\Gamma)^{d^{*}}$ inner-product to the trace space and its dual. By
density (see e.g. [34, Theorem 3.1]), we obtain

$\displaystyle\langle\operatorname{Tr}\bm{\Xi}(\tilde{v},\bm{\psi}),\hat{w}\rangle_{\Gamma}=0\qquad\forall\hat{w}\in
H^{1}(\Omega)^{d^{*}}:w|_{\Gamma_{f}}\equiv 0.$

As a result, $\operatorname{Im}\bm{\Xi}$ is contained in the closed subspace
of $\bm{H}(\mathrm{curl};\Omega)$ consisting of functions with vanishing
tangential trace on $\Gamma\setminus\Gamma_{f}$:

$\displaystyle\bm{H}_{\Gamma}(\mathrm{curl};\Omega):=\\{\bm{\theta}\in\bm{H}(\mathrm{curl};\Omega):\langle\operatorname{Tr}\bm{\theta},\hat{w}\rangle_{\Gamma}=0\quad\forall\hat{w}\in
H^{1}(\Omega)^{d^{*}}:w|_{\Gamma_{f}}\equiv 0\\}.$

The following result shows that in fact $\operatorname{Im}\bm{\Xi}$ can be
identified with $\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$:

######  Lemma 3.1.

For every $\bm{\theta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$, there exists
$(\tilde{v},\bm{\psi})\in H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$
such that

$\displaystyle\bm{\Xi}(\tilde{v},\bm{\psi})=\bm{\theta}\quad\text{and}\quad
C_{1}\left(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\right)\leq\|\bm{\theta}\|_{\operatorname{curl}}\leq
C_{2}\left(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\right),$ (3.5)

where $C_{1}$ and $C_{2}$ are positive constants independent of $\bm{\theta}$.
Consequently,
$\operatorname{Im}\bm{\Xi}=\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$.

###### Proof.

Let $d=3$. Thanks to [27, Theorem 5.2], for every
$\bm{\theta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$, there exists
$(\tilde{v},\bm{\psi})\in H^{1}_{\Gamma}(\Omega)\times
H^{1}_{\Gamma}(\Omega)^{3}\subset
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ such that 3.5 holds. The
same arguments can be appropriately adjusted for the case $d=2$. Thus,
$\bm{H}_{\Gamma}(\mathrm{curl};\Omega)\subseteq\operatorname{Im}\bm{\Xi}$. The
result now follows on noting that we have already shown that
$\operatorname{Im}\bm{\Xi}\subseteq\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$. ∎

Thanks to Lemma 3.1, we have
$\operatorname{Im}\bm{\Xi}=\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$, and so
$\bm{\Phi}_{\Gamma}(\Omega)=(\operatorname{Im}\bm{\Xi})^{*}=\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$,
which we equip with the standard dual norm

$\displaystyle\|\bm{\eta}\|_{*}:=\sup_{\bm{0}\neq\bm{\theta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)}\frac{\langle\bm{\theta},\bm{\eta}\rangle}{\|\bm{\theta}\|_{\operatorname{curl}}}\qquad\forall\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}.$

The following inf-sup condition will be useful in establishing the well-
posedness of 3.3:

######  Corollary 3.1.

There holds

$\displaystyle\beta:=\inf_{\begin{subarray}{c}\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}\\\
\bm{\eta}\neq\bm{0}\end{subarray}}\sup_{\begin{subarray}{c}(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\\\
(\tilde{v},\bm{\psi})\neq(0,\bm{0})\end{subarray}}\frac{\langle\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\eta}\rangle}{(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1})\|\bm{\eta}\|_{*}}>0.$
(3.6)

###### Proof.

Let $\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$ be given and let
$\bm{\theta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ be its Riesz
representer. By Lemma 3.1, there exists $(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ such that
$\bm{\Xi}(\tilde{v},\bm{\psi})=\bm{\theta}$ and
$\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\leq
C\|\bm{\theta}\|_{\operatorname{curl}}=\|\bm{\eta}\|_{*}$. Inequality 3.6 now
follows since

$\displaystyle\frac{\langle\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\eta}\rangle}{(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1})\|\bm{\eta}\|_{*}}\geq
C^{-1}\frac{\langle\bm{\theta},\bm{\eta}\rangle}{\|\bm{\eta}\|_{*}^{2}}=C^{-1}.$

∎

###  3.2 Characterizing $\operatorname{Ker}\bm{\Xi}$

The kernel of $\bm{\Xi}$ is defined by

$\displaystyle\operatorname{Ker}\bm{\Xi}:=\\{(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega):\langle\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\eta}\rangle=0\
\forall\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}\\}$

and has the following simple characterization:

######  Lemma 3.2.

There holds

$\displaystyle\operatorname{Ker}\bm{\Xi}=\\{(w,\operatorname{\mathbf{grad}}w):w\in
H^{2}_{\Gamma}(\Omega)\\}.$ (3.7)

###### Proof.

Let $(\tilde{v},\bm{\psi})\in\operatorname{Ker}\bm{\Xi}$. Then, as shown
above,
$\bm{\Xi}(\tilde{v},\bm{\psi})\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$, and
$\langle\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\eta}\rangle=0$ for all
$\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$. Hence,
$\bm{\Xi}(\tilde{v},\bm{\psi})=\bm{0}$ or, equally well,
$\operatorname{\mathbf{grad}}\tilde{v}=\bm{\psi}$, and so $\tilde{v}\in
H^{2}_{\Gamma}(\Omega)$. Thus,
$\operatorname{Ker}\bm{\Xi}\subseteq\\{(w,\operatorname{\mathbf{grad}}w):w\in
H^{2}_{\Gamma}(\Omega)\\}$. The reverse inclusion follows by definition, and
so 3.7 holds. ∎

###  3.3 Well-posedness

Thanks to the well-posedness of 2.1, there exists $w\in
H^{2}_{\Gamma}(\Omega)$ satisfying 2.1. Consequently,
$(\tilde{w},\bm{\gamma})=(w,\operatorname{\mathbf{grad}}w)\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ satisfies 3.3a for all
$(\tilde{v},\bm{\psi})\in\operatorname{Ker}\bm{\Xi}$ and satisfies 3.3b. The
following result establishes the well-posedness of the mixed formulation 3.3
in the case of general data:

######  Theorem 3.1.

Let $L\in(H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega))^{*}$ and
$\bm{g}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$. Then, there exists a unique
solution to the following mixed problem: Find $(\tilde{w},\bm{\gamma})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ and
$\bm{\phi}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$ such that

$\displaystyle
a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})+\langle\bm{\Xi}(v,\bm{\psi}),\bm{\phi}\rangle$
$\displaystyle=L(\tilde{v},\bm{\psi})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega),$ (3.8a)
$\displaystyle\langle\bm{\Xi}(\tilde{w},\bm{\gamma}),\bm{\eta}\rangle$
$\displaystyle=\langle\bm{g},\bm{\eta}\rangle\qquad$
$\displaystyle\forall\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}.$
(3.8b)

Additionally, the solution satisfies

$\displaystyle\|\tilde{w}\|_{1}+\|\bm{\gamma}\|_{1}+\|\bm{\phi}\|_{*}\leq
C\left(\|L\|_{(H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega))^{*}}+\|\bm{g}\|_{\operatorname{curl}}\right)\quad\text{and}\quad\bm{\Xi}(\tilde{w},\bm{\gamma})=\bm{g},$
(3.9)

where $C$ is independent of $L$ and $\bm{g}$.

###### Proof.

The proof is a standard application of Babuška-Brezzi theory. Lemma 3.2 and
2.3 show that the bilinear form $a(\cdot,\cdot)+c(\cdot,\cdot)$ satisfies an
inf-sup condition on $\operatorname{Ker}\bm{\Xi}$. The inf-sup condition 3.6
then gives the existence and uniqueness of solutions to 3.8 and 3.9; see e.g.
[35, Theorem 4.2.3]. ∎

The next result shows, in the special case where the data in 3.8 is of the
form in 3.3, that $\tilde{w}\in H^{1}_{\Gamma}(\Omega)$ in fact belongs to
$H^{2}_{\Gamma}(\Omega)$ and coincides with the solution $w\in
H^{2}_{\Gamma}(\Omega)$ to the primal problem 2.1:

######  Corollary 3.2.

Let $F_{1}\in\bm{G}_{\Gamma}(\Omega)^{*}$ and $F_{2}\in
H^{1}_{\Gamma}(\Omega)^{*}$. Then, there exists a unique solution to the
following mixed problem: Find $(\tilde{w},\bm{\gamma})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ and
$\bm{\phi}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$ such that

$\displaystyle
a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})+\langle\bm{\Xi}(v,\bm{\psi}),\bm{\phi}\rangle$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega),$ (3.10a)
$\displaystyle\langle\bm{\Xi}(\tilde{w},\bm{\gamma}),\bm{\eta}\rangle$
$\displaystyle=0\qquad$
$\displaystyle\forall\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}.$
(3.10b)

Moreover, the solution satisfies $\tilde{w}=w$ and
$\bm{\gamma}=\operatorname{\mathbf{grad}}w$, where $w\in
H^{2}_{\Gamma}(\Omega)$ is the solution to problem 2.1.

###### Proof.

The existence, uniqueness, and stability of
$(\tilde{w},\bm{\gamma},\bm{\phi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\times\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$
and the relation $\operatorname{\mathbf{grad}}\tilde{w}=\bm{\gamma}$ follows
from Theorem 3.1. Thanks to Lemma 3.2, $\tilde{w}\in H^{2}_{\Gamma}(\Omega)$
and testing 3.10a with
$(\tilde{v},\bm{\psi})=(v,\operatorname{\mathbf{grad}}v)$ for $v\in
H^{2}_{\Gamma}(\Omega)$ shows that $\tilde{w}$ satisfies 2.1. Since solutions
to 2.1 are unique, $\tilde{w}=w$ and
$\bm{\gamma}=\operatorname{\mathbf{grad}}\tilde{w}=\operatorname{\mathbf{grad}}w$.
∎

The variational formulation 3.10 can be expressed in a form that avoids the
presence of the dual spaces altogether. Specifically, replacing
$\bm{\phi}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)^{*}$ by its Riesz
representer $\bm{q}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ and identifying
the duality pairing with the inner product on
$\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ leads to the following restatement of
Corollary 3.2:

######  Corollary 3.3.

Let $F_{1}\in\bm{G}_{\Gamma}(\Omega)^{*}$ and $F_{2}\in
H^{1}_{\Gamma}(\Omega)^{*}$. Then, there exists a unique solution to the
following mixed problem: Find $(\tilde{w},\bm{\gamma})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ and
$\bm{q}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ such that

$\displaystyle
a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})+(\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{q})_{\operatorname{curl}}$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega),$ (3.11a)
$\displaystyle(\bm{\Xi}(\tilde{w},\bm{\gamma}),\bm{\eta})_{\operatorname{curl}}$
$\displaystyle=0\qquad$
$\displaystyle\forall\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega),$
(3.11b)

where $(\cdot,\cdot)_{\operatorname{curl}}$ is the inner-product on
$\bm{H}(\mathrm{curl};\Omega)$:

$\displaystyle(\bm{\theta},\bm{\eta})_{\operatorname{curl}}:=(\bm{\theta},\bm{\eta})+(\operatorname{curl}\bm{\theta},\operatorname{curl}\bm{\eta})\qquad\forall\bm{\theta},\bm{\eta}\in\bm{H}(\mathrm{curl};\Omega).$

Moreover, the solution satisfies $\tilde{w}=w$ and
$\bm{\gamma}=\operatorname{\mathbf{grad}}w$, where $w\in
H^{2}_{\Gamma}(\Omega)$ is the solution to problem 2.1.

The system 3.11 is not the only possible mixed formulation of problem 2.1.
Indeed, simpler mixed formulations are possible [18, 19] depending on the
particular form of $B(\cdot,\cdot)$ and the boundary conditions. For example,
in the case of the biharmonic equation 2.6 with boundary conditions $w=\Delta
w=0$ on $\Gamma$, one can use a mixed formulation based on introducing an
auxiliary variable $\sigma=-\Delta w$: Find $(w,\sigma)\in
H^{1}_{\Gamma}(\Omega)^{2}$ such that

$\displaystyle(\sigma,v)+(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v)$
$\displaystyle=0\qquad$ $\displaystyle\forall v\in H^{1}_{\Gamma}(\Omega),$
(3.12a)
$\displaystyle(\operatorname{\mathbf{grad}}\sigma,\operatorname{\mathbf{grad}}u)$
$\displaystyle=G(u)\qquad$ $\displaystyle\forall u\in H^{1}_{\Gamma}(\Omega),$
(3.12b)

which can be discretized using only $H^{1}(\Omega)$-conforming finite element
spaces. However, the apparently more complicated mixed formulation 3.11 has an
unexpected advantage over 3.12, and other, mixed schemes in that under quite
mild conditions, the resulting discretization produces the
$H^{2}_{\Gamma}(\Omega)$-conforming solution to the original problem 2.1. This
is not the case with other mixed schemes such as 3.12. The question arises of
whether a finite element discretization of the new mixed form might inherit
this property at the discrete level and thus provide a way to compute the
$H^{2}_{\Gamma}(\Omega)$-conforming approximation $w_{\mathcal{T}}$ defined by
2.5.

##  4 Discretization of novel mixed formulation

In order to construct conforming discretizations of the mixed problem 3.11, we
select finite dimensional spaces $\tilde{\mathbb{W}}_{\Gamma}\subset
H_{\Gamma}^{1}(\Omega)$,
$\boldsymbol{\mathbb{G}}_{\Gamma}\subset\bm{G}_{\Gamma}(\Omega)$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}\subset\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$
and consider the following scheme: Find
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\boldsymbol{\mathbb{Q}}_{\Gamma}$
such that

$\displaystyle
a(\bm{\gamma}_{X},\bm{\psi})+c(\tilde{w}_{X},\tilde{v})+(\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{q}_{X})_{\operatorname{curl}}$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma},$
(4.1a)
$\displaystyle(\bm{\Xi}(\tilde{w}_{X},\bm{\gamma}_{X}),\bm{\eta})_{\operatorname{curl}}$
$\displaystyle=0\qquad$
$\displaystyle\forall\bm{\eta}\in\boldsymbol{\mathbb{Q}}_{\Gamma}.$ (4.1b)

Here, and in what follows, the subscript “$X$” indicates the dependence of the
discrete variables on the finite dimensional spaces
$\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}$.

Of course, in order for this scheme to be well-posed, some additional
conditions must be imposed on $\tilde{\mathbb{W}}_{\Gamma}$,
$\boldsymbol{\mathbb{G}}_{\Gamma}$, and $\boldsymbol{\mathbb{Q}}_{\Gamma}$.
Fortunately, the derivation of the mixed problem 3.11 in the previous section
provides some guidance as to how to choose these spaces. We shall assume the
subspaces $\tilde{\mathbb{W}}_{\Gamma}$ and $\boldsymbol{\mathbb{G}}_{\Gamma}$
satisfy the following condition:

(A1)

$\tilde{\mathbb{W}}\subset H^{1}(\Omega)$ and $\boldsymbol{\mathbb{G}}\subset
H^{1}(\Omega)^{d}$ are conforming finite-dimensional subspaces and

$\displaystyle\tilde{\mathbb{W}}_{\Gamma}:=\tilde{\mathbb{W}}\cap
H^{1}_{\Gamma}(\Omega)\quad\text{and}\quad\boldsymbol{\mathbb{G}}_{\Gamma}:=\boldsymbol{\mathbb{G}}\cap\bm{G}_{\Gamma}(\Omega).$

Examples of suitable choices for $\tilde{\mathbb{W}}_{\Gamma}$ and
$\boldsymbol{\mathbb{G}}_{\Gamma}$ will be given in section 4.3.

The choice of
$\boldsymbol{\mathbb{Q}}_{\Gamma}\subset\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$
is less obvious. The space $\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ arose from
considering the mapping properties of the operator $\bm{\Xi}$ in 3.3 which led
to the conclusion
$\operatorname{Im}\bm{\Xi}=\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$. The
situation in the discrete setting 4.1 mirrors that of the continuous problem
3.3 with the main difference being that the operator $\bm{\Xi}$ is replaced by
its restriction to the subspace
$\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\subset
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)$ defined by the rule
$\bm{\Xi}_{X}:\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\to\operatorname{Im}\bm{\Xi}_{X}$,

$\displaystyle\bm{\Xi}_{X}(\tilde{v},\bm{\psi}):=\operatorname{\mathbf{grad}}\tilde{v}-\bm{\psi}\qquad\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}.$
(4.2)

The analysis in the continuous setting suggests choosing
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ as follows:

(A2)

$\boldsymbol{\mathbb{Q}}_{\Gamma}:=\operatorname{Im}\bm{\Xi}_{X}$.

While attractive from the theoretical viewpoint, in order to implement this
choice in practice, it will be useful to study the operator $\bm{\Xi}_{X}$
more closely.

###  4.1 Characterization of $\operatorname{Ker}\bm{\Xi}_{X}$

The kernel of $\bm{\Xi}_{X}$ given by

$\displaystyle\operatorname{Ker}\bm{\Xi}_{X}:=\\{(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}:(\bm{\Xi}_{X}(\tilde{v},\bm{\psi}),\bm{\eta})_{\operatorname{curl}}=0\
\forall\bm{\eta}\in\boldsymbol{\mathbb{Q}}_{\Gamma}\\},$

has the same structure as the kernel of $\bm{\Xi}$ 3.7. In particular, we have
the following analogue of Lemma 3.2:

######  Lemma 4.1.

Let $\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ satisfy (A1) and (A2) and define

$\displaystyle\mathbb{W}_{\Gamma}:=\\{v\in\tilde{\mathbb{W}}_{\Gamma}:\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}_{\Gamma}\\}.$
(4.3)

Then, $\mathbb{W}_{\Gamma}$ is a finite dimensional subspace of
$H_{\Gamma}^{2}(\Omega)$, and there holds

$\displaystyle\operatorname{Ker}\bm{\Xi}_{X}=\\{(v,\operatorname{\mathbf{grad}}v):v\in\mathbb{W}_{\Gamma}\\}.$
(4.4)

###### Proof.

Since $\tilde{W}_{\Gamma}\subset H^{1}_{\Gamma}(\Omega)$ and
$\boldsymbol{\mathbb{G}}_{\Gamma}\subset\bm{G}_{\Gamma}(\Omega)$ are finite
dimensional by (A1), $\mathbb{W}_{\Gamma}$ is a finite dimensional subspace of
$H_{\Gamma}^{2}(\Omega)$ thanks to 3.2.

Now suppose that $(\tilde{v},\bm{\psi})\in\operatorname{Ker}\bm{\Xi}_{X}$. The
condition
$(\bm{\Xi}_{X}(\tilde{v},\bm{\psi}),\bm{\eta})_{\operatorname{curl}}=0$ for
all $\bm{\eta}\in\operatorname{Im}\bm{\Xi}_{X}$ then means that
$\operatorname{\mathbf{grad}}\tilde{v}=\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$.
By definition, $\tilde{v}\in\mathbb{W}_{\Gamma}$, and so
$\operatorname{Ker}\bm{\Xi}_{X}\subseteq\\{(v,\operatorname{\mathbf{grad}}v):v\in\mathbb{W}_{\Gamma}\\}$.
The reverse inclusion follows by definition, and so 4.4 holds. ∎

###  4.2 Well-posedness

The space $\mathbb{W}_{\Gamma}$ that arose in 4.3 from considering the
discrete kernel provides a convenient way to express necessary and sufficient
conditions for the discrete mixed problem 4.1 to be well-posed.

######  Lemma 4.2.

Let $\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ satisfy (A1) and (A2) with
$\mathbb{W}_{\Gamma}$ defined as in 4.3. Then, the mixed problem 4.1 is
uniquely solvable for any $F_{1}\in\boldsymbol{\mathbb{G}}_{\Gamma}^{*}$ and
$F_{2}\in\tilde{\mathbb{W}}_{\Gamma}^{*}$ if and only if $\mathbb{W}_{\Gamma}$
satisfies the inf-sup condition:

(A3)

There exists $\alpha_{X}>0$ such that

$\displaystyle\alpha_{X}$ $\displaystyle:=\inf_{0\neq
u\in\mathbb{W}_{\Gamma}}\sup_{0\neq
v\in\mathbb{W}_{\Gamma}}\frac{B(u,v)}{\|u\|_{2}\|v\|_{2}}.$ (4.5)

###### Proof.

Suppose first that (A3) holds. Since 4.1 is a finite dimensional linear
system, existence follows from uniqueness. To this end, let
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\boldsymbol{\mathbb{Q}}_{\Gamma}$
be a solution to 4.1 with $F_{1}\equiv 0$ and $F_{2}\equiv 0$. Choosing
$\bm{\eta}=\bm{\Xi}_{X}(\tilde{w}_{X},\bm{\gamma}_{X})$ in 4.1b shows that
$\bm{\Xi}_{X}(\tilde{w}_{X},\bm{\gamma}_{X})\equiv 0$. Consequently,
$\tilde{w}_{X}\in\mathbb{W}_{\Gamma}$ by Lemma 4.1 and
$\operatorname{\mathbf{grad}}\tilde{w}_{X}=\bm{\gamma}_{X}$. Choosing
$(\tilde{v},\bm{\psi})=(v,\operatorname{\mathbf{grad}}v)$ for
$v\in\mathbb{W}_{\Gamma}$ in 4.1a and applying 4.5 gives $\tilde{w}_{X}\equiv
0$ and so $\bm{\gamma}_{X}\equiv\bm{0}$. Finally, choosing any
$(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
such that $\bm{\Xi}_{X}(\tilde{v},\bm{\psi})=\bm{q}_{X}$ in 4.1a gives
$\bm{q}_{X}\equiv\bm{0}$. Therefore, solutions to 4.1 are unique and thus
exist.

Now suppose that 4.1 is uniquely solvable. Thanks to Lemma 4.1, the problem

$\displaystyle w_{X}\in\mathbb{W}_{\Gamma}:\qquad
B(w_{X},v)=F_{1}(\operatorname{\mathbf{grad}}v)+F_{2}(v)\qquad\forall
v\in\mathbb{W}_{\Gamma}$ (4.6)

is also solvable. Again, since 4.6 is a finite dimensional square system,
solutions to 4.6 are therefore also unique, and as a result, $\alpha_{X}$
defined in 4.5 is positive. Indeed, if this were not the case, then there
would exist $w_{X}\in\mathbb{W}_{\Gamma}\setminus\\{0\\}$ such that
$B(w_{X},v)=0$ for all $v\in\mathbb{W}_{\Gamma}$, which would contradict the
uniqueness of solutions to 4.6. ∎

Lemma 4.2 shows that given spaces that satisfy (A1) and (A2), the well-
posedness of the mixed problem 4.1 boils down to checking the inf-sup
condition 4.5. As shown in the proof of Lemma 4.2, the strict positivity of
$\alpha_{X}$ is equivalent to the well-posedness of 4.6.

The variational problem 4.6 naturally arose from the novel mixed formulation
and is nothing more than a $H^{2}_{\Gamma}(\Omega)$-conforming Galerkin
approximation to the original primal problem 2.1. The following result shows
that the component $\tilde{w}_{X}$ of the solution to 4.1 coincides exactly
with this $\mathbb{W}_{\Gamma}$-conforming approximation to 2.5, and mirrors
the result at the continuous level given in Corollary 3.3.

######  Theorem 4.1.

Let $\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ satisfy (A1)-(A3). Then, for any
$F_{1}\in\boldsymbol{\mathbb{G}}_{\Gamma}^{*}$ and
$F_{2}\in\tilde{\mathbb{W}}_{\Gamma}^{*}$, the unique solution
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\boldsymbol{\mathbb{Q}}_{\Gamma}$
to 4.1 satisfies $\tilde{w}_{X}=w$ and
$\bm{\gamma}_{X}=\operatorname{\mathbf{grad}}w_{X}$, where $w_{X}$ is the
$H^{2}$-conforming $\mathbb{W}_{\Gamma}$ 4.3 approximation defined by 4.6. In
particular, $w_{X}$ is independent of the splitting 2.10 of the bilinear form
$B(\cdot,\cdot)$.

###### Proof.

Lemma 4.2 shows that 4.1 is well-posed. The same arguments used in the proof
of Lemma 4.2 also show that for any
$F_{1}\in\boldsymbol{\mathbb{G}}_{\Gamma}^{*}$ and
$F_{2}\in\tilde{\mathbb{W}}_{\Gamma}^{*}$,
$\operatorname{\mathbf{grad}}\tilde{w}_{X}=\bm{\gamma}_{X}$. By definition,
$\tilde{w}_{X}\in\mathbb{W}_{\Gamma}\subset H^{2}_{\Gamma}(\Omega)$. Choosing
$(\tilde{v},\bm{\psi})=(v,\operatorname{\mathbf{grad}}v)$ for
$v\in\mathbb{W}_{\Gamma}$ in 4.1a then shows that
$B(\tilde{w}_{X},v)=F_{1}(\operatorname{\mathbf{grad}}v)+F_{2}(v)$ for all
$v\in\mathbb{W}_{\Gamma}$. Since (A3) ensures that solutions to 4.6 are
unique, $\tilde{w}_{X}=w_{X}$. ∎

Theorem 3.1 shows that under assumptions (A1)-(A3), the mixed formulation 4.1
recovers the $\mathbb{W}_{\Gamma}$-conforming approximation 4.6 of the
original $H^{2}_{\Gamma}(\Omega)$ problem, where $\mathbb{W}_{\Gamma}$ is
defined by 4.3.

######  Remark 1.

Let $[\cdot,\cdot]$ denote any inner-product on
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ and consider the following generalization
of 4.1: Find
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\boldsymbol{\mathbb{Q}}_{\Gamma}$
such that

$\displaystyle
a(\bm{\gamma}_{X},\bm{\psi})+c(\tilde{w}_{X},\tilde{v})+[\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{q}_{X}]$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma},$
(4.7a) $\displaystyle[\bm{\Xi}(\tilde{w}_{X},\bm{\gamma}_{X}),\bm{\eta}]$
$\displaystyle=0\qquad$
$\displaystyle\forall\bm{\eta}\in\boldsymbol{\mathbb{Q}}_{\Gamma}.$ (4.7b)

Then, Lemma 4.1, Lemma 4.2, and Theorem 4.1 all hold with 4.1 replaced by 4.7
with the exact same proof. The flexibility to choose the inner-product will
turn out to be helpful later when we consider a time-dependent problem in
section 7.

Remarkably, one obtains $w\in\mathbb{W}_{\Gamma}$ without having to construct
a basis for the $H^{2}_{\Gamma}(\Omega)$-conforming space
$\mathbb{W}_{\Gamma}$. As such, the preceding framework provides a vehicle
whereby one can compute an $H^{2}_{\Gamma}(\Omega)$-conforming approximation
while avoiding having to deal with $H^{2}_{\Gamma}(\Omega)$-conforming spaces.
Of course, this begs the question of what exactly one is computing given the
rather indirect construction of the space $\mathbb{W}_{\Gamma}$. Ideally, one
would prefer to start by specifying the $H^{2}_{\Gamma}(\Omega)$-conforming
space $\mathbb{W}_{\Gamma}$. The problem then is how to choose appropriate
spaces $\tilde{\mathbb{W}}_{\Gamma}$ and $\boldsymbol{\mathbb{G}}_{\Gamma}$
satisfying (A1) that give rise to the desired space $\mathbb{W}_{\Gamma}$.

###  4.3 Choices of $\tilde{\mathbb{W}}_{\Gamma}$ and
$\boldsymbol{\mathbb{G}}_{\Gamma}$ for common $H^{2}$-conforming spaces
$\mathbb{W}_{\Gamma}$

We now show how to choose finite dimensional spaces
$\tilde{\mathbb{W}}_{\Gamma}$ and $\boldsymbol{\mathbb{G}}_{\Gamma}$ so that
the space $\mathbb{W}_{\Gamma}$ defined by 4.3 coincides with existing finite
element spaces. A useful tool will be the following result that removes the
need to explicitly consider boundary conditions:

######  Lemma 4.3.

Suppose that $\mathbb{W}\subset H^{2}(\Omega)$, $\tilde{\mathbb{W}}\subset
H^{1}(\Omega)$, and $\boldsymbol{\mathbb{G}}\subset\bm{H}^{1}(\Omega)$ satisfy

$\displaystyle\mathbb{W}=\\{\tilde{v}\in\tilde{\mathbb{W}}:\operatorname{\mathbf{grad}}\tilde{v}\in\boldsymbol{\mathbb{G}}\\}.$
(4.8)

Then, $\mathbb{W}_{\Gamma}:=\mathbb{W}\cap H^{2}_{\Gamma}(\Omega)$,
$\tilde{\mathbb{W}}_{\Gamma}:=\tilde{\mathbb{W}}\cap H^{1}_{\Gamma}(\Omega)$,
and
$\boldsymbol{\mathbb{G}}_{\Gamma}:=\boldsymbol{\mathbb{G}}\cap\bm{G}_{\Gamma}(\Omega)$
satisfy 4.3.

###### Proof.

Let $v\in\mathbb{W}_{\Gamma}$ be given. Thanks to 4.8, there holds
$v\in\tilde{\mathbb{W}}$ and
$\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}$. Since
$v|_{\Gamma_{cs}}=0$ and $\partial_{n}v|_{\Gamma_{c}}=0$, there holds
$v\in\tilde{\mathbb{W}}_{\Gamma}$ and
$\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}_{\Gamma}$. Thus,
$\mathbb{W}_{\Gamma}\subseteq\\{\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}:\operatorname{\mathbf{grad}}\tilde{v}\in\boldsymbol{\mathbb{G}}_{\Gamma}\\}$.

Now let $\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}$ with
$\operatorname{\mathbf{grad}}\tilde{v}\in\boldsymbol{\mathbb{G}}_{\Gamma}$. By
4.8, $\tilde{v}\in\mathbb{W}$. Since $\tilde{v}|_{\Gamma_{cs}}=0$ and
$\operatorname{\mathbf{grad}}\tilde{v}|_{\Gamma_{c}}=\bm{0}$, there holds
$\tilde{v}\in\mathbb{W}_{\Gamma}$, and so
$\\{\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}:\operatorname{\mathbf{grad}}\tilde{v}\in\boldsymbol{\mathbb{G}}_{\Gamma}\\}\subseteq\mathbb{W}_{\Gamma}$,
which completes the proof. ∎

####  4.3.1 Argyris/TUBA element

The classical $H^{2}$-conforming finite element in two dimensions is the
Argyris/TUBA element [14] consisting of degree $p\geq 5$ polynomials with
$C^{2}$ degrees of freedom at element vertices. The element has been extended
to three dimensions by Ženíšek [36] with $p=9$ and generalized to elements of
order $p>9$ by Lai and Schumaker [15, Chapter 18.2]:

$\displaystyle\begin{aligned} \mathbb{W}=\\{&v\in
C^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\ \forall K\in\mathcal{T}\text{ and
}\\\ &\text{$v$ is $C^{2}$ at element vertices and $p\geq 5$ if $d=2$, or }\\\
&\text{$v$ is $C^{4}$ at element vertices, $C^{2}$ on element edges, and
$p\geq 9$ if $d=3$}\\}.\end{aligned}$ (4.9)

The space $\tilde{\mathbb{W}}$ may be taken to be the standard
$H^{1}$-conforming continuous piecewise polynomial space, but the space
$\boldsymbol{\mathbb{G}}$ reflects the additional continuity in the space:

######  Lemma 4.4.

Let $p\geq 5$ if $d=2$ and $p\geq 9$ if $d=3$ and $\mathbb{W}$ be given by
4.9. If

$\displaystyle\tilde{\mathbb{W}}$ $\displaystyle=\\{v\in
C(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\ \forall K\in\mathcal{T}\\},$ (4.10a)
$\displaystyle\boldsymbol{\mathbb{G}}$ $\displaystyle=\begin{aligned}
\\{&\bm{\theta}\in\bm{C}(\Omega):\bm{\theta}|_{K}\in[\mathcal{P}_{p-1}(K)]^{d}\
\forall K\in\mathcal{T}\\\ &\text{$\bm{\theta}$ is $\bm{C}^{1}$ at element
vertices if $d=2$, or }\\\ &\text{$\bm{\theta}$ is $\bm{C}^{3}$ on element
vertices and $\bm{C}^{1}$ at element edges if $d=3$}\\},\end{aligned}$ (4.10b)

then 4.8 holds.

###### Proof.

The inclusion
$\mathbb{W}\subseteq\\{v\in\tilde{W}:\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}\\}$
follows by definition. Now let $v\in\tilde{W}$ satisfy
$\operatorname{\mathbf{grad}}v\in\mathbb{G}$. Then, $v\in C^{1}(\Omega)$ and
$v$ is $C^{2}$ at element vertices if $d=2$ or $v$ is $C^{4}$ at element
vertices and $C^{2}$ at element edges if $d=3$, and so $v\in\mathbb{W}$.
Consequently,
$\\{v\in\tilde{W}:\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}\\}\subseteq\mathbb{W}$,
which completes the proof of 4.8. ∎

####  4.3.2 $C^{1}$ spline space

Let $\mathbb{W}$ to be the space of $C^{1}$ polynomials on a simplicial
partitioning $\mathcal{T}$ satisfying the conditions in section 2:

$\displaystyle\mathbb{W}=\\{v\in C^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}\\},\qquad p\in\mathbb{N}.$ (4.11)

In two dimensions, the space $\mathbb{W}$ is often called the Morgan-Scott
space [26]. It is known that $\mathbb{W}$ has a local (elementwise) basis for
polynomial orders $p\geq 5$ in two dimensions and for $p\geq 9$ in three
dimensions [15, Theorem 17.29]. For smaller values of $p$, these spaces need
not have a local basis; see e.g. [37] for the case $d=2$ and $p=4$.
Regardless, $\mathbb{W}$ admits the decomposition 4.8 with
$\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ consisting of standard
continuous piecewise polynomials for all polynomial orders $p\in\mathbb{N}$:

######  Lemma 4.5.

Let $p\geq 1$ and $\mathbb{W}$ be given by 4.11. If

$\displaystyle\tilde{\mathbb{W}}$ $\displaystyle=\\{v\in
C(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\ \forall K\in\mathcal{T}\\},$ (4.12a)
$\displaystyle\boldsymbol{\mathbb{G}}$
$\displaystyle=\\{\bm{\theta}\in\bm{C}(\Omega):\bm{\theta}|_{K}\in[\mathcal{P}_{p-1}(K)]^{d}\
\forall K\in\mathcal{T}\\},$ (4.12b)

then 4.8 holds.

###### Proof.

The inclusion
$\mathbb{W}\subseteq\\{v\in\tilde{W}:\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}\\}$
follows by definition. Now let $v\in\tilde{\mathbb{W}}$ satisfy
$\operatorname{\mathbf{grad}}v\in\mathbb{G}$. Then, $v\in C^{1}(\Omega)$ and
so $v\in\mathbb{W}$. Consequently,
$\\{v\in\tilde{\mathbb{W}}:\operatorname{\mathbf{grad}}v\in\boldsymbol{\mathbb{G}}\\}\subseteq\mathbb{W}$,
which completes the proof of 4.8. ∎

####  4.3.3 $H^{2}$-conforming macroelements

$H^{2}$-conforming spaces are often constructed using macroelements (see e.g.
[15, Chapters 6 & 18]). Let $\mathcal{T}$ be a partition as before, and let
$\mathcal{T}_{A}$ denote the Alfeld splitting obtained by subdividing every
simplex of $\mathcal{T}$ into $d+1$ subsimplices by connecting the barycenter
(or other internal point) of each element to the vertices of the element. Let

$\displaystyle\mathbb{W}=\\{v\in C^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}_{A}\\},\qquad p\geq 3,$

so that, in the 2D case with $p=3$, $\mathbb{W}$ coincides with the Hseih-
Clough-Tocher (HCT) space [38]. Applying Lemma 4.5 to the partition
$\mathcal{T}_{A}$ shows that this choice of $\mathbb{W}$ satisfies 4.8 with

$\displaystyle\tilde{\mathbb{W}}$ $\displaystyle=\\{v\in
C(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\ \forall K\in\mathcal{T}_{A}\\},$
$\displaystyle\boldsymbol{\mathbb{G}}$
$\displaystyle=\\{\bm{\theta}\in\bm{C}(\Omega):\bm{\theta}|_{K}\in[\mathcal{P}_{p-1}(K)]^{d}\
\forall K\in\mathcal{T}_{A}\\}.$

In the 3D case, one could also use the Worsey-Farin splitting
$\mathcal{T}_{WF}$ obtained by dividing every tetrahedron in $\mathcal{T}$
into 12 subtetrahedra; see [15, Chapter 16.7] for the precise definition. The
Worsey-Farin $C^{1}$-spline space for $p\geq 3$ is then defined by

$\displaystyle\mathbb{W}=\\{v\in C^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}_{WF},\ \text{$v$ is $C^{2}$ at the incenter of $T$}\
\forall T\in\mathcal{T}\\}.$

Theorem 18.11 of [15] shows that $\mathbb{W}$ has a local basis when $p=3$,
while similar arguments show that the result holds more generally for $p\geq
3$. In fact, it is not necessary to explicitly require $C^{2}$-continuity at
the incenter since, as shown in [39, Remark 9] and [40, Theorem 4], the
$C^{2}$-continuity automatically holds, and so $\mathbb{W}$ may be written in
the equivalent form

$\displaystyle\mathbb{W}=\\{v\in C^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}_{WF}\\}.$

Applying Lemma 4.5 to the partition $\mathcal{T}_{WF}$ shows that the Worsey-
Farin space may be characterized in the form 4.8 with $\tilde{\mathbb{W}}$ and
$\boldsymbol{\mathbb{G}}$ given by 4.13 after replacing $\mathcal{T}_{A}$ with
$\mathcal{T}_{WF}$.

##  5 Algorithmic implementation

Summarizing thus far, we have succeeded in our primary goal of producing a
finite element scheme involving at most $H^{1}$-conforming spaces that
delivers $H^{2}$-conforming approximations to 2.1. In principle, by choosing
spaces that satisfy (A1)-(A3) and solving a mixed problem 4.7 (which makes no
mention of any $H^{2}$-conforming spaces), we can recover the conforming
approximation to 2.1. However, it is not so straightforward to compute the
solution to 4.7. One problem is that a basis for the space
$\operatorname{Im}\bm{\Xi}_{X}$ in (A2) is often not known or available, which
means that standard direct discretizations of the mixed problem 4.7 cannot be
applied. Instead, we propose to apply the iterated penalty method [29, 41, 42]
to solve 4.7 which has the advantage of not requiring the construction of a
basis for $\operatorname{Im}\bm{\Xi}_{X}$.

Let $\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and
$\boldsymbol{\mathbb{Q}}_{\Gamma}$ satisfy (A1)-(A3), let
$\mathbb{W}_{\Gamma}$ be given by 4.3, and let $[\cdot,\cdot]$ be any inner-
product on $\operatorname{Im}\bm{\Xi}_{X}$. We define the following bilinear
form on
$(\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma})\times(\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma})$:

$\displaystyle A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi})$
$\displaystyle:=a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})+\lambda[\bm{\Xi}_{X}(\tilde{w},\bm{\gamma}),\bm{\Xi}_{X}(\tilde{v},\bm{\psi})].$
(5.1)

Given $\lambda>0$ and an initial approximation
$(\tilde{u}^{0},\bm{\phi}^{0})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$,
the iterated penalty method for 4.7 reads as follows: For $n=0,1,\ldots$,
define
$(\tilde{w}^{n},\bm{\gamma}^{n}),(\tilde{u}^{n+1},\bm{\phi}^{n+1})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
by

$\displaystyle A_{\lambda}(\tilde{w}^{n},\bm{\gamma}^{n};\tilde{v},\bm{\psi})$
$\displaystyle=F_{1}(\bm{\psi})+F_{2}(\tilde{v})\qquad$
$\displaystyle\qquad-[\bm{\Xi}_{X}(\tilde{u}^{n},\bm{\phi}^{n}),\bm{\Xi}_{X}(\tilde{v},\bm{\psi})]\qquad$
$\displaystyle\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma},$
(5.2a) $\displaystyle\tilde{u}^{n+1}$
$\displaystyle=\tilde{u}^{n}+\lambda\tilde{w}^{n},\qquad$ (5.2b)
$\displaystyle\bm{\phi}^{n+1}$
$\displaystyle=\bm{\phi}^{n}+\lambda\bm{\gamma}^{n}.\qquad$ (5.2c)

Under appropriate conditions (given below) the iterates converge to the
solution to 4.7:

$\displaystyle\tilde{w}^{n}\to\tilde{w}_{X},\quad\bm{\gamma}^{n}\to\bm{\gamma}_{X},\quad\text{and}\quad\bm{\Xi}_{X}(\tilde{u}^{n},\bm{\phi}^{n})\to\bm{q}_{X}.$

The rate of convergence will depend on the choice of spaces
$\tilde{\mathbb{W}}_{\Gamma}$ and $\boldsymbol{\mathbb{G}}_{\Gamma}$ and the
choice of inner-product $[\cdot,\cdot]$ and can be quantified in terms of the
following constants

$\displaystyle\beta_{X}$
$\displaystyle:=\inf_{\begin{subarray}{c}\bm{\eta}\in\operatorname{Im}\bm{\Xi}_{X}\\\
\bm{\eta}\neq\bm{0}\end{subarray}}\sup_{\begin{subarray}{c}(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\\\
(\tilde{v},\bm{\psi})\neq(0,\bm{0})\end{subarray}}\frac{[\bm{\Xi}(\tilde{v},\bm{\psi}),\bm{\eta}]}{(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1})\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\eta}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|},\qquad$
$\displaystyle\text{where
}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\eta}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|^{2}:=[\bm{\eta},\bm{\eta}],$
(5.3) $\displaystyle\Upsilon$
$\displaystyle:=\sup_{(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}}\frac{\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{v},\bm{\psi})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|}{\|\tilde{v}\|_{1}+\|\bm{\gamma}\|_{1}}.\qquad$

as follows:

######  Theorem 5.1.

Let
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\operatorname{Im}\bm{\Xi}_{X}$
satisfy 4.7 and let

$\displaystyle\lambda_{0}:=\frac{5M}{4\beta_{X}^{2}}\left(1+\frac{2M}{\alpha_{X}}\right)^{2},$
(5.4)

where $M$ and $\alpha_{X}$ are defined in 2.11 and 4.5, respectively. For
$\lambda>\lambda_{0}$, the iterates
$(\tilde{w}^{n},\bm{\gamma}^{n}),(\tilde{u}^{n},\bm{\phi}^{n})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
given by 5.2 are well-defined, and satisfy

$\displaystyle\|\tilde{w}^{n}-\tilde{w}_{X}\|_{1}+\|\bm{\gamma}^{n}-\bm{\gamma}_{X}\|_{1}$
$\displaystyle\leq\frac{1}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|,$
(5.5)
$\displaystyle\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{u}^{n},\bm{\phi}^{n})-\bm{q}_{X}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|$
$\displaystyle\leq\frac{1}{\beta_{X}}\left\\{\frac{M}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)+\Upsilon\lambda\right\\}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|,$
(5.6)

where

$\displaystyle\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\leq\Upsilon\left\\{\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)\right\\}^{n}\left(\|\tilde{w}^{0}-\tilde{w}_{X}\|_{1}+\|\bm{\gamma}^{0}-\bm{\gamma}_{X}\|_{1}\right).$
(5.7)

Theorem 5.1 shows that if $\lambda$ is sufficiently large, then the iterative
process 5.2 is well-defined and converges at a geometric rate to the solution
to 4.7. The remainder of this section is concerned with the proof of Theorem
5.1, which extends the analysis in [29, Chapter 13] to cases where the
bilinear form $B(\cdot,\cdot)$ need not be positive semidefinite.

###  5.1 Auxiliary results

For the remainder of this section, let $[\cdot,\cdot]$ be an inner-product on
$\operatorname{Im}\bm{\Xi}_{X}$ and let
$\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\cdot\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|$
denote the induced norm 5.3. The following result is a consequence of [29,
Lemma 12.5.10]:

######  Lemma 5.1.

Let $\bm{\Xi}_{X}$ and $\beta_{X}$ be defined as in 4.2 and 5.3. Then
$\bm{\Xi}_{X}$ has a bounded right inverse
$\bm{L}_{X}:\operatorname{Im}\bm{\Xi}_{X}\to\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
satisfying

$\displaystyle\bm{\Xi}_{X}\bm{L}_{X}\bm{\eta}=\bm{\eta}\quad\text{and}\quad\|\bm{L}_{X}\bm{\eta}\|_{1}\leq\beta_{X}^{-1}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\eta}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\qquad\forall\bm{\eta}\in\operatorname{Im}\bm{\Xi}_{X}.$

Next, we examine the inclusion
$\ker\bm{\Xi}_{X}\subset\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
where $\ker\bm{\Xi}_{X}$ is defined in 4.4. If the bilinear forms
$a(\cdot,\cdot)$ and $c(\cdot,\cdot)$ are symmetric positive definite, then
one can define the orthogonal complement of $\ker\bm{\Xi}_{X}$ in
$\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$ in the
usual way. The next result extends this notion to more general bilinear forms
$a(\cdot,\cdot)$ and $c(\cdot,\cdot)$ satisfying conditions 2.11.

######  Lemma 5.2.

The space $\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
admits the following decomposition:

$\displaystyle\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}=\mathbb{K}^{\circ}\oplus\operatorname{Ker}\bm{\Xi}_{X},$
(5.8)

where

$\displaystyle\mathbb{K}^{\circ}:=\\{(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}:a(\bm{\gamma},\bm{\psi})+c(\tilde{w},\tilde{v})=0\
\forall(\tilde{v},\bm{\psi})\in\operatorname{Ker}\bm{\Xi}_{X}\\}.$

In particular, for every
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$,
there exists unique $(\tilde{v},\bm{\psi})\in\mathbb{K}^{\circ}$ and
$(\tilde{n},\bm{\eta})\in\operatorname{Ker}\bm{\Xi}_{X}$ satisfying
$\tilde{w}=\tilde{v}+\tilde{n}$ and $\bm{\gamma}=\bm{\psi}+\bm{\eta}$, with

$\displaystyle\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}$
$\displaystyle\leq\frac{1}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|,$
(5.9a) $\displaystyle\|\tilde{n}\|_{1}+\|\bm{\eta}\|_{1}$
$\displaystyle\leq\|\tilde{w}\|_{1}+\|\bm{\gamma}\|_{1}+\frac{1}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|.$
(5.9b)

###### Proof.

Let
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
be given. Thanks to Lemma 5.1, there exist
$(\tilde{u},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
satisfying

$\displaystyle\bm{\Xi}_{X}(\tilde{u},\bm{\gamma})=\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})\quad\text{and}\quad\|\tilde{u}\|_{1}+\|\bm{\gamma}\|_{1}\leq\beta_{X}^{-1}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|.$

The well-posedness of 4.6 then gives the existence of
$z\in\mathbb{W}_{\Gamma}$ such that

$\displaystyle
B(z,v)=a(\bm{\gamma},\operatorname{\mathbf{grad}}v)+c(\tilde{u},v)\qquad\forall
v\in\mathbb{W}_{\Gamma}$

and

$\displaystyle\|z\|_{2}\leq
2M\alpha_{X}^{-1}\left(\|\tilde{u}\|_{1}+\|\bm{\gamma}\|_{1}\right).$

Thanks to 4.4,
$(\tilde{v},\bm{\psi}):=(\tilde{u}-z,\bm{\gamma}-\operatorname{\mathbf{grad}}z)\in\mathbb{K}^{\circ}$
satisfies
$\bm{\Xi}_{X}(\tilde{v},\bm{\psi})=\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})$ and

$\displaystyle\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\leq\frac{1}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w},\bm{\gamma})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|.$

Choosing $\tilde{n}=\tilde{w}-\tilde{v}$ and $\bm{\eta}=\bm{\gamma}-\bm{\psi}$
gives $(\tilde{n},\bm{\eta})\in\operatorname{Ker}\bm{\Xi}_{X}$ satisfying
5.9b. ∎

Next, we give a stability result for the $A_{\lambda}(\cdot,\cdot)$ bilinear
form:

######  Lemma 5.3.

Suppose $\lambda>\lambda_{0}$, where $\lambda_{0}$ is defined in 5.4. Let
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
be decomposed according to 5.8: i.e. $\tilde{w}=w+\tilde{u}$ and
$\bm{\gamma}=\operatorname{\mathbf{grad}}w+\bm{\phi}$ with
$w\in\mathbb{W}_{\Gamma}$ and $(\tilde{u},\bm{\phi})\in\mathbb{K}^{\circ}$.
Then, there exists
$(\tilde{v},\bm{\psi})\in\mathbb{W}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
such that

$\displaystyle\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda+\lambda_{0}}\right)\|w\|_{2}^{2}+\left(\|\tilde{u}\|_{1}+\|\bm{\phi}\|_{1}\right)^{2}\leq\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi}),$
(5.10)

and

$\displaystyle\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\leq
2\alpha_{X}^{-1}\|w\|+M^{-1}\left(\|\tilde{u}\|_{1}+\|\bm{\phi}\|_{1}\right),$
(5.11)

where $A_{\lambda}(\cdot,\cdot;\cdot,\cdot)$ is defined in 5.1.

Consequently, for $\lambda>\lambda_{0}$ and
$G\in(\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma})^{*}$,
there exists a unique solution to the problem: Find
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
such that

$\displaystyle A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi})$
$\displaystyle=G(\tilde{v},\bm{\psi})\qquad\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}.$
(5.12)

###### Proof.

Let $\lambda>\lambda_{0}$,
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$,
$w\in\mathbb{W}_{\Gamma}$, and $(\tilde{u},\bm{\phi})\in\mathbb{K}^{\circ}$ be
as in the statement of the lemma. The well-posedness of 2.5 means that there
exists a unique solution $v\in\mathbb{W}_{\Gamma}$ of the adjoint problem

$\displaystyle B(z,v)=(w,z)_{2}\qquad\forall z\in\mathbb{W}_{\Gamma}$

that depends continuously on the data $\|v\|_{2}\leq\alpha_{X}^{-1}\|w\|_{2}$,
where $\alpha_{X}$ is the inf-sup constant defined in 4.5. Let
$(\tilde{v},\bm{\psi})$ be given by

$\displaystyle\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\ni(\tilde{v},\bm{\psi})=(v,\operatorname{\mathbf{grad}}v)+M^{-1}(\tilde{u},\bm{\phi}).$
(5.13)

Then, 5.11 holds and direct computation gives

$\displaystyle A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi})$
$\displaystyle=a(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v)+c(w,v)+\epsilon\lambda\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{u},\bm{\phi})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|^{2}$
$\displaystyle\qquad+a(\bm{\phi},\operatorname{\mathbf{grad}}v)+c(\tilde{u},v)+\epsilon(a(\operatorname{\mathbf{grad}}w,\bm{\phi})+c(w,\tilde{u}))$
$\displaystyle\qquad+\epsilon(a(\bm{\phi},\bm{\phi})+c(\tilde{u},\tilde{u}))$
$\displaystyle=\|w\|_{2}^{2}+\epsilon\lambda\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{u},\bm{\phi})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|^{2}+\epsilon(a(\operatorname{\mathbf{grad}}w,\bm{\phi})+c(w,\tilde{u}))$
$\displaystyle\qquad+\epsilon(a(\bm{\phi},\bm{\phi})+c(\tilde{u},\tilde{u}))$

with $\epsilon:=M^{-1}$. Applying 5.9a gives

$\displaystyle A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi})$
$\displaystyle\geq\|w\|_{2}^{2}+\frac{\lambda\beta_{X}^{2}}{M}\left(1+\frac{2M}{\alpha_{X}}\right)^{-2}\left(\|\tilde{u}\|_{1}+\|\bm{\phi}\|_{1}\right)^{2}$
$\displaystyle\qquad-\|w\|_{1}(\|\bm{\phi}\|_{1}+\|\tilde{u}\|_{1})-\|\bm{\phi}\|_{1}^{2}-\|\tilde{u}\|^{2}$
$\displaystyle\geq\left(1-\frac{1}{4\delta}\right)\|w\|_{2}^{2}+\frac{5}{4}\left(\frac{\lambda}{\lambda_{0}}-\frac{4}{5}(1+\delta)\right)\left(\|\tilde{u}\|_{1}+\|\bm{\phi}\|_{1}\right)^{2}$

for any $\delta>0$. Inequality 5.10 now follows on choosing
$\delta=\frac{1}{8}(1+\frac{\lambda}{\lambda_{0}})$.

In order to show uniqueness of solutions to 5.12, suppose that
$(\tilde{w},\bm{\gamma})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
satisfies $A_{\lambda}(\tilde{w},\bm{\gamma};\tilde{v},\bm{\psi})=0$ for all
$(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$.
Then 5.10 implies $(\tilde{w},\bm{\gamma})=(0,\bm{0})$ and uniqueness follows.
∎

###  5.2 Proof of Theorem 5.1

Let $\tilde{\epsilon}^{n}:=\tilde{w}_{X}-\tilde{w}^{n}$,
$\bm{e}^{n}:=\bm{\gamma}_{X}-\bm{\gamma}^{n}$, and
$\bm{r}^{n}:=\bm{q}_{X}-\bm{\Xi}_{X}(\tilde{u}^{n},\bm{\phi}^{n})$.
Subtracting 5.2 from 4.7a and using that
$\bm{\Xi}_{X}(\tilde{w}_{X},\bm{\gamma}_{X})\equiv\bm{0}$ by 4.7b gives, for
$n\in\mathbb{N}_{0}$,

$\displaystyle
A_{\lambda}(\tilde{\epsilon}^{n},\bm{e}^{n};\tilde{v},\bm{\psi})=-[\bm{r}^{n},\bm{\Xi}_{X}(\tilde{v},\bm{\psi})]\qquad\forall(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$
(5.14)

and
$\bm{r}^{n+1}=\bm{r}^{n}-\lambda\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})=\bm{r}^{n}+\lambda\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n})$.
Consequently, we have

$\displaystyle
A_{\lambda}(\tilde{\epsilon}^{n+1},\bm{e}^{n+1};\tilde{v},\bm{\psi})$
$\displaystyle=-[\bm{r}^{n},\bm{\Xi}_{X}(\tilde{v},\bm{\psi}))]-\lambda[\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n}),\bm{\Xi}_{X}(\tilde{v},\bm{\psi})]$
$\displaystyle=A_{\lambda}(\tilde{\epsilon}^{n},\bm{e}^{n};\tilde{v},\bm{\psi})-\lambda[\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n}),\bm{\Xi}_{X}(\tilde{v},\bm{\psi})]$
$\displaystyle=a(\bm{e}^{n},\bm{\psi})+c(\tilde{\epsilon}^{n},\tilde{v})$

for all
$(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}$.
Since $(\tilde{\epsilon}^{n+1},\bm{e}^{n+1})\in\mathbb{K}^{\circ}$ by 5.14,
Lemma 5.3 and 5.13 shows that the choice
$\tilde{v}=M^{-1}\tilde{\epsilon}^{n+1}$ and $\bm{\psi}=M^{-1}\bm{e}^{n+1}$
satisfies

$\displaystyle\left(\|\tilde{\epsilon}^{n+1}\|_{1}+\|\bm{e}^{n+1}\|_{1}\right)^{2}$
$\displaystyle\leq\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)A_{\lambda}(\tilde{\epsilon}^{n+1},\bm{e}^{n+1};\tilde{v},\bm{\psi})$
$\displaystyle=\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)\left(a(\bm{e}^{n},\bm{\psi})+c(\tilde{\epsilon}^{n},\tilde{v})\right)$
$\displaystyle\leq\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)\left(\|\bm{e}^{n}\|_{1}\|\bm{e}^{n+1}\|_{1}+\|\tilde{\epsilon}^{n}\|_{1}\|\tilde{\epsilon}^{n+1}\|_{1}\right)$
$\displaystyle\leq\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)\left(\|\tilde{\epsilon}^{n}\|_{1}+\|\bm{e}^{n}\|_{1}\right)\left(\|\tilde{\epsilon}^{n+1}\|_{1}+\|\bm{e}^{n+1}\|_{1}\right).$

Therefore,

$\displaystyle\|\tilde{\epsilon}^{n}\|_{1}+\|\bm{e}^{n}\|_{1}\leq\left\\{\frac{8}{9}\left(\frac{\lambda_{0}}{\lambda-\lambda_{0}}\right)\right\\}^{n}\left(\|\tilde{\epsilon}^{0}\|_{1}+\|\bm{e}^{0}\|_{1}\right),$

and so 5.7 follows from the relations

$\displaystyle\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|=\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\leq\Upsilon(\|\tilde{\epsilon}^{n}\|_{1}+\|\bm{e}^{n}\|_{1}).$

Again using that $(\tilde{\epsilon}^{n},\bm{e}^{n})\in\mathbb{K}^{\circ}$ and
applying 5.9a gives 5.5. Thanks to the inf-sup condition 5.3, there holds

$\displaystyle\beta_{X}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{r}^{n}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|$
$\displaystyle\leq\sup_{\begin{subarray}{c}(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\\\
(\tilde{v},\bm{\psi})\neq(0,\bm{0})\end{subarray}}\frac{[\bm{\Xi}_{X}(\tilde{v},\bm{\psi}),\bm{r}^{n}]}{\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}}=\sup_{\begin{subarray}{c}(\tilde{v},\bm{\psi})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\\\
(\tilde{v},\bm{\psi})\neq(0,\bm{0})\end{subarray}}\frac{A_{\lambda}(\tilde{\epsilon}^{n},\bm{e}^{n};\tilde{v},\bm{\psi})}{\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}}.$

Since

$\displaystyle
A_{\lambda}(\tilde{\epsilon}^{n},\bm{e}^{n};\tilde{v},\bm{\psi})$
$\displaystyle\leq
M\|\bm{e}^{n}\|_{1}\|\bm{\psi}\|_{1}+M\|\tilde{e}^{n}\|_{1}\|\tilde{v}\|_{1}+\lambda\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{v},\bm{\psi})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|$
$\displaystyle\leq\left\\{M(\|\tilde{e}^{n}\|_{1}+\|\bm{e}^{n}\|_{1})+\Upsilon\lambda\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\right\\}(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1})$
$\displaystyle\leq\left\\{\frac{M}{\beta_{X}}\left(1+\frac{2M}{\alpha_{X}}\right)+\Upsilon\lambda\right\\}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\left|\bm{\Xi}_{X}(\tilde{\epsilon}^{n},\bm{e}^{n})\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}\right|(\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}).$

Inequality 5.6 now follows. ∎

##  6 Numerical examples

Firedrake [6, 43] is a freely available package that provides a wide variety
of finite element schemes. However, it does not offer high order
$C^{1}$-finite element schemes and, as such, it is typical of the types of
packages which the current methodology is targeted. In this section, we use
Firedrake to compute high order $C^{1}$-conforming approximations of some
representative problems. The code is available at [44].

###  6.1 Kirchhoff plate with point load

Let $\Omega\times[-\tau/2,\tau/2]$ be an isotropic steel Kirchhoff plate with
density $\rho=7820\unit[per-mode=symbol]{\per^{3}}$ and thickness
$\tau=10^{-2}\unit{}$, where $\Omega$ is an L-shaped plate with 3 holes as in
Figure 1(a). The plate is simply-supported on the outer boundary and free on
the boundary of the holes. We apply a constant point-load to the mesh vertex
located at $\bm{z}:=(0.66,0.33)$ and let the plate reach equilibrium. We wish
to compute the approximation to the transverse displacement
$w_{X}\in\mathbb{W}_{\Gamma}$ that would be obtained using the Morgan-Scott
space 4.11 of degree $p\geq 2$ as follows:

$\displaystyle w_{X}\in\mathbb{W}_{\Gamma}:\qquad
a_{s}(\operatorname{\mathbf{grad}}w_{X},\operatorname{\mathbf{grad}}v)$
$\displaystyle=10^{3}\cdot v(\bm{z})\qquad\forall v\in\mathbb{W}_{\Gamma},$
(6.1)

where $a_{s}(\cdot,\cdot)$ denotes the bilinear form defined in 2.7, the
Young’s modulus $E=2.1\cdot 10^{11}\unit{}$, the Poisson ration $\nu=0.3$, and
the bending stiffness $D=E\tau^{3}/(12(1-\nu^{2}))$.

We compute the solution to 6.1 by solving the mixed problem 4.1 with
$\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ chosen as in 4.12 with
$a(\cdot,\cdot)=a_{s}(\cdot,\cdot)/D$, $c(\cdot,\cdot)\equiv 0$, $F_{1}\equiv
0$, and $F_{2}(\tilde{v})=(10^{3}/D)\cdot\tilde{v}(\bm{z})$ using the iterated
penalty method 5.2 with penalty parameter $\lambda=10^{3}$ and terminating
when
$\|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\|_{\operatorname{curl}}<10^{-8}$.
The numbers of iterated penalty iterations to achieve convergence were 5 for
$p=3$ and 3 for $4\leq p\leq 15$. The method did not converge in 100
iterations for $p=2$. These results are consistent with the convergence
estimate 5.7 given that the inf-sup constant $\beta_{X}$ 5.3 is independent of
$p$ as shown in Theorem 8.1. The $p=10$ solution $\tilde{w}_{X}$ given by the
iterated penalty method and its gradient
$\operatorname{\mathbf{grad}}\tilde{w}_{X}$, displayed in Figure 1, show that
the upper left corner of the plate displaces in the opposite direction of the
load and that $\tilde{w}_{X}$ is indeed $C^{1}$.

(a) $\tilde{w}_{X}$

(b) $|\operatorname{\mathbf{grad}}\tilde{w}_{X}|$

Figure 1: (a) The displacement $\tilde{w}_{X}$ and (b)
$|\operatorname{\mathbf{grad}}\tilde{w}_{X}|$ for the $p=10$ solution to the
Kirchhoff plate problem 6.1.

###  6.2 Three dimensional example

Let $\Omega=(0,1)^{3}$ be the unit cube and let $\mathbb{W}$ be the
$C^{1}$-spline space 4.11 defined on a sequence of Freudenthal partitions
$\\{\mathcal{T}_{m}\\}$ generated as follows: Given $m\in\mathbb{N}$,
subdivide $\Omega$ into $m^{3}$ congruent subcubes and divide each of these
cubes into six tetrahedra (see e.g. [12] for a precise description), so that
$\mathcal{T}_{m}$ is quasi-uniform of size $h:=1/m$. We consider the
$H^{2}$-projection problem

$\displaystyle w_{X}\in\mathbb{W}_{\Gamma}:\qquad B(w_{X},v)$
$\displaystyle=B(w,v)\qquad\forall v\in\mathbb{W}_{\Gamma},$ (6.2)

where $\Gamma_{f}=\Gamma$, $w=\sin(\pi x)\sin(\pi y)\sin(\pi z)$, and

$\displaystyle
B(u,v):=(D^{2}u,D^{2}v)+(\operatorname{\mathbf{grad}}u,\operatorname{\mathbf{grad}}v)+(u,v)\qquad\forall
u,v\in H^{2}(\Omega).$

Very few finite element packages provide a capability to use
$C^{1}$-conforming elements in the three dimensional setting. Nevertheless,
the current approach can be used to obtain the solution to 6.2 using Firedrake
[6].

We decompose the bilinear form $B(\cdot,\cdot)$ as in 2.10 by choosing

$\displaystyle a(\bm{\theta},\bm{\psi})$
$\displaystyle:=(\operatorname{\mathbf{grad}}\bm{\theta},\operatorname{\mathbf{grad}}\bm{\psi})\qquad$
$\displaystyle\forall\bm{\theta},\bm{\psi}\in\bm{H}^{1}(\Omega),$
$\displaystyle c(u,v)$
$\displaystyle:=(\operatorname{\mathbf{grad}}u,\operatorname{\mathbf{grad}}v)+(u,v)\qquad$
$\displaystyle\forall u,v\in H^{1}(\Omega).$

The linear functional $F(v):=B(w,v)$ is decomposed as in 2.12 in the same
manner:

$\displaystyle
F_{1}(\bm{\psi}):=a(\operatorname{\mathbf{grad}}w,\bm{\psi})\qquad\forall\bm{\psi}\in\bm{H}^{1}(\Omega)\quad\text{and}\quad
F_{2}(v):=c(w,v)\qquad\forall v\in H^{1}(\Omega).$

When $\mathbb{W}$ consists of piecewise degree $p\geq 9$ polynomials, the
optimal error estimate $\|w-w_{X}\|_{2}\leq Ch^{p-1}$ holds on an arbitrary
shape-regular partition [15, Theorem 18.3], in part because a local basis can
be constructed. Whether this bound holds for $p<9$ in general is unclear.

Nevertheless, using the method developed in the current work, we are in a
position to compute the approximation in the cases $p<9$. For
$p\in\\{2,3,\ldots,9\\}$ and the sequence of meshes $\\{\mathcal{T}_{m}\\}$,
we compute the solution to 6.2 by solving the mixed problem 4.1 with
$\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ chosen as in 4.12 using the
iterated penalty method 5.2 with penalty parameter $\lambda=10^{4}$ and
terminating when
$\|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\|_{\operatorname{curl}}<10^{-8}$.
The relative $H^{2}$ errors $\|w_{X}-w\|_{2}/\|w\|_{2}$ displayed in Figure
2(a) show that the optimal error estimate appears to holds for $p\geq 6$,
while the sub-optimal error estimate $\|w-w_{X}\|_{2}\leq
Ch^{\max\\{p-3,0\\}}$ appears to hold for $p\in\\{2,\ldots,5\\}$. The number
of iterated penalty iterations displayed in table 1 tell a similar story – the
number of iterations is bounded as the mesh is refined for $p\geq 6$, but
grows for $p\in\\{2,3,\ldots,5\\}$. Using the same methods and parameters, we
also compute the solution to 6.2 for $p=\\{2,3,\ldots,12\\}$ on the mesh
$\mathcal{T}_{2}$. The relative $H^{2}$ errors in Figure 2(b) appear to decay
at an exponential rate, while the number of iterated penalty iterations in
table 1 remains bounded as $p$ grows.

(a)

(b)

Figure 2: Relative $H^{2}$ errors and for the projection problem 6.2 with degree-$p$ $C^{1}$-spline elements 4.11 (a) on a sequence of meshes with $2\leq p\leq 9$ and (b) on the single mesh $\mathcal{T}_{2}$ with $2\leq p\leq 12$. The number next to each dashed line in (a) indicates its slope. $m$ $p$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$
---|---|---|---|---|---|---|---|---|---|---|---
1 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 3 | 2 | 2 | 2
2 | 3 | 4 | 4 | 4 | 4 | 3 | 3 | 2 | 2 | 1 | 1
3 | 3 | 5 | 5 | 5 | 4 | 3 | 2 | 2 | 1 | 1 |
4 | 3 | 6 | 6 | 8 | 3 | 3 | 2 | 1 |  |  |
5 | 3 | 8 | 8 | 11 | 3 | 3 | 1 | 1 |  |  |
6 | 3 | 11 | 11 | 17 | 3 | 3 |  |  |  |  |
7 | 3 | 15 | 16 | 27 | 3 |  |  |  |  |  |
8 | 4 | 21 | 22 | 40 | 3 |  |  |  |  |  |
Table 1: Iterated penalty iterations for the projection problem 6.2 with
degree-$p$ $C^{1}$-spline elements 4.11, $2\leq p\leq 12$, on the Freudenthal
meshes $\\{\mathcal{T}_{m}\\}$.

##  7 Extension to time-dependent problems?

The numerical examples in the previous section show that the
$H^{2}$-conforming finite element approximation 2.5 can be readily computed
without having to implement $C^{1}$-continuous elements. In this section, we
explore the possibility of applying the method to time-dependent problems even
though such problems do not satisfy the previous assumptions.

Right off the bat, we run into the problem that the computation of the initial
data entails computing the $L^{2}$-projection of the initial data $g$ onto the
$H^{2}$-conforming space:

$\displaystyle w_{0}\in\mathbb{W}_{\Gamma}:\qquad(w_{0},v)$
$\displaystyle=(g,v)\qquad\forall v\in\mathbb{W}_{\Gamma},$ (7.1)

where $g\in L^{2}(\Omega)$. The $L^{2}$-inner product $(\cdot,\cdot)$ is not
$H^{2}_{\Gamma}(\Omega)$-elliptic, and hence does not satisfy condition 2.3.
However, thanks to equivalence of norms on the finite dimensional space
$\mathbb{W}_{\Gamma}(\Omega)$, there exists $\varpi_{X}>0$, depending on the
dimension of $\mathbb{W}_{\Gamma}$, such that

$\displaystyle\varpi_{X}\|v\|_{2}^{2}\leq\|v\|^{2}\qquad\forall
v\in\mathbb{W}_{\Gamma}.$ (7.2)

This means that, at the discrete level, the $L^{2}$-inner product on
$\mathbb{W}_{\Gamma}$ satisfies assumptions 2.2 and 4.5 meaning that we can
apply our method and solve the mixed problem 4.7 with
$c(\cdot,\cdot)=(\cdot,\cdot)$ and $a(\cdot,\cdot)\equiv 0$ to recover the
solution to 7.1. However, we expect the number of iterated penalty iterations
to degenerate as the discretization is refined since $\varpi_{X}$ is not
uniformly bounded away from zero with respect to the dimension of
$\mathbb{W}_{\Gamma}$.

A similar situation arises in time-stepping, where advancing the approximation
in time involves solving problems of the form

$\displaystyle w_{n}\in\mathbb{W}_{\Gamma}:\qquad(w_{n},v)+(\Delta
t)^{m}B(w_{n},v)=F_{n}(v)\qquad\forall v\in\mathbb{W}_{\Gamma},$ (7.3)

where $B(\cdot,\cdot)$ satisfies conditions 2.2, 2.3, 2.4, 2.10, and 4.5, $m$
is the order of the time derivative of the problem, and
$F_{n}\in\mathbb{W}_{\Gamma}^{*}$ depends on the data from previous time
steps. If the bilinear form $B(\cdot,\cdot)$ is elliptic, i.e.
$B(v,v)\geq\alpha\|v\|_{2}^{2}$ for all $v\in H^{2}_{\Gamma}(\Omega)$ for some
$\alpha>0$, then there holds

$\displaystyle(\varpi_{X}+\alpha(\Delta
t)^{m})\|v\|_{2}^{2}\leq\|v\|^{2}+(\Delta t)^{m}B(v,v)\qquad\forall
v\in\mathbb{W}_{\Gamma},$

where $\varpi_{X}$ is given by 7.2. Once again, this means that the conditions
needed for our approach are satisfied at the discrete level, so that the
method can be used to solve the mixed problem 4.7 to recover the solution to
7.3. However, the fact that the constants in the assumptions depend on the
dimension of the discrete space manifests itself through growth in the number
of iterated penalty iterations as $\Delta t\to 0$ or as the discretization is
refined. Nevertheless, the method can be applied to time-dependent problems
subject to the above proviso. We illustrate the performance in the case of a
dynamic Kirchhoff plate.

###  7.1 Dynamic Kirchhoff plate

We continue the example in section 6.1 by removing the point load that was
responsible for the initial deflection and consider the resulting vibrations
of the plate which are governed by the problem: Find
$w(t)\in\mathbb{W}_{\Gamma}$ such that

$\displaystyle\frac{d^{2}}{dt^{2}}\left(w(t),v\right)+\frac{1}{\rho\tau}a_{s}(\operatorname{\mathbf{grad}}w(t),\operatorname{\mathbf{grad}}v)$
$\displaystyle=0\quad\forall v\in\mathbb{W}_{\Gamma},$

subject to initial conditions $w(0)=w_{X}$ and $\partial_{t}w(0)=0$, where
$w_{X}$ is the initial deflection given by 6.1. As in the previous section,
$a_{s}(\cdot,\cdot)$ denotes the bilinear form defined in 2.7.

We discretize in time with a Newmark method (see e.g. [45]) as follows. Let
$\Delta t>0$ be a given time step and $\beta,\delta\geq 0$ be given
parameters. For $n=1,2,\ldots,$ the approximation to $\partial_{tt}w$ at time
$t_{n}:=n\cdot\Delta t$ is updated by the following variational formulation:
Find $w_{n}^{[2]}\in\mathbb{W}_{\Gamma}$ such that

$\displaystyle(w_{n}^{[2]},v)+\frac{\beta(\Delta
t)^{2}}{\rho\tau}a_{s}(\operatorname{\mathbf{grad}}w_{n}^{[2]},\operatorname{\mathbf{grad}}v)=-\frac{1}{\rho\tau}a_{s}(\operatorname{\mathbf{grad}}\hat{w}_{n-1},\operatorname{\mathbf{grad}}v)\qquad\forall
v\in\mathbb{W}_{\Gamma},$ (7.4)

where

$\displaystyle\hat{w}_{n-1}:=w_{n-1}+(\Delta
t)w_{n-1}^{[1]}+\left(\frac{1}{2}-\beta\right)(\Delta t)^{2}w_{n-1}^{[2]}.$

The approximations to $w$ and $\partial_{t}w$ at time $t_{n}$ are then updated
by

$\displaystyle w_{n}$ $\displaystyle=\hat{w}_{n-1}+\beta(\Delta
t)^{2}w_{n}^{[2]}\quad\text{and}\quad w_{n}^{[1]}=w_{n-1}^{[1]}+(\Delta
t)\left((1-\delta)w_{n-1}^{[1]}+\delta w_{n}^{[2]}\right).$ (7.5)

The initial displacement $w_{0}:=w_{X}$ is solution from section 6.1 and the
initial velocity $w_{0}^{[1]}=0$, while the initial acceleration $w_{0}^{[2]}$
is determined by

$\displaystyle
w_{0}^{[2]}\in\mathbb{W}_{\Gamma}:\qquad(w_{0}^{[2]},v)=-\frac{1}{\rho\tau}a_{s}(\operatorname{\mathbf{grad}}w_{0},\operatorname{\mathbf{grad}}v)\qquad\forall
v\in\mathbb{W}_{\Gamma},$ (7.6)

which is precisely the form of problem in 7.1. If $\delta=1/2$ and
$\beta=1/4$, then [45] shows that the energy

$\displaystyle\mathcal{E}_{n}:=\frac{1}{2}(w_{n}^{[1]},w_{n}^{[1]})+\frac{1}{2\rho\tau}a_{s}(\operatorname{\mathbf{grad}}w_{n},\operatorname{\mathbf{grad}}w_{n}),\qquad
n\geq 0,$ (7.7)

remains constant for all $n\geq 0$.

####  7.1.1 $L^{2}$-projection

In order to carry out the timestepping of the scheme, it is necessary to solve
7.6 to determine the initial acceleration. A standard inverse estimate
$\|u\|_{1}\leq Ch^{-1}p^{2}\|u\|$ for $u\in\\{u\in
L^{2}(\Omega):u|_{K}\in\mathcal{P}_{p}(K)\ \forall\mathcal{T}\\}$ (see e.g.
[46]) gives

$\displaystyle
Ch^{4}p^{-8}\|v\|_{2}^{2}\leq\|v\|^{2}\leq\|v\|_{2}^{2}\qquad\forall
v\in\mathbb{W}_{\Gamma},$ (7.8)

where $C$ is a positive constant independent of $h$ and $p$, which means that
the bilinear form appearing in 7.6 satisfies assumptions 2.2 and 4.5, with a
stability constant $\alpha_{X}:=Ch^{4}p^{-8}$ that degenerates as the
discretization is refined. To help compensate for this degeneration, we choose
the inner-product in 4.7 to be the $L^{2}$ inner-product; i.e.
$[\cdot,\cdot]=(\cdot,\cdot)$.

We compute the solution to 7.6 by solving the mixed problem 4.7 with
$\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ chosen as in 4.12 with
$a(\cdot,\cdot)\equiv 0$, $c(\cdot,\cdot)=(\cdot,\cdot)$,
$F_{1}(\cdot)=-a_{s}(\operatorname{\mathbf{grad}}w_{0},\cdot)/(\rho\tau)$,
$F_{2}(\cdot)\equiv 0$, and $[\cdot,\cdot]=(\cdot,\cdot)$ using the iterated
penalty method 5.2 with penalty parameter $\lambda=2\cdot 10^{4}$ and
terminating when $\|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\|<10^{-8}$.
The iterated penalty method required 2 iterations to converge for $3\leq p\leq
7$ and 3 for $8\leq p\leq 12$, but failed to converge for $p\geq 13$.
Consequently, the degeneracy predicted by 7.8 does indeed manifest in practice
for sufficiently large $p$. However, the flexibility of the inner-product
$[\cdot,\cdot]$ does aid to some extent – if we instead choose $[\cdot,\cdot]$
to be the $\bm{H}(\mathrm{curl};\Omega)$ inner-product
$(\cdot,\cdot)_{\operatorname{curl}}$, then the iterated penalty method fails
to converge for $p\geq 5$. The $p=10$ solution, denoted $\tilde{w}_{0}^{[2]}$
and displayed in Figure 3, is concentrated at the point load and is indeed
$C^{1}$-conforming.

(a) $\tilde{w}_{0}^{[2]}$

(b) $|\operatorname{\mathbf{grad}}\tilde{w}_{0}^{[2]}|$

Figure 3: (a) The initial acceleration $\tilde{w}_{0}^{[2]}$ 7.6 and (b)
$|\operatorname{\mathbf{grad}}\tilde{w}_{0}^{[2]}|$ for the $p=10$ solution to
the dynamic Kirchhoff plate problem.

####  7.1.2 Time-stepping

The approximation is advanced in time by solving the linear system 7.4 with
$\Delta t=2\cdot 10^{-4}$. The bilinear form appearing in problem 7.4 is of
the form 7.3 with

$\displaystyle C\left(h^{4}p^{-8}+\frac{\beta(\Delta
t)^{2}}{\rho\tau}\right)\|v\|_{2}^{2}\leq\|v\|^{2}+\frac{\beta(\Delta
t)^{2}}{\rho\tau}a_{s}(\operatorname{\mathbf{grad}}v,\operatorname{\mathbf{grad}}v)\qquad\forall
v\in H_{\Gamma}^{2}(\Omega),$ (7.9)

where $C$ is a positive constant independent of the discretization parameters
$h$, $p$, and $\Delta t$. We compute the solution to 7.4 by solving the mixed
problem 4.7 with $\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ chosen as
in 4.12 with $a(\cdot,\cdot)=(\beta(\Delta
t)^{2}/(\rho\tau))a_{s}(\cdot,\cdot)$, $c(\cdot,\cdot)=(\cdot,\cdot)$,
$F_{1}(\cdot)=-a_{s}(\hat{\bm{\gamma}}_{n-1},\cdot)/(\rho\tau)$,
$F_{2}(\cdot)\equiv 0$, and $[\cdot,\cdot]=(\cdot,\cdot)+(\Delta
t)^{2}(\operatorname{curl}\cdot,\operatorname{curl}\cdot)$ using the iterated
penalty method 5.2 with penalty parameter $\lambda=10^{4}$ and terminating
when

$\displaystyle\left(\|\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\|^{2}+(\Delta
t)^{2}\|\operatorname{curl}\bm{\Xi}_{X}(\tilde{w}^{n},\bm{\gamma}^{n})\|^{2}\right)^{1/2}<10^{-8}.$

Here,

$\displaystyle\hat{\bm{\gamma}}_{n-1}:=\bm{\gamma}_{n-1}+(\Delta
t)\bm{\gamma}_{n-1}^{[1]}+\left(\frac{1}{2}-\beta\right)(\Delta
t)^{2}\bm{\gamma}_{n-1}^{[2]},$

where $\bm{\gamma}_{n}^{[2]}$ is the approximation to
$\operatorname{\mathbf{grad}}w_{n}^{[2]}$ given by the iterated penalty method
and $\bm{\gamma}_{n}$ and $\bm{\gamma}_{n}^{[1]}$ are updated by the same
expressions as in 7.5 for $n\geq 1$. The initial value $\bm{\gamma}_{0}$ is
the approximation to $\operatorname{\mathbf{grad}}w_{0}$ given by the iterated
penalty method from the computation in section 6.1 and
$\bm{\gamma}_{0}^{[1]}:=\bm{0}$.

(a)

(b)

Figure 4: (a) The number of iterated penalty (IP) iterations to compute the
initial acceleration $w_{0}^{[2]}$ given by 7.6 and the max number of
iterations to advance the acceleration $w_{n}^{[2]}$ via 7.4; and (b) the
maximum relative energy deviation 7.10 computed using
$\operatorname{\mathbf{grad}}\tilde{w}_{n}$ and $\bm{\gamma}_{n}$. The metrics
in (a) and (b) are computed over 251 time steps.

The largest number of iterated penalty iterations over 251 time steps,
displayed in Figure 4(a), show that at most 4 iterations are required for
$3\leq p\leq 10$, but the iterated penalty method failed to converge for
$p\geq 11$. Thus, the degeneracy in 7.9 also arises in practice for
sufficiently large $p$. If we instead choose
$[\cdot,\cdot]=(\cdot,\cdot)_{\operatorname{curl}}$, then the iterated penalty
method fails to converge for $p\geq 5$, which again shows that the flexibility
in the choice of $[\cdot,\cdot]$ is helpful is pushing the boundaries of the
method. A snapshot of the displacement $\tilde{w}_{n}$ and velocity
$\tilde{w}_{n}^{[1]}$ at time $t=0.0502$ is displayed in Figure 5 and the
plots of the gradients confirm that the solutions are indeed $C^{1}$.

If the linear system 7.4 is solved exactly for all $n\geq 1$, then the energy
7.7 is exactly conserved. To measure the quantify the effect of computing the
solution to 7.4 using the iterated penalty method with the above tolerances,
we display the maximum relative energy deviation

$\displaystyle\max_{1\leq n\leq
251}\frac{|\mathcal{E}_{n}-\mathcal{E}_{0}|}{\mathcal{E}_{0}}$ (7.10)

in Figure 4(b) for $3\leq p\leq 10$. We compute $\mathcal{E}_{n}$ in two ways:
(i) using $\operatorname{\mathbf{grad}}\tilde{w}_{n}$ in 7.7 and (ii) using
$\bm{\gamma}_{n}$ in place of $\operatorname{\mathbf{grad}}\tilde{w}_{n}$. The
maximum relative energy deviation is on the order of $2\cdot 10^{-6}$ when
$\operatorname{\mathbf{grad}}\tilde{w}_{n}$ is used, while the relative energy
deviation is on the order of $10^{-8}$ for $3\leq p\leq 5$, but rises to
$4\cdot 10^{-7}$ as $p$ increases to $11$ due to the accumulation of roundoff
errors when $\bm{\gamma}_{n}$ is used.

(a) $\tilde{w}_{n}$

(b) $|\operatorname{\mathbf{grad}}\tilde{w}_{n}|$

(c) $\tilde{w}_{n}^{[1]}$

(d) $|\operatorname{\mathbf{grad}}\tilde{w}_{n}^{[1]}|$

Figure 5: The $p=10$ displacement $\tilde{w}_{n}$ and its gradient (a-b) and
velocity $\tilde{w}_{n}^{[1]}$ and its gradient (c-d) for the dynamic
Kirchhoff plate problem at time $t=0.0502$.

##  8 Uniform stability of the Morgan-Scott space

Throughout this section, let $d=2$ and $\mathbb{W}$ be Morgan-Scott space
4.11, and let $\tilde{\mathbb{W}}$ and $\boldsymbol{\mathbb{G}}$ be chosen as
in 4.12:

$\displaystyle\tilde{\mathbb{W}}$ $\displaystyle=\\{v\in
C(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\ \forall K\in\mathcal{T}\\},$
$\displaystyle\boldsymbol{\mathbb{G}}$
$\displaystyle=\\{\bm{\theta}\in\bm{C}(\Omega):\bm{\theta}|_{K}\in[\mathcal{P}_{p-1}(K)]^{d}\
\forall K\in\mathcal{T}\\}.$

We shall show that when the polynomial degree is sufficiently large ($p\geq
5$) the space $\boldsymbol{\mathbb{Q}}_{\Gamma}=\operatorname{Im}\bm{\Xi}_{X}$
can be explicitly characterized and the inf-sup constant $\beta_{X}$ 5.3 is
independent of both the mesh size $h:=\max_{K\in\mathcal{T}}\mathrm{diam}(K)$
and the polynomial degree $p$:

######  Theorem 8.1.

For $p\geq 5$, there holds

$\displaystyle\beta_{X}\geq\beta_{0}\xi_{\mathcal{T}}$ (8.1)

where $\beta_{X}$ is the inf-sup constant defined in 5.3 with
$[\cdot,\cdot]=(\cdot,\cdot)_{\operatorname{curl}}$, $\beta_{0}>0$ is
independent of $h$ and $p$, and $\xi_{\mathcal{T}}>0$ is a quantity depending
only on the topology of the mesh.

Theorem 8.1 is an immediate consequence of Theorem 8.3 below. The quantity
$\xi_{\mathcal{T}}$ in 8.1 depends on the topology of the mesh but not the
mesh size or polynomial degree. While $\xi_{\mathcal{T}}$ may degenerate on
certain mesh configurations, we shall assume that $\xi_{\mathcal{T}}$ is
bounded away from zero for present purposes. Further details and an explicit
formula for $\xi_{\mathcal{T}}$ can be found in [16, eq. (A.3)].

The degree $p-1$ Brezzi-Douglas-Marini space [47] is defined by

$\displaystyle\bm{BDM}_{\Gamma}^{p-1}:=\left\\{\bm{\gamma}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega):\bm{\gamma}|_{K}\in\mathcal{P}_{p-1}(K)^{2}\
\forall K\in\mathcal{T}\right\\}.$

Then, $\operatorname{Im}\bm{\Xi}_{X}$ is contained in the proper subspace of
$\bm{BDM}_{\Gamma}^{p-1}$ consisting of functions whose curl belongs to
$\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$ since

$\displaystyle\operatorname{curl}\operatorname{Im}\bm{\Xi}_{X}=\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\implies\operatorname{Im}\bm{\Xi}_{X}\subseteq\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}.$

The above inclusion is in fact an equality. In order to see this, we construct
a right inverse for the the $\bm{\Xi}_{X}$ operator. To this end, it will be
useful to start by considering the curl operator.

###  8.1 Right inverse for the curl operator

Let $\\{\Gamma_{f}^{(i)}\\}_{i=1}^{N_{f}}$ denote the $N_{f}\geq 0$ connected
components of $\Gamma_{f}$. Then, the following is a generalization of [16,
Lemma A.3].

######  Theorem 8.2.

Suppose that $\Gamma_{f}$ has $N_{f}$ connected components
$\\{\Gamma_{f}^{(i)}\\}_{i=1}^{N_{f}}$. Let

$\displaystyle r\in L^{2}_{\Gamma}(\Omega):=\\{r\in
L^{2}(\Omega):(r,1)=0\text{ if }|\Gamma_{f}|=0\\}.$ (8.2)

and $\vec{\omega}\in\mathbb{R}^{N_{f}}$ satisfy

$\displaystyle\sum_{i=1}^{N_{f}}\omega_{i}=\int_{\Omega}r\ d\bm{x}.$ (8.3)

Then, there exists $\bm{\psi}\in\bm{G}_{\Gamma}(\Omega)$ satisfying

$\displaystyle\operatorname{curl}\bm{\psi}=r,\quad(\bm{t}\cdot\bm{\psi},1)_{\Gamma_{f}^{(i)}}=\omega_{i},\quad
1\leq i\leq N_{f},$ (8.4)

and

$\displaystyle\|\bm{\psi}\|_{1}\leq C\left(\|r\|+|\vec{\omega}|\right),$

where $|\vec{\omega}|=\sum_{i=1}^{N_{f}}|\omega_{i}|$ and $C>0$ is independent
of $r$ and $\vec{\omega}$.

###### Proof.

First note that Lemma A.1 of [48], while stated for a simply-connected domain
$\Omega$, extends to the present case (with the exact same proof). Then, the
exact same arguments used for [16, Lemma A.3] completes the proof. ∎

The following corollary of Theorem 8.2 shows that the operator
$\operatorname{curl}:\boldsymbol{\mathbb{G}}_{\Gamma}\to\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$
admits a continuous right inverse.

######  Corollary 8.1.

Let $p\geq 5$. Let $r\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$
and $\vec{\omega}\in\mathbb{R}^{N_{f}}$ satisfy 8.3. Then, there exists
$\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$ satisfying 8.4 and

$\displaystyle\|\bm{\psi}\|_{1}\leq
C\left(\xi_{\mathcal{T}}^{-1}\|r\|+|\vec{\omega}|\right),$

where $C>0$ is independent of $r$, $\vec{\omega}$, $h$, $p$, and
$\xi_{\mathcal{T}}$.

###### Proof.

First note that Corollary 5.1 of [48], stated for simply-connected $\Omega$,
also holds for a connected Lipschitz polygon with the exact same proof. The
result now follows from the exact same arguments as in the proof of [16, Lemma
A.4] using Theorem 8.2 in place of [16, Lemma A.3]. ∎

###  8.2 Right inverse for the $\bm{\Xi}_{X}$ operator

…$\Gamma_{cs}^{(1)}$$\Gamma_{f}^{(1)}$$\Gamma_{cs}^{(2)}$$\Gamma_{f}^{(2)}$$\Gamma_{f}^{(N)}$$\Gamma_{cs}^{(N)}$
Figure 6: Ordering of $\Gamma_{cs}^{(i)}$ and $\Gamma_{f}^{(i)}$,
$i=1,\ldots,N$ for simply-connected $\Omega\subset\mathbb{R}^{2}$.

We now turn to the construction of a right inverse of the $\bm{\Xi}_{X}$
operator. The first result is a type of inverse trace theorem.

######  Lemma 8.1.

Let $\\{\Gamma_{cs}^{(i)}\\}_{i=1}^{N_{cs}}$ denote the $N_{cs}$ connected
components of $\Gamma_{cs}$. For every $\vec{\kappa}\in\mathbb{R}^{N_{cs}}$,
there exists $w\in H^{2}(\Omega)$ satisfying

$\displaystyle w|_{\Gamma_{cs}^{(i)}}=\kappa_{i},\quad 1\leq j\leq
N_{cs},\quad\partial_{n}w|_{\Gamma}=0,\quad\text{and}\quad\|w\|_{2}\leq
C|\vec{\kappa}|,$ (8.5)

where $C$ is independent of $\vec{\kappa}$.

###### Proof.

Consider first the case where $\Omega$ is simply-connected so that
$N_{f}=N_{cs}$ and label the components $\\{\Gamma_{f}^{(i)}\\}$ and
$\\{\Gamma_{cs}^{(i)}\\}$ in a counterclockwise orientation so that
$\Gamma_{f}^{(i)}$ is located between $\Gamma_{cs}^{(i)}$ and
$\Gamma_{cs}^{(i+1)}$, $i=1,2,\ldots,N_{f}$, where
$\Gamma_{cs}^{(N_{f}+1)}:=\Gamma_{cs}^{(1)}$ ; see Figure 6. Define $f\in
L^{2}(\Gamma)$ as follows: $f$ is piecewise constant on $\Gamma$, taking the
value $\kappa_{i}$ on $\Gamma_{cs}^{(i)}$ and $\kappa_{i+1}$ on
$\Gamma_{f}^{(i)}$, where $\kappa_{N_{cs}+1}:=\kappa_{1}$, except on the
$N_{f}$ edges laying on $\Gamma_{f}^{(i)}$ that have a vertex at
$\bar{\Gamma}_{f}^{(i)}\cap\bar{\Gamma}_{cs}^{(i)}$, where $f$ is a smooth
transition from $\kappa_{i}$ to $\kappa_{i+1}$ with vanishing tangential
derivatives at the endpoints of the edges.

Note that by construction $f$ is continuous, $f|_{\gamma}\in
C^{\infty}(\gamma)$ for all edges $\gamma$ of $\Gamma$, and the tangential
derivative of $f$ vanishes at all vertices of $\Gamma$. Thanks to [49, Theorem
6.1], there exists $w\in H^{2}(\Omega)$ with

$\displaystyle
w|_{\Gamma}=f,\quad\partial_{n}w|_{\Gamma}=0,\quad\text{and}\quad\|w\|_{2}\leq
C\left(\|f\|_{\Gamma}+\|f^{\prime}\bm{t}\|_{\frac{1}{2},\Gamma}\right)\leq
C|\vec{\kappa}|.$

We now return to the more general case where $\Omega$ satisfies the conditions
in section 2. Applying the above argument on each $\Omega_{i}$, we obtain
$w_{i}\in H^{2}(\Omega_{i})$, $0\leq i\leq N_{H}$, satisfying

$\displaystyle w_{i}|_{\Gamma_{cs}^{(j)}}=\kappa_{j}\quad\text{if
$|\Gamma_{cs}^{(j)}\cap\partial\Omega_{i}|>0$},\quad\partial_{n}w|_{\partial\Omega_{i}}=0,\quad\text{and}\quad\|w_{i}\|_{2}\leq
C|\vec{\kappa}|,$

for all $1\leq j\leq N_{cs}$. Now, for $1\leq i\leq N_{H}$, [50] shows there
exists an extension $\tilde{w}_{i}\in H^{2}(\mathbb{R}^{2})$ satisfying
$\tilde{w}_{i}=w_{i}$ on $\Omega_{i}$ and
$\|\tilde{w}_{i}\|_{2,\mathbb{R}^{2}}\leq C\|w_{i}\|_{2,\Omega_{i}}$.
Moreover, for $1\leq i\leq N_{H}$, there exist $\zeta_{i}\in
C^{\infty}(\mathbb{R}^{2})$ satisfying $\zeta_{i}\equiv 1$ on $\Omega_{i}$ and
$\zeta_{i}\equiv 0$ on $\bar{\Omega}_{j}$ for $1\leq j\leq N_{H}$, $i\neq j$.
Finally, there exists $\zeta_{0}\in C^{\infty}(\mathbb{R}^{2})$ satisfying
$\zeta_{0}\equiv 1$ on $\partial\Omega_{0}$ and $\zeta_{0}\equiv 0$ on
$\Omega_{i}$ for $1\leq i\leq N_{H}$. The function
$w:=\sum_{i=0}^{N_{H}}\zeta_{i}\tilde{w}_{i}$ then satisfies 8.5. ∎

The next result shows the existence of a continuous right inverse of the
operator $\bm{\Xi}_{X}$.

######  Theorem 8.3.

Let $p\geq 5$. For every
$\bm{\eta}\in\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}$,
there exists $\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}$ and
$\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$ satisfying

$\displaystyle\bm{\Xi}_{X}(\tilde{v},\bm{\psi})=\bm{\eta}\quad\text{and}\quad\|\tilde{v}\|_{1}+\|\bm{\psi}\|_{1}\leq
C\xi_{\mathcal{T}}^{-1}\|\bm{\eta}\|_{\operatorname{curl}},$ (8.6)

where $C$ is independent of $\bm{\eta}$, $\xi_{\mathcal{T}}$, $h$, and $p$.
Consequently,
$\operatorname{Im}\bm{\Xi}_{X}=\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}$
and 8.1 holds with $\beta_{0}=C^{-1}\xi_{\mathcal{T}}$.

###### Proof.

Let
$\bm{\eta}\in\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}$
be given, and let $\\{\Gamma^{(j)}\\}_{j=1}^{J}$ denote the connected
components of $\Gamma$. Let
$r=\operatorname{curl}\bm{\eta}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$
and let $\vec{\omega}\in\mathbb{R}^{N_{f}}$ be any vector satisfying

$\displaystyle\sum_{\begin{subarray}{c}1\leq i\leq N_{f}\\\
\Gamma_{f}^{(i)}\subseteq\Gamma^{(j)}\end{subarray}}\omega_{i}=\langle\bm{t}\cdot\bm{\eta},1\rangle_{\Gamma^{(j)}},\quad
1\leq j\leq J,\quad\text{and}\quad|\vec{\omega}|\leq
C\sum_{j=0}^{J}|\langle\bm{t}\cdot\bm{\eta},1\rangle_{\Gamma^{(j)}}|.$

Then, the pair $(r,\vec{\omega})$ satisfies

$\displaystyle\sum_{i=1}^{N_{f}}\omega_{i}=\sum_{j=0}^{J}\langle\bm{t}\cdot\bm{\eta},1\rangle_{\Gamma^{(j)}}=\langle\bm{t}\cdot\bm{\eta},1\rangle_{\Gamma}=\operatorname{curl}\bm{\eta}=r\quad\text{and}\quad\|r\|+|\vec{\omega}|\leq
C\|\bm{\eta}\|_{\operatorname{curl}},$

where we used trace theorem to conclude $|\vec{\omega}|\leq
C\|\bm{\eta}\|_{\operatorname{curl}}$. Applying Corollary 8.1 then shows that
there exists $\bm{\theta}\in\boldsymbol{\mathbb{G}}_{\Gamma}$ satisfying

$\displaystyle\operatorname{curl}(\bm{\theta}+\bm{\eta})=0,\quad\langle\bm{t}\cdot(\bm{\theta}+\bm{\eta}),1\rangle_{\Gamma^{(j)}}=0,\quad
1\leq j\leq J,\quad\text{and}\quad\|\bm{\theta}\|_{1}\leq
C\xi_{\mathcal{T}}^{-1}\|\bm{\eta}\|_{\operatorname{curl}}.$

Thanks to [31, p. 37 Theorem 3.1], there exists $\tilde{w}\in H^{1}(\Omega)$
satisfying $\operatorname{\mathbf{grad}}\tilde{w}=\bm{\theta}+\bm{\eta}$.
First note that
$\operatorname{\mathbf{grad}}\tilde{w}\in\bm{BDM}_{\Gamma}^{p-1}$, and so

$\displaystyle\tilde{w}\in\\{v\in H^{1}(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}\\}=\\{v\in C(\Omega):v|_{K}\in\mathcal{P}_{p}(K)\
\forall K\in\mathcal{T}\\}.$

Moreover,

$\displaystyle\langle\partial_{t}\tilde{w},u\rangle_{\Gamma}=\langle\bm{t}\cdot(\bm{\theta}+\bm{\eta}),u\rangle_{\Gamma}=0\qquad\forall
u\in H^{1}(\Omega):u|_{\Gamma_{f}}=0,$

and so $\tilde{w}|_{\Gamma_{cs}^{(i)}}\in\mathbb{R}$ for $1\leq i\leq N_{cs}$.
The function $\tilde{u}:=\tilde{w}-\tilde{w}|_{\Gamma_{cs}^{(1)}}$ then
satisfies

$\displaystyle\|\tilde{u}\|_{1}\leq
C\|\operatorname{\mathbf{grad}}\tilde{u}\|=C\|\operatorname{\mathbf{grad}}\tilde{w}\|\leq
C\xi_{\mathcal{T}}^{-1}\|\bm{\eta}\|_{\operatorname{curl}}.$

To correct the trace of $\tilde{u}$, we appeal to Lemma 8.1 for the existence
of $\phi\in H^{2}(\Omega)$ satisfying

$\displaystyle\phi|_{\Gamma_{cs}}=\tilde{u}|_{\Gamma_{cs}},\quad\partial_{n}\phi|_{\Gamma}=0,\quad\text{and}\quad\|\phi\|_{2}\leq
C\sum_{i=1}^{N}|\tilde{u}|_{\Gamma_{cs}^{(i)}}|\leq C\|\tilde{u}\|_{1}\leq
C\xi_{\mathcal{T}}^{-1}\|\bm{\eta}\|_{\operatorname{curl}}.$

Let $\tilde{\phi}\in W^{5}$ denote the piecewise degree 5 Girault-Scott
interpolant [51] of $\phi$ which satisfies

$\displaystyle\tilde{\phi}|_{\Gamma_{cs}}=\phi|_{\Gamma_{cs}},\quad\partial_{n}\tilde{\phi}|_{\Gamma}=0,\quad\text{and}\quad\|\tilde{\phi}\|_{2}\leq
C\|\phi\|_{2}\leq C\xi_{\mathcal{T}}^{-1}\|\bm{\eta}\|_{\operatorname{curl}}.$

The functions $\tilde{v}=\tilde{u}-\tilde{\phi}$ and
$\bm{\psi}=\bm{\theta}-\operatorname{\mathbf{grad}}\tilde{\phi}$ then satisfy
$\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}$,
$\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$, and 8.6. Consequently,
$\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}\subseteq\operatorname{Im}\bm{\Xi}_{X}$,
and so
$\operatorname{Im}\bm{\Xi}_{X}=\\{\bm{\gamma}\in\bm{BDM}_{\Gamma}^{p-1}:\operatorname{curl}\bm{\gamma}\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\\}$.
The bound $\beta_{X}\geq C^{-1}\xi_{\mathcal{T}}$ then follows from 8.6. ∎

##  9 Relationship to previous work

In previous work [16], we considered the special case that $d=2$,
$c(\cdot,\cdot)\equiv 0$ for which the mixed problem 3.11 can be written in an
alternative form that avoids the vector-valued space
$\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$. In this section, we show that the
current method can be regarded as a generalization of the method in [16] to
the cases $d\in\\{2,3\\}$ and where $c(\cdot,\cdot)$ need not vanish. We begin
with two properties of the space $\bm{G}_{\Gamma}(\Omega)$. The first is an
immediate consequence of Theorem 8.2:

(P1)

For all $r\in L^{2}_{\Gamma}(\Omega)$ and
$\vec{\kappa}\in\mathbb{R}^{N_{f}-1}$, there exists
$\bm{\psi}\in\bm{G}_{\Gamma}(\Omega)$ satisfying
$\operatorname{curl}\bm{\psi}=r$,
$(\bm{t}\cdot\bm{\psi},1)_{\Gamma_{f}^{(i)}}=\kappa_{i}$, $1\leq i\leq
N_{f}-1$, and $\|\bm{\psi}\|_{1}\leq C(\|r\|+|\vec{\kappa}|)$, where
$L^{2}_{\Gamma}(\Omega)$ is defined as in 8.2.

The second property, which follows from [31, p. 37, Theorem 3.1],
characterizes the gradients of $H^{2}_{\Gamma}(\Omega)$ functions in terms of
curl-free vector-fields in $\bm{G}_{\Gamma}(\Omega)$:

(P2)

$\operatorname{\mathbf{grad}}H^{2}_{\Gamma}(\Omega)=\\{\bm{\gamma}\in\bm{G}_{\Gamma}(\Omega):\operatorname{curl}\bm{\gamma}\equiv
0\text{ and }(\bm{t}\cdot\bm{\gamma},1)_{\Gamma_{f}^{(i)}}=0,\ 1\leq i\leq
N_{f}-1\\}$.

With these properties in hand, we have the following representation result:

######  Lemma 9.1.

For every pair $(r,\vec{\kappa})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$, there exists a unique
vector-field

$\displaystyle\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}:=\\{\bm{\phi}\in\operatorname{Im}\bm{\Xi}:(\bm{\phi},\operatorname{\mathbf{grad}}\tilde{v})_{\operatorname{curl}}=0\quad\forall\tilde{v}\in
H^{1}_{\Gamma}(\Omega)\\},$ (9.1)

satisfying

$\displaystyle\tilde{b}(\bm{\eta};r,\vec{\kappa}):=(r,\operatorname{curl}\bm{\eta})+\sum_{i=1}^{N_{f}-1}(\bm{t}\cdot\bm{\eta},\kappa_{i})_{\Gamma_{f}^{(i)}}=-(\bm{\zeta},\bm{\eta})_{\operatorname{curl}}\qquad\forall\bm{\eta}\in\operatorname{Im}\bm{\Xi}.$
(9.2)

Conversely, for every
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$,
there exists a unique pair $(r,\vec{\kappa})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$ satisfying 9.2.

###### Proof.

Let $(r,\vec{\kappa})\in L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$ be
given. Thanks to the trace theorem, there holds

$\displaystyle|\tilde{b}(\bm{\eta};r,\vec{\kappa})|\leq
C(\|r\|+|\vec{\kappa}|)\|\bm{\eta}\|_{\operatorname{curl}}\qquad\forall\bm{\eta}\in\operatorname{Im}\bm{\Xi}=\bm{H}_{\Gamma}(\mathrm{curl};\Omega),$

and so the mapping
$\bm{H}_{\Gamma}(\mathrm{curl};\Omega)\ni\bm{\eta}\mapsto\tilde{b}(\bm{\eta};r,\vec{\kappa})$
is a continuous linear functional on $\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$.
By the Riesz representation theorem, there exists
$\bm{\zeta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$ satisfying 9.2 and
$\|\bm{\zeta}\|_{\operatorname{curl}}\leq C(\|r\|+|\vec{\kappa}|)$. Since
$\tilde{b}(\operatorname{\mathbf{grad}}\tilde{v};r,\vec{\kappa})=0$ for all
$\tilde{v}\in H^{1}_{\Gamma}(\Omega)$, there holds
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$.
Moreover, the the linear mapping $(r,\vec{\kappa})\mapsto\bm{\zeta}$ is
injective since if $\tilde{b}(\bm{\eta};r,\vec{\kappa})=0$ for all
$\bm{\eta}\in\bm{H}_{\Gamma}(\mathrm{curl};\Omega)$, then (P1) shows that
$r\equiv 0$ and $\vec{\kappa}=\vec{0}$.

Now let
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$ be
given. Thanks to (P1), there exists $\tilde{\beta}>0$ such that

$\displaystyle\inf_{\begin{subarray}{c}(r,\vec{\kappa})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}\\\
(r,\vec{\kappa})\neq(0,\vec{0})\end{subarray}}\sup_{\begin{subarray}{c}\bm{\theta}\in\bm{G}_{\Gamma}(\Omega)\\\
\bm{\theta}\neq\bm{0}\end{subarray}}\frac{\tilde{b}(\bm{\theta};r,\vec{\kappa})}{(\|r\|+|\vec{\kappa}|)\|\bm{\theta}\|_{1}}\geq\tilde{\beta},$

and so there exists $r\in L^{2}_{\Gamma}(\Omega)$ and
$\vec{\kappa}\in\mathbb{R}^{N_{f}-1}$ satisfying

$\displaystyle\tilde{b}(\bm{\psi};r,\vec{\kappa})=-(\bm{\zeta},\bm{\psi})_{\operatorname{curl}}\qquad\forall\bm{\psi}\in\bm{G}_{\Gamma}(\Omega).$

Now let $\bm{\eta}\in\operatorname{Im}\bm{\Xi}$ be decomposed as in Lemma 3.1:
i.e. $\bm{\eta}=\operatorname{\mathbf{grad}}\tilde{v}+\bm{\psi}$, where
$\tilde{v}\in H^{1}_{\Gamma}(\Omega)$ and
$\bm{\psi}\in\bm{G}_{\Gamma}(\Omega)$. Then, there holds

$\displaystyle\tilde{b}(\bm{\eta};r,\vec{\kappa})=\tilde{b}(\operatorname{\mathbf{grad}}\tilde{v};r,\vec{\kappa})+\tilde{b}(\bm{\psi};r,\vec{\kappa})=-(\bm{\zeta},\bm{\psi})_{\operatorname{curl}}=-(\bm{\zeta},\bm{\eta})_{\operatorname{curl}},$

where we used that $\tilde{b}(\cdot;r,\vec{\kappa})$ and
$(\bm{\zeta},\cdot)_{\operatorname{curl}}$ vanish on
$\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega)$. As a result, 9.2 holds,
and we have shown that the continuous mapping
$(r,\vec{\kappa})\mapsto\bm{\zeta}$ is bijective. ∎

The next result shows that solutions to 3.11 satisfy a system of four
equations consisting of two elliptic projections and a Stokes-like equation:

######  Lemma 9.2.

Suppose that $c(\cdot,\cdot)\equiv 0$. Let
$F_{1}\in\bm{G}_{\Gamma}(\Omega)^{*}$, $F_{2}\in H^{1}_{\Gamma}(\Omega)^{*}$,
$(\tilde{w},\bm{\gamma},\bm{q})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\times\operatorname{Im}\bm{\Xi}$
be the solution to 3.11, and $\tilde{z}\in H^{1}_{\Gamma}(\Omega)$ satisfy

$\displaystyle(\operatorname{\mathbf{grad}}\tilde{z},\operatorname{\mathbf{grad}}\tilde{v})$
$\displaystyle=F_{2}(\tilde{v})\qquad$ $\displaystyle\forall\tilde{v}\in
H^{1}_{\Gamma}(\Omega).$ (9.3)

Then, there exists unique $(r,\vec{\kappa})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$ such that

$\displaystyle a(\bm{\gamma},\bm{\psi})+\tilde{b}(\bm{\psi};r,\vec{\kappa})$
$\displaystyle=(\operatorname{\mathbf{grad}}\tilde{z},\bm{\psi})+F_{1}(\bm{\psi})\qquad$
$\displaystyle\forall\bm{\psi}\in\bm{G}_{\Gamma}(\Omega),$ (9.4a)
$\displaystyle\tilde{b}(\bm{\gamma};s,\vec{\mu})$ $\displaystyle=0\qquad$
$\displaystyle\forall(s,\vec{\mu})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1},$ (9.4b)

and $\tilde{w}$ satisfies

$\displaystyle(\operatorname{\mathbf{grad}}\tilde{w},\operatorname{\mathbf{grad}}\tilde{u})=(\bm{\gamma},\operatorname{\mathbf{grad}}\tilde{u})\qquad\forall\tilde{u}\in
H^{1}_{\Gamma}(\Omega).$ (9.5)

###### Proof.

Let
$\bm{\zeta}=\bm{q}-\operatorname{\mathbf{grad}}\tilde{z}\in\operatorname{Im}\bm{\Xi}$.
Choosing $\bm{\psi}\equiv\bm{0}$ in 3.11a gives

$\displaystyle(\bm{\zeta},\operatorname{\mathbf{grad}}\tilde{v})_{\operatorname{curl}}=(\bm{q},\operatorname{\mathbf{grad}}\tilde{v})_{\operatorname{curl}}-(\operatorname{\mathbf{grad}}\tilde{z},\operatorname{\mathbf{grad}}\tilde{v})_{\operatorname{curl}}=0\qquad\forall\tilde{v}\in
H^{1}_{\Gamma}(\Omega).$

Consequently,
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$
9.1, and so there exists unique $(r,\vec{\kappa})\in
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$ satisfying 9.2 thanks to
Lemma 9.1. Choosing $\tilde{v}\equiv 0$ in 3.11a then gives

$\displaystyle
F_{1}(\bm{\psi})=a(\bm{\gamma},\bm{\psi})-(\bm{\psi},\bm{q})_{\operatorname{curl}}=a(\bm{\gamma},\bm{\psi})-(\bm{\zeta},\bm{\psi})_{\operatorname{curl}}-(\operatorname{\mathbf{grad}}\tilde{z},\bm{\psi})\qquad\forall\bm{\psi}\in\bm{G}_{\Gamma}(\Omega),$

and 9.4a follows.

Now let $(s,\vec{\mu})\in L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}$.
Thanks to Lemma 9.1, there exists
$\bm{\eta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$
such that
$b(\bm{\gamma};s,\vec{\mu})=(\bm{\eta},\operatorname{\mathbf{grad}}\tilde{w}-\bm{\gamma})_{\operatorname{curl}}=0$,
and 9.4b follows. Finally, choosing
$\bm{\eta}=\operatorname{\mathbf{grad}}\tilde{u}$ for $\tilde{u}\in
H^{1}_{\Gamma}(\Omega)$ in 3.11b gives

$\displaystyle(\operatorname{\mathbf{grad}}\tilde{w},\operatorname{\mathbf{grad}}\tilde{u})=(\operatorname{\mathbf{grad}}\tilde{w},\operatorname{\mathbf{grad}}\tilde{u})_{\operatorname{curl}}=(\bm{\gamma},\operatorname{\mathbf{grad}}\tilde{u})_{\operatorname{curl}}=(\bm{\gamma},\operatorname{\mathbf{grad}}\tilde{u}),$

and 9.5 follows. ∎

Finally, we show that the converse of Lemma 9.2 holds:

######  Lemma 9.3.

Let $c(\cdot,\cdot)\equiv 0$. For every $F_{1}\in\bm{G}_{\Gamma}(\Omega)^{*}$
and $F_{2}\in H^{1}_{\Gamma}(\Omega)^{*}$, the following variational problem
is uniquely solvable: Find
$(\tilde{z},\bm{\gamma},r,\vec{\kappa},\tilde{w})\in
H^{1}_{\Gamma}(\Omega)\times\bm{G}_{\Gamma}(\Omega)\times
L^{2}_{\Gamma}(\Omega)\times\mathbb{R}^{N_{f}-1}\times H^{1}_{\Gamma}(\Omega)$
such that 9.3, 9.4, and 9.5 holds. Moreover, there exists unique
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$
such that

$\displaystyle(\bm{\zeta},\bm{\eta})_{\operatorname{curl}}=-\tilde{b}(\bm{\eta};r,\vec{\kappa})\qquad\forall\bm{\eta}\in\operatorname{Im}\bm{\Xi},$
(9.6)

and the triplet
$(\tilde{w},\bm{\gamma},\bm{\zeta}+\operatorname{\mathbf{grad}}\tilde{z})$ is
the unique solution to 3.11.

###### Proof.

Existence of solutions to 9.3, 9.4, and 9.5 follow from Lemma 9.2. We now turn
to uniqueness of solutions. Suppose that $F_{1}\equiv 0$ and $F_{2}\equiv 0$.
Then 9.3 shows that $\tilde{z}\equiv 0$. Equation 9.4b and (P2) mean that
$\bm{\gamma}=\operatorname{\mathbf{grad}}w$ for some $w\in
H^{2}_{\Gamma}(\Omega)$. Testing 9.4a with
$\bm{\psi}=\operatorname{\mathbf{grad}}v$ for $v\in H^{2}_{\Gamma}(\Omega)$
gives $a(\operatorname{\mathbf{grad}}w,\operatorname{\mathbf{grad}}v)=0$ for
all $v\in H^{2}_{\Gamma}(\Omega)$, and so $w\equiv 0$ thanks to 2.3. Testing
9.4a with $\bm{\psi}$ chosen as in (P1) then gives $r=0$ and $\vec{\kappa}=0$.
Since $\bm{\gamma}=\operatorname{\mathbf{grad}}w=\bm{0}$, 9.5 shows that
$\tilde{w}\equiv 0$.

The existence of
$\bm{\zeta}\in(\operatorname{\mathbf{grad}}H^{1}_{\Gamma}(\Omega))^{\perp}$
satisfying 9.6 follows from Lemma 9.1 and that the triplet
$(\tilde{w},\bm{\gamma},\bm{\zeta}+\operatorname{\mathbf{grad}}\tilde{z})$
satisfies 3.11 follows from similar arguments as in the proof of Lemma 9.2. ∎

We now repeat the same arguments at the discrete level to avoid the appearance
of the space $\operatorname{Im}\bm{\Xi}_{X}$. We start by stating the
following discrete analogues of (P1) and (P2) respectively:

(B1)

For all $\vec{\kappa}\in\mathbb{R}^{N_{f}-1}$, there exists
$\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$ satisfying

$\displaystyle\operatorname{curl}\bm{\psi}\equiv
0\quad\text{and}\quad(\bm{t}\cdot\bm{\psi},1)_{\Gamma_{f}^{(i)}}=\kappa_{i},\quad
1\leq i\leq N_{f}-1.$

In particular, note that (B1) means that for all
$r\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$ and
$\vec{\kappa}\in\mathbb{R}^{N_{f}-1}$, there exists
$\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma}$ satisfying

$\displaystyle\operatorname{curl}\bm{\psi}\equiv
r\quad\text{and}\quad(\bm{t}\cdot\bm{\psi},1)_{\Gamma_{f}^{(i)}}=\kappa_{i},\quad
1\leq i\leq N_{f}-1.$

Since $\boldsymbol{\mathbb{G}}_{\Gamma}\times\mathbb{R}^{N_{f}-1}$ is finite
dimensional, $\bm{\psi}$ can also be chosen so that $\|\bm{\psi}\|_{1}\leq
C_{X}(\|r\|+|\vec{\kappa}|)$, where $C_{X}>0$ is a constant independent of $r$
and $\vec{\kappa}$ possibly depending on the dimension of
$\boldsymbol{\mathbb{G}}_{\Gamma}\times\mathbb{R}^{N_{f}-1}$. Thus, (B1)
implies that (P2) holds with $\bm{G}_{\Gamma}(\Omega)$ and
$L^{2}_{\Gamma}(\Omega)$ replaced by $\boldsymbol{\mathbb{G}}_{\Gamma}$ and
$\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}$.

The discrete analogue of (P2) is precisely (P2) with $H^{2}_{\Gamma}(\Omega)$
and $\bm{G}_{\Gamma}(\Omega)$ replaced by $\mathbb{W}_{\Gamma}$ and
$\boldsymbol{\mathbb{G}}_{\Gamma}$:

(B2)

$\operatorname{\mathbf{grad}}\mathbb{W}_{\Gamma}=\\{\bm{\gamma}\in\boldsymbol{\mathbb{G}}_{\Gamma}:\operatorname{curl}\bm{\gamma}\equiv
0\text{ and }(\bm{t}\cdot\bm{\gamma},1)_{\Gamma_{f}^{(i)}}=0,\ 1\leq i\leq
N_{f}-1\\}$.

Then, exactly the same arguments that led to Lemmas 9.2 and 9.3 give the
following result:

######  Theorem 9.1.

Suppose that $c(\cdot,\cdot)\equiv 0$. Let $\tilde{\mathbb{W}}_{\Gamma}$,
$\boldsymbol{\mathbb{G}}_{\Gamma}$, and $\boldsymbol{\mathbb{Q}}_{\Gamma}$ be
given spaces satisfying (A1)-(A3) and (B1)-(B2). Suppose that
$F_{2}\in\tilde{\mathbb{W}}_{\Gamma}^{*}$,
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{q}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\boldsymbol{\mathbb{Q}}_{\Gamma}$
is the solution to 4.1, and $\tilde{z}_{X}\in\tilde{\mathbb{W}}_{\Gamma}$
satisfies

$\displaystyle(\operatorname{\mathbf{grad}}\tilde{z}_{X},\operatorname{\mathbf{grad}}\tilde{v})$
$\displaystyle=F_{2}(\tilde{v})\qquad$
$\displaystyle\forall\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}.$ (9.7)

Then, there exists unique
$(r_{X},\vec{\kappa})\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\times\mathbb{R}^{N_{f}-1}$
such that

$\displaystyle
a(\bm{\gamma}_{X},\bm{\psi})+\tilde{b}(\bm{\psi};r_{X},\vec{\kappa})$
$\displaystyle=(\operatorname{\mathbf{grad}}\tilde{z}_{X},\bm{\psi})+F_{1}(\bm{\psi})\qquad$
$\displaystyle\forall\bm{\psi}\in\boldsymbol{\mathbb{G}}_{\Gamma},$ (9.8a)
$\displaystyle\tilde{b}(\bm{\gamma}_{X};s,\vec{\mu})$ $\displaystyle=0\qquad$
$\displaystyle\forall(s,\vec{\mu})\in\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\times\mathbb{R}^{N_{f}-1},$
(9.8b)

and $\tilde{w}_{X}$ satisfies

$\displaystyle(\operatorname{\mathbf{grad}}\tilde{w}_{X},\operatorname{\mathbf{grad}}\tilde{u})=(\bm{\gamma}_{X},\operatorname{\mathbf{grad}}\tilde{u})\qquad\forall\tilde{u}\in\tilde{\mathbb{W}}_{\Gamma}.$
(9.9)

Conversely, for all $F_{1}\in\boldsymbol{\mathbb{G}}_{\Gamma}^{*}$ and
$F_{2}\in\tilde{\mathbb{W}}_{\Gamma}^{*}$, the following variational problem
is uniquely solvable: Find
$(\tilde{z}_{X},\bm{\gamma}_{X},r_{X},\vec{\kappa},\tilde{w}_{X})\in\tilde{\mathbb{W}}_{\Gamma}\times\boldsymbol{\mathbb{G}}_{\Gamma}\times\operatorname{curl}\boldsymbol{\mathbb{G}}_{\Gamma}\times\mathbb{R}^{N_{f}-1}\times\tilde{\mathbb{W}}_{\Gamma}$
such that 9.7, 9.8, and 9.9 holds. Moreover, there exists unique vector-field

$\displaystyle\bm{\zeta}\in(\operatorname{\mathbf{grad}}\tilde{\mathbb{W}}_{\Gamma})^{\perp}:=\\{\bm{\phi}\in\operatorname{Im}\bm{\Xi}_{X}:(\bm{\phi},\operatorname{\mathbf{grad}}\tilde{v})_{\operatorname{curl}}=0\quad\forall\tilde{v}\in\tilde{\mathbb{W}}_{\Gamma}\\},$

such that

$\displaystyle(\bm{\zeta}_{X},\bm{\eta})_{\operatorname{curl}}=-\tilde{b}(\bm{\eta};r_{X},\vec{\kappa})\qquad\forall\bm{\eta}\in\boldsymbol{\mathbb{Q}}_{\Gamma},$

and the triple
$(\tilde{w}_{X},\bm{\gamma}_{X},\bm{\zeta}_{X}+\operatorname{\mathbf{grad}}\tilde{z}_{X})$
is the unique solution to 4.1.

Incidentally, 9.7, 9.8, and 9.9 is precisely the method we derived in [16]
under the assumption that $\Omega$ is simply-connected and $F_{1}\equiv 0$.
Theorem 9.1 shows that the method still produces the
$\mathbb{W}_{\Gamma}$-conforming finite element approximation 2.5 under the
more general domain assumptions in section 2 and for nonzero
$F_{1}\in\boldsymbol{\mathbb{G}}_{\Gamma}^{*}$.

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