remarkRemark hypothesisHypothesis claimClaim Preconditioning Stokes FlowM.
Ainsworth and C. Parker

# A Mass Conserving Mixed $hp$-FEM Scheme for Stokes Flow. Part III:
Implementation and Preconditioning ††thanks: Submitted to the editors DATE.

Mark Ainsworth  Division of Applied Mathematics, Brown University, Providence,
RI<EMAIL_ADDRESS><EMAIL_ADDRESS>Charles
Parker22footnotemark: 2

###### Abstract

This is the third part in a series on a mass conserving, high order, mixed
finite element method for Stokes flow. In this part, we study a block-diagonal
preconditioner for the indefinite Schur complement system arising from the
discretization of the Stokes equations using these elements. The underlying
finite element method is uniformly stable in both the mesh size $h$ and
polynomial order $p$, and we prove bounds on the eigenvalues of the
preconditioned system which are independent of $h$ and grow modestly in $p$.
The analysis relates the Schur complement system to an appropriate variational
setting with subspaces for which exact sequence properties and inf-sup
stability hold. Several numerical examples demonstrate agreement with the
theoretical results.

###### keywords:

preconditioning mixed $hp$-finite elements, Stokes flow, domain decomposition

65N30, 65N55, 76M10

##  1 Introduction

This paper is the third part in a series discussing a mass conserving, high
order, mixed finite element method for Stokes flow on a simply connected
polygon $\Omega$ with boundary $\Gamma=\partial\Omega$: Find
$(\bm{u},p)\in\bm{H}^{1}_{0}(\Omega)\times L^{2}_{0}(\Omega)$ such that

(1a) $\displaystyle a(\bm{u},\bm{v})+b(\bm{v},p)$
$\displaystyle=(\bm{f},\bm{v})$
$\displaystyle\forall\bm{v}\in\bm{H}^{1}_{0}(\Omega)$ (1b) $\displaystyle
b(\bm{u},q)$ $\displaystyle=0$ $\displaystyle\forall q\in L^{2}_{0}(\Omega),$

where $\bm{u}=(u_{1},u_{2})$ is the fluid velocity, $p$ the pressure,
$\bm{f}\in\bm{L}^{2}(\Omega)$ the body force,
$a(\bm{u},\bm{v}):=\nu(\nabla\bm{u},\nabla\bm{v})$ and
$b(\bm{v},p):=-(\operatorname{div}\bm{v},p)$. Without loss of generality, by
rescaling, we may reduce Eq. 1 to the case where the kinematic viscosity
$\nu=1$. Here, $H^{s}(\Omega)$ and $H^{s}_{0}(\Omega)$ denote the usual
Sobolev spaces [1], $\bm{H}^{s}(\Omega)$, $\bm{H}^{s}_{0}(\Omega)$ the vector
valued Sobolev spaces, i.e. $\bm{H}^{s}(\Omega):=[H^{s}(\Omega)]^{2}$, and
$L^{2}_{0}(\Omega)$ denotes the (closed) subspace of square integrable
functions with vanishing average value:

$\displaystyle L^{2}_{0}(\Omega):=\left\\{q\in L^{2}(\Omega):\int_{\Omega}q\
d\bm{x}=0\right\\}.$

Problem Eq. 1 is approximated a using mixed, high order, finite element scheme
on a mesh $\mathcal{T}$ as follows: Find
$(\bm{u}_{hk},p_{hk})\in\bm{V}_{0}\times Q_{0}$ such that

(2a) $\displaystyle a(\bm{u}_{hk},\bm{v})+b(\bm{v},p_{hk})$
$\displaystyle=(\bm{f},\bm{v})$ $\displaystyle\forall\bm{v}\in\bm{V}_{0}$ (2b)
$\displaystyle b(\bm{u}_{hk},q)$ $\displaystyle=0$ $\displaystyle\forall q\in
Q_{0},$

where the finite element spaces are chosen to be [6, 7, 16]:

$\displaystyle V$ $\displaystyle:=\\{v\in
H^{1}(\Omega):v|_{K}\in\mathcal{P}_{k}(K)\ \forall K\in\mathcal{T},\ v\text{
is ${C}^{1}$ at noncorner vertices}\\},$ $\displaystyle{Q}$
$\displaystyle:=\\{q\in{L}^{2}(\Omega):q|_{K}\in\mathcal{P}_{k-1}(K)\ \forall
K\in\mathcal{T},\ q\text{ is $C^{0}$ at noncorner vertices}\\},$

$V_{0}:=V\cap H^{1}_{0}(\Omega)$, $\bm{V}_{0}=V_{0}\times V_{0}$, $Q_{0}=Q\cap
L^{2}_{0}(\Omega)$, $\mathcal{P}_{k}$ denotes the space of all polynomials of
degree at most $k$, and a corner vertex is a vertex of the physical domain
$\Omega$. The local degrees of freedom of the spaces $V$ and $Q$ are
illustrated in Fig. 1.

(a)

(b)

Figure 1: Local degrees of freedom for the finite element spaces (a) $V$ and
(b) $Q$ in the case $k=5$. Dots indicate degrees of freedom corresponding to
evaluation at the point located at the dot while circles indicate gradient
evaluation.

In Part I [6], it was shown that that these elements are uniformly inf-sup
stable in the mesh size $h$ and polynomial order $k$ if the mesh $\mathcal{T}$
is corner-split which, roughly speaking, means that every element
$K\in\mathcal{T}$ has at most one edge lying on the domain boundary $\Gamma$;
for a precise definition, see [6, p. 12].

######  Theorem 1.1 (Theorem 3.1 & Corollary 3.2 [6]).

If the mesh $\mathcal{T}$ is corner-split, then for every $q\in Q_{0}$, there
exists a $\bm{v}\in\bm{V}_{0}$ such that $\operatorname{div}\bm{v}=q$ and

$\displaystyle\|\bm{v}\|_{\bm{H}^{1}(\Omega)}\leq\beta^{-1}\|q\|_{L^{2}(\Omega)},$

where $0<\beta<1$ is independent of $k$ and $h$. Thus, the spaces
$\bm{V}_{0}\times Q_{0}$ are uniformly inf-sup stable:

(3) $\displaystyle\inf_{0\neq q\in
Q_{0}}\sup_{\bm{0}\neq\bm{v}\in\bm{V}_{0}}\frac{b(\bm{v},q)}{\|\bm{v}\|_{\bm{H}^{1}(\Omega)}\|q\|_{L^{2}(\Omega)}}\geq\beta.$

Strictly speaking, [6, Corollary 3.2] shows that $\beta$ depends on the mesh-
dependent quantity $\Theta(\mathcal{T})$ defined in [6, eq. (3.2)], but is
nevertheless bounded independently of the mesh size $h$ and polynomial degree
$k$. Moreover, the finite element solution $\bm{u}_{hk}$ will be pointwise
divergence free [6, §1 and Theorem 2.6]. In Part II [7], it was shown that
these elements have optimal approximation properties in both the mesh size $h$
and the polynomial order $k$. On locally quasi-uniform meshes, the finite
element solution to Eq. 2 converges at the optimal algebraic rate to the
solution to Eq. 1 [7, Theorem 2.2]. Moreover, if the data $\bm{f}$ belongs to
a particular countably normed space, then the finite element method with
properly geometrically graded meshes converges exponentially fast as both the
mesh is refined and the polynomial degree is increased [7, Corollary 2.5]. The
spaces $\bm{V}_{0}\times Q_{0}$ are currently the only known triangular finite
element spaces that are uniformly inf-sup stable in $h$ and $k$, give
pointwise divergence free velocities, and posses optimal approximation
properties.

In the current work, we turn to issues relating to the practical application
of the method. In particular, we give explicit bases for the spaces $\bm{V}$
and $Q$ that result in an efficient preconditioner for the solution of the
resulting linear system for Eq. 2, which may be used in conjunction with an
iterative solver for indefinite systems, such as MINRES [30]. The
preconditioner consists of a standard static condensation, or elimination of
the interior degrees of freedom, along with an Additive Schwarz preconditioner
(ASM) [35, 37] for the resulting Schur complement system associated with the
interface degrees of freedom. Thanks to a judicious choice of basis, the
condition number grow at most as $\log^{3}k$ as $k$ is increased, and is
uniform in the mesh size.

The current work finds inspiration in the early works of [12, 19, 34, 38] for
$h$-version methods, [8, 22] for $hp$-version finite element methods, and [20,
24, 31, 32] for spectral element methods, each of which developed block
diagonal and/or block triangular preconditioners in terms of existing
preconditioners for second order elliptic problems. Unfortunately, these types
of approaches do not readily extend to the mixed finite element scheme Eq. 2
owing to the additional smoothness requirements imposed at element vertices
for both the velocity and pressure spaces. Our treatment of these degrees of
freedom is similar to the treatment of the second order derivative degrees of
freedom in preconditioning the stiffness matrix for $H^{2}(\Omega)$-conforming
methods [5] and the treatment of the vertex degrees of freedom in
preconditioning the mass matrix for $H^{1}(\Omega)$ problems [4].

##  2 General Form of a Block-Diagonal Preconditioner

By fixing bases for the spaces $\bm{V}_{0}$ and $Q$, we may express
$\bm{u}\in\bm{V}_{0}$ and $p\in Q$ as

$\displaystyle\bm{u}=\vec{u}_{E}^{T}\vec{\Phi}_{E}+\vec{u}_{I}^{T}\vec{\Phi}_{I}\quad\text{and}\quad
p=\vec{p}_{e}^{T}\vec{\psi}_{e}+\vec{p}_{\iota}^{T}\vec{\psi}_{\iota},$

for suitable $\vec{u}_{E}$, $\vec{u}_{I}$, $\vec{p}_{e}$, $\vec{p}_{\iota}$,
where $\vec{\Phi}_{E}$ is the vector of exterior velocity basis functions
(vertex and edge functions), $\vec{\Phi}_{I}$ the vector of interior velocity
basis functions, $\vec{\psi}_{e}$ the vector of exterior pressure basis
functions, and $\vec{\psi}_{\iota}$ the vector of interior pressure basis
functions. Here, the exterior pressure functions consist of vertex functions
and a function corresponding to the average value over each element. The
variational problem Eq. 2 in matrix form then reads

(4)
$\displaystyle\begin{bmatrix}\begin{array}[]{cc|cc}\bm{A}_{EE}&\bm{B}_{Ee}&\bm{A}_{EI}&\bm{B}_{E\iota}\\\
\bm{B}_{eE}&\bm{0}&\bm{B}_{eI}&\bm{0}\\\
\hline\cr\bm{A}_{IE}&\bm{B}_{Ie}&\bm{A}_{II}&\bm{B}_{I\iota}\\\ \bm{B}_{\iota
E}&\bm{0}&\bm{B}_{\iota
I}&\bm{0}\end{array}\end{bmatrix}\begin{bmatrix}\begin{array}[]{c}\vec{u}_{E}\\\
\vec{p}_{e}\\\ \hline\cr\vec{u}_{I}\\\
\vec{p}_{\iota}\end{array}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\begin{array}[]{c}\vec{f}_{E}\\\ \vec{0}\\\
\hline\cr\vec{f}_{I}\\\ \vec{0}\end{array}\end{bmatrix}.$

The matrix appearing in Eq. 4 is symmetric but indefinite, owing to the zero
subblocks. The pressure variable in problem Eq. 2 is unique up to a constant,
meaning that the matrix in Eq. 4 has a one-dimensional null space.
Nevertheless, the system Eq. 4 is consistent since the components of the load
vector corresponding to pressure basis functions vanish identically and, a
fortiori, are orthogonal to constant pressure modes. Consequently, the system
Eq. 4 is uniquely solvable up to the addition of a constant in the pressure
thanks to the inf-sup condition Eq. 3 and the uniform ellipticity of
$a(\cdot,\cdot)$.

The conditioning of the matrix, in common with standard $hp$-finite elements,
degenerates rapidly with both the mesh size $h$ and the polynomial order $k$
of the elements. Indeed, almost every practical choice of basis function
results in a rapid deterioration of the condition number $k$, even for
symmetric, positive definite systems [3, 29]. We seek a preconditioner for the
symmetric, indefinite system Eq. 4 which controls the growth of the
conditioning in both $h$ and $k$.

The first step towards preconditioning is to eliminate, or statically
condense, the interior degrees of freedom to arrive at the Schur complement
system

(5) $\displaystyle\bm{S}\begin{bmatrix}\vec{u}_{E}\\\
\vec{p}_{e}\end{bmatrix}=\begin{bmatrix}\vec{f}_{E}^{*}\\\
\vec{g}_{e}^{*}\end{bmatrix}:=\begin{bmatrix}\vec{f}_{E}\\\
\vec{0}\end{bmatrix}-\begin{bmatrix}\bm{A}_{EI}&\bm{B}_{E\iota}\\\
\bm{B}_{eI}&\bm{0}\end{bmatrix}\begin{bmatrix}\bm{A}_{II}&\bm{B}_{I\iota}\\\
\bm{B}_{\iota I}&\bm{0}\end{bmatrix}^{-1}\begin{bmatrix}\vec{f}_{I}\\\
\vec{0}\end{bmatrix},$

where

(6) $\displaystyle\bm{S}$
$\displaystyle=\begin{bmatrix}\widetilde{\bm{A}}&\widetilde{\bm{B}}^{T}\\\
\widetilde{\bm{B}}&\bm{0}\end{bmatrix}=\begin{bmatrix}\bm{A}_{EE}&\bm{B}_{Ee}\\\
\bm{B}_{eE}&\bm{0}\end{bmatrix}-\begin{bmatrix}\bm{A}_{EI}&\bm{B}_{E\iota}\\\
\bm{B}_{eI}&\bm{0}\end{bmatrix}\begin{bmatrix}\bm{A}_{II}&\bm{B}_{I\iota}\\\
\bm{B}_{\iota
I}&\bm{0}\end{bmatrix}^{-1}\begin{bmatrix}\bm{A}_{IE}&\bm{B}_{Ie}\\\
\bm{B}_{\iota E}&\bm{0}\end{bmatrix}$

and we have used the fact (see Lemma 5.1) that the $(2,2)$ block of Schur
complement matrix $\bm{S}$ reduces to the zero matrix. The inverse of the
matrix appearing in Eqs. 5 and 6 is well-defined by Theorem 3.4. After the
degrees of freedom on the element interfaces are in hand, the interior degrees
of freedom can be recovered by back substitution using the relation

$\displaystyle\begin{bmatrix}\vec{u}_{I}\\\
\vec{p}_{\iota}\end{bmatrix}=\begin{bmatrix}\bm{A}_{II}&\bm{B}_{I\iota}\\\
\bm{B}_{\iota I}&\bm{0}\end{bmatrix}^{-1}\left(\begin{bmatrix}\vec{f}_{I}\\\
\vec{0}\end{bmatrix}-\begin{bmatrix}\bm{A}_{IE}&\bm{B}_{Ie}\\\ \bm{B}_{\iota
E}&\bm{0}\end{bmatrix}\begin{bmatrix}\vec{u}_{E}\\\
\vec{p}_{e}\end{bmatrix}\right).$

The element interface degrees of freedom are obtained by solving the Schur
complement system Eq. 5. The matrix $\bm{S}$ defined in Eq. 6 is symmetric and
indefinite, and inherits the one dimensional null space from the full system
matrix Eq. 4, again corresponding to the constant pressure mode. Similarly,
the right hand side in Eq. 5 inherits the consistency of the load vector
meaning that Eq. 5 is uniquely solvable up to a constant pressure mode. The
indefiniteness of the problem coupled with the presence of a low dimensional
null space suggests using a MINRES iterative solver [30] in conjunction with a
suitable preconditioner.

We seek a block diagonal matrix of the form

(7) $\displaystyle\bm{P}=\begin{bmatrix}\bar{\bm{A}}&\bm{0}\\\
\bm{0}&\bar{\bm{M}}\end{bmatrix}$

to precondition $\bm{S}$, where $\bar{\bm{A}}$ and $\bar{\bm{M}}$ are
symmetric positive definite matrices. The convergence of the MINRES algorithm
with preconditioner $\bm{P}^{-1}$ depends on the location of the nonzero
eigenvalues of $\bm{P}^{-1}\bm{S}$ [14, Remark 4.13 and §4.2.4]. In
particular, let $\delta$, $\Delta$, $\theta$, and $\Theta$ be nonnegative
constants such that

$\displaystyle\delta\leq\frac{\vec{u}_{E}^{T}\widetilde{\bm{A}}\vec{u}_{E}}{\vec{u}_{E}^{T}\bar{\bm{A}}\vec{u}_{E}}\leq\Delta\
\ \
\forall\bm{u}\in\bm{V}_{0}\quad\text{and}\quad\theta\leq\frac{\vec{q}_{e}^{T}\widetilde{\bm{B}}\widetilde{\bm{A}}^{-1}\widetilde{\bm{B}}^{T}\vec{q}_{e}}{\vec{q}_{e}^{T}\bar{\bm{M}}\vec{q}_{e}}\leq\Theta\
\ \ \forall q\in Q_{0}.$

Then, by [14, Theorem 4.7 and eq. (4.37)], the eigenvalues of
$\bm{P}^{-1}\bm{S}$ lie in the set

(8)
$\displaystyle\left[-\Theta^{2},\frac{1}{2}\left(\delta-\sqrt{\delta^{2}+4\delta\theta^{2}}\right)\right]\cup\\{0\\}\cup\left[\delta,\frac{1}{2}\left(\Delta+\sqrt{\Delta^{2}+4\Delta\Theta^{2}}\right)\right].$

In order to use variational techniques like Additive Schwarz Methods to
construct $\bar{\bm{A}}$ and $\bar{\bm{M}}$, we must first identify the
appropriate variational setting of the Schur complement system Eq. 5. In
particular, the Schur complement is posed over the subspaces spanned by the
external degrees of freedom of $\bm{V}_{0}\times Q_{0}$, which are rather non-
standard owing to the additional continuity imposed at noncorner vertices.
Section 3 gives a precise characterization of these spaces including new
results showing that they form a discrete exact sequence property (Theorem
3.8) and that they, like the spaces $\bm{V}_{0}\times Q_{0}$, are uniformly
inf-sup stable in both $h$ and $k$ (Theorem 3.11).

Section 4 defines the Stokes extension operator and its relation to the
subspace splittings. Section 5 uses the results of the previous two sections
to relate the matrix form of the Schur complement system to a variational
problem. In Section 6, we present an explicit set of basis functions on the
reference element for the spaces $\bm{V}$ and $Q$ and then detail how these
are used in the construction of the global basis functions. We develop the
additive Schwarz theory and construct the matrices $\bar{\bm{A}}$ and
$\bar{\bm{M}}$ in Section 7, which is then applied to two numerical examples
demonstrating in Section 8. Appendix A contains technical lemmas related to
the additive Schwarz theory.

##  3 Subspace Splittings, Exact Sequences, and Stability

A key property of the mixed finite element pair $\bm{V}_{0}\times Q_{0}$ is
the exactness of the sequence [6, Theorem 2.6] and [16, §3.2]:

(9) $\displaystyle 0\xrightarrow{\ \ \ \subset\ \ \ }\Sigma_{0}\xrightarrow{\
\ \operatorname{\mathbf{curl}}\ \ }\bm{V}_{0}\xrightarrow{\ \
\operatorname{div}\ \ }Q_{0}\xrightarrow{\qquad}0,$

where $\operatorname{\mathbf{curl}}=(\partial_{y},-\partial_{x})^{T}$ and
$\Sigma_{0}$ is the space of $H^{2}(\Omega)$-conforming piecewise polynomials
(see [6, §2]) given by

$\displaystyle\Sigma_{0}$
$\displaystyle=\\{\phi\in{H}_{0}^{2}(\Omega):\phi|_{K}\in\mathcal{P}_{k+1}(K)\
\forall K\in\mathcal{T},\ \phi\text{ is $C^{2}$ at noncorner vertices}\\}.$

The exact sequence property Eq. 9 was used in [7, Theorem 2.2] to obtain
optimal error estimates for the velocity that were independent of the pressure
error. In the remainder of this section, we seek exact sequences analogous to
Eq. 9 that respect the separation of interior and exterior degrees of freedom.
Such sequences will be used to both identify the variational problem
associated with the Schur complement system Eq. 5 and prove its uniform
stability.

Before we begin, we introduce some notation. Let $\mathcal{V}$ denote the set
of all element vertices, and partition $\mathcal{V}$ into: $\mathcal{V}_{C}$,
the set of element vertices located at a vertex of the polygonal domain
$\Omega$; $\mathcal{V}_{B}$, the set of remaining element vertices on the
domain boundary $\Gamma$ which are not corner vertices; and $\mathcal{V}_{I}$,
the set of element vertices in the interior of domain $\Omega$. Let
$\mathcal{E}$ be the set of all element edges. Given an element
$K\in\mathcal{T}$, $\mathcal{E}_{K}$ denotes the edges of $K$ and
$\mathcal{V}_{K}$ denotes the vertices of $K$. Likewise, given a vertex
$\bm{a}\in\mathcal{V}$, $\mathcal{E}_{\bm{a}}$ denotes the set of edges having
$\bm{a}$ as an endpoint and $\mathcal{T}_{\bm{a}}$ the set of elements having
$\bm{a}$ as a vertex. We assume that $\mathcal{T}$ is a partition of the
domain $\Omega$ into triangles such that the nonempty intersection of any two
distinct elements from $\mathcal{T}$ is either a single common vertex or a
single common edge of both elements, and there exists $\kappa>0$ independent
of $\mathcal{T}$ such that

(10) $\displaystyle\rho_{K}\geq\kappa h_{K}\quad\forall K\in\mathcal{T},$

where $h_{K}:=\mathrm{diam}(K)$ and $\rho_{K}$ is the diameter of the largest
inscribed circle of $K$. The mesh size $h$ denotes the diameter of the largest
element, i.e. $h:=\max_{K\in\mathcal{T}}h_{K}$.

###  3.1 Interior Subspaces

We first examine the subspaces associated with the interior degrees of freedom
given by

$\displaystyle\Sigma_{I}$ $\displaystyle=\\{\phi\in\Sigma_{0}:\phi|_{\partial
K}=\partial_{n}\phi|_{\partial K}=0,\ \forall K\in\mathcal{T}\\}$
$\displaystyle\bm{V}_{I}$
$\displaystyle=\\{\bm{v}\in\bm{V}_{0}:\bm{v}|_{\partial K}=\bm{0},\ \forall
K\in\mathcal{T}\\}$ $\displaystyle Q_{I}$ $\displaystyle=\left\\{q\in
Q_{0}:\int_{K}q\ d\bm{x}=0,\ q|_{K}(\bm{a})=0,\
\forall\bm{a}\in\mathcal{V}_{K},\ \forall K\in\mathcal{T}\right\\},$

which, in turn, may be decomposed into contributions from individual elements:

(11)
$\displaystyle\Sigma_{I}=\bigoplus_{K\in\mathcal{T}}\Sigma_{I}(K),\quad\bm{V}_{I}=\bigoplus_{K\in\mathcal{T}}\bm{V}_{I}(K),\quad\text{and}\quad\quad
Q_{I}=\bigoplus_{K\in\mathcal{T}}Q_{I}(K),$

where

$\displaystyle\Sigma_{I}(K)$ $\displaystyle:=\mathcal{P}_{k+1}(K)\cap
H^{2}_{0}(K),\qquad\bm{V}_{I}(K):=\bm{\mathcal{P}}_{k}(K)\cap\bm{H}^{1}_{0}(K),$
$\displaystyle Q_{I}(K)$ $\displaystyle:=\left\\{q\in\mathcal{P}_{k-1}(K)\cap
L^{2}_{0}(K):q(\bm{a})=0,\ \forall\bm{a}\in\mathcal{V}_{K}\right\\}.$

Both the element-level interior spaces and the corresponding interior spaces
on a mesh form exact sequences:

######  Theorem 3.1.

The following sequences are exact:

(12) $\displaystyle 0\xrightarrow{\ \ \ \subset\ \ \
}\Sigma_{I}(K)\xrightarrow{\ \ \operatorname{\mathbf{curl}}\ \
}\bm{V}_{I}(K)\xrightarrow{\ \ \operatorname{div}\ \
}Q_{I}(K)\xrightarrow{\qquad}0,\quad\forall K\in\mathcal{T},$

and

(13) $\displaystyle 0\xrightarrow{\ \ \ \subset\ \ \ }\Sigma_{I}\xrightarrow{\
\ \operatorname{\mathbf{curl}}\ \ }\bm{V}_{I}\xrightarrow{\ \
\operatorname{div}\ \ }Q_{I}\xrightarrow{\qquad}0.$

######  Proof 3.2.

In view of $\operatorname{\mathbf{curl}}H^{2}_{0}(K)\subset\bm{H}^{1}_{0}(K)$,
we have the relations
$\operatorname{\mathbf{curl}}\Sigma_{I}(K)\subset\bm{V}_{I}(K)$,
$\operatorname{\mathbf{curl}}\Sigma_{I}(K)\subseteq\ker\operatorname{div}$,
and by [6, Theorem 3.4], $\operatorname{div}\bm{V}_{I}(K)=Q_{I}(K)$. Here, we
consider div as a linear operator $\bm{V}_{I}(K)\to Q_{I}(K)$. Moreover, for
$\phi\in\Sigma_{I}(K)$, $\operatorname{\mathbf{curl}}\phi\equiv 0$ if and only
if $\phi\equiv 0$ and so
$\dim\operatorname{\mathbf{curl}}\Sigma_{I}(K)=\dim\Sigma_{I}(K)$. The
dimension counts $\dim\Sigma_{I}(K)=\frac{k^{2}-7k+12}{2}$,
$\dim\bm{V}_{I}(K)=k^{2}-3k+2$, and $\dim Q_{I}(K)=\frac{k^{2}+k-8}{2}$ reveal
that $\dim\Sigma_{I}(K)+\dim Q_{I}(K)-\dim\bm{V}_{I}(K)=0$. By the rank-
nullity theorem, we have

$\displaystyle\dim\bm{V}_{I}(K)=\dim\mathrm{Im}\operatorname{div}+\dim\ker\operatorname{div}\geq\dim
Q_{I}(K)+\dim\Sigma_{I}(K)=\dim\bm{V}_{I}(K).$

and so $\ker\operatorname{div}=\operatorname{\mathbf{curl}}\Sigma_{I}(K)$.
Thus, the element-level sequence Eq. 12 is exact. The exactness of the global
spaces Eq. 13 may be proved along similar lines using the exactness of the
element level sequence Eq. 12.

Theorem 3.1 gives a useful decomposition of the interior spaces in terms of
the $\operatorname{\mathbf{curl}}$ operator:

######  Corollary 3.3.

The spaces $\bm{V}_{I}(K)$ and $\bm{V}_{I}$ admit the following
decompositions:

$\displaystyle\bm{V}_{I}(K)$
$\displaystyle=\operatorname{\mathbf{curl}}\Sigma_{I}(K)\oplus\\{\bm{v}\in\bm{\mathcal{P}}_{k-1}(K)\cap\bm{H}^{1}_{0}(K),\
a(\bm{v},\operatorname{\mathbf{curl}}\phi)=0,\ \forall\phi\in\Sigma_{I}(K)\\}$

and

(14) $\displaystyle\bm{V}_{I}$
$\displaystyle=\operatorname{\mathbf{curl}}\Sigma_{I}\oplus\\{\bm{v}\in\bm{V}_{0}:\bm{v}|_{\partial
K}=\bm{0},\ \forall K\in\mathcal{T},\
a(\bm{v},\operatorname{\mathbf{curl}}\phi)=0,\ \forall\phi\in\Sigma_{I}\\}.$

The next result concerns the stability of the interior mixed finite element
pair $\bm{V}_{I}\times Q_{I}$.

######  Theorem 3.4.

Let $K\in\mathcal{T}$ and let $\beta$ be the discrete inf-sup constant
appearing in Eq. 3. If $q\in Q_{I}(K)$, then there exists
$\bm{v}\in\bm{V}_{I}(K)$ such that $\operatorname{div}\bm{v}=q$ and

(15) $\displaystyle
h_{K}^{-1}\|\bm{v}\|_{\bm{L}^{2}(K)}+|\bm{v}|_{\bm{H}^{1}(K)}\leq\beta^{-1}\|q\|_{L^{2}(K)}.$

Consequently, (i) the spaces $\bm{V}_{I}\times Q_{I}$ are uniformly inf-sup
stable:

(16) $\displaystyle\inf_{0\neq q\in
Q_{I}}\sup_{\bm{0}\neq\bm{v}\in\bm{V}_{I}}\frac{b(\bm{v},q)}{|\bm{v}|_{\bm{H}^{1}(\Omega)}\|q\|_{L^{2}(\Omega)}}\geq\beta,$

and (ii) the matrix $\begin{bmatrix}\bm{A}_{II}&\bm{B}_{I\iota}\\\
\bm{B}_{\iota I}&\bm{0}\end{bmatrix}$ is invertible, and hence the Schur
complement $\bm{S}$ appearing in Eq. 5 is well-defined by the formula Eq. 6.

######  Proof 3.5.

Let $q\in Q_{I}(K)$. Then, [6, Theorem 3.5] gives the existence of
$\bm{v}\in\bm{V}_{I}(K)$ with $\operatorname{div}\bm{v}=q$ satisfying the
estimate Eq. 15.

Now let $q\in Q_{I}$ be decomposed as in Eq. 11 so that
$q=\sum_{K\in\mathcal{T}}q_{K}$ with $q_{K}\in Q_{I}(K)$. By the first
statement in the theorem, there exists $\bm{v}_{K}\in\bm{V}_{I}(K)$ with
$\operatorname{div}\bm{v}_{K}=-q_{K}$ satisfying Eq. 15. Hence,
$\bm{v}:=\sum_{K\in\mathcal{T}}\bm{v}_{K}$ satisfies
$\operatorname{div}\bm{v}=-q$ and

$\displaystyle|\bm{v}|_{\bm{H}^{1}(\Omega)}^{2}$
$\displaystyle=\sum_{K\in\mathcal{T}}|\bm{v}_{K}|_{\bm{H}^{1}(\Omega)}^{2}\leq\beta^{-2}\sum_{K\in\mathcal{T}}\|q_{K}\|_{L^{2}(\Omega)}^{2}=\beta^{-2}\|q\|_{L^{2}(\Omega)}^{2},$

from which Eq. 16 immediately follows. (ii) follows at once thanks to the
ellipticity of $a(\cdot,\cdot)$ and the inf-sup condition Eq. 16.

###  3.2 Boundary Subspaces

The subspaces $\tilde{\Sigma}_{E}$, $\tilde{\bm{V}}_{E}$, and $\tilde{Q}_{E}$
are defined as follows

(17a) $\displaystyle\tilde{\Sigma}_{E}$
$\displaystyle:=\\{\phi\in\Sigma_{0}:a(\operatorname{\mathbf{curl}}\phi,\operatorname{\mathbf{curl}}\psi)=0,\
\forall\psi\in\Sigma_{I}\\}$ (17b) $\displaystyle\tilde{\bm{V}}_{E}$
$\displaystyle:=\\{\bm{v}\in\bm{V}_{0}:\operatorname{div}\bm{v}\in\tilde{Q}_{E},\
a(\bm{v},\operatorname{\mathbf{curl}}\psi)=0,\ \forall\psi\in\Sigma_{I}\\}$
(17c) $\displaystyle\tilde{Q}_{E}$ $\displaystyle:=\\{q\in Q_{0}:(q,r)=0,\
\forall r\in Q_{I}\\},$

and correspond to degrees of freedom on the element boundaries. More
precisely, we have:

######  Theorem 3.6.

The following decompositions hold:

(18)
$\displaystyle\Sigma_{0}=\Sigma_{I}\oplus\tilde{\Sigma}_{E},\quad\bm{V}_{0}=\bm{V}_{I}\oplus\tilde{\bm{V}}_{E},\quad\text{and}\quad
Q_{0}=Q_{I}\oplus\tilde{Q}_{E}.$

######  Proof 3.7.

The decompositions of $\Sigma_{0}$ and $Q_{0}$ follow immediately using the
orthogonality conditions in the definition of the spaces Eqs. 17a and 17c. The
decomposition of the velocity space $\bm{V}_{0}$ is more involved. We first
use the exact sequence Eq. 9 to write:

(19) $\displaystyle\bm{V}_{0}$
$\displaystyle=\operatorname{\mathbf{curl}}\Sigma_{0}\oplus(\operatorname{\mathbf{curl}}\Sigma_{0})^{\perp},$

where
$(\operatorname{\mathbf{curl}}\Sigma_{0})^{\perp}:=\\{\bm{u}\in\bm{V}_{0}:a(\bm{u},\operatorname{\mathbf{curl}}\psi)=0,\
\forall\psi\in\Sigma_{0}\\}.$ Now, let $\bm{v}\in\bm{V}_{0}$ be given. By the
decomposition Eq. 19 and the decomposition of $\Sigma_{0}$ in Eq. 18, there
exists $\phi_{I}\in\Sigma_{I}$, $\tilde{\phi}_{E}\in\tilde{\Sigma}_{E}$, and
$\bm{v}_{\perp}\in(\operatorname{\mathbf{curl}}\Sigma_{0})^{\perp}$ such that
$\bm{v}=\operatorname{\mathbf{curl}}(\phi_{I}+\tilde{\phi}_{E})+\bm{v}_{\perp}$.
We decompose the divergence analogously:
$\operatorname{div}\bm{v}=q_{I}+\tilde{q}_{E}$ with $q_{I}\in Q_{I}$ and
$\tilde{q}_{E}\in\tilde{Q}_{E}$. Thanks to the exact sequence Eq. 13 and the
decomposition Eq. 14, there exists $\bm{w}\in\bm{V}_{I}$ such that
$\operatorname{div}\bm{w}=q_{I}$ and $\bm{w}\in\\{\bm{v}\in\bm{V}_{I}:\
a(\bm{v},\operatorname{\mathbf{curl}}\psi)=0,\ \forall\psi\in\Sigma_{I}\\}$.
Then, $\bm{v}_{I}:=\operatorname{\mathbf{curl}}\phi_{I}+\bm{w}$ satisfies
$\bm{v}_{I}\in\bm{V}_{I}$ and $\operatorname{div}\bm{v}_{I}=q_{I}$.
Consequently,
$\tilde{\bm{v}}_{E}:=\bm{v}-\bm{v}_{I}=\operatorname{\mathbf{curl}}\tilde{\phi}_{E}+\bm{v}_{\perp}-\bm{w}$
satisfies
$\operatorname{div}\tilde{\bm{v}}_{E}=\operatorname{div}(\bm{v}-\bm{v}_{I})=\tilde{q}_{E}\in\tilde{Q}_{E}$
and

$\displaystyle
a(\tilde{\bm{v}}_{E},\operatorname{\mathbf{curl}}\psi)=a(\operatorname{\mathbf{curl}}\tilde{\phi}_{E},\operatorname{\mathbf{curl}}\psi)+a(\bm{v}_{\perp},\operatorname{\mathbf{curl}}\psi)+a(\bm{w},\operatorname{\mathbf{curl}}\psi)=0,\
\forall\psi\in\Sigma_{I}$

by construction. Thus, $\tilde{\bm{v}}_{E}\in\tilde{\bm{V}}_{E}$, which
completes the proof.

Theorem 3.6 means that the decompositions in the columns of the following
complex Eq. 20 are valid. The next result shows that the rows form exact
sequences:

######  Theorem 3.8.

Each row the of the following complex is an exact sequence

(20a) $\displaystyle 0\quad$ $\displaystyle\xrightarrow{\ \ \ \subset\ \ \
}\quad$ $\displaystyle\Sigma_{0}\quad$ $\displaystyle\xrightarrow{\ \
\operatorname{\mathbf{curl}}\ \ }\quad$ $\displaystyle\bm{V}_{0}\quad$
$\displaystyle\xrightarrow{\ \ \operatorname{div}\ \ }\quad$ $\displaystyle
Q_{0}\quad$ $\displaystyle\xrightarrow{\qquad}\quad$ $\displaystyle 0$
$\displaystyle\shortparallel_{\phantom{0}}$
$\displaystyle\shortparallel_{\phantom{0}}$
$\displaystyle\shortparallel_{\phantom{0}}$ (20b) $\displaystyle 0$
$\displaystyle\xrightarrow{\ \ \ \subset\ \ \ }$ $\displaystyle\Sigma_{I}$
$\displaystyle\xrightarrow{\ \ \operatorname{\mathbf{curl}}\ \ }$
$\displaystyle\bm{V}_{I}$ $\displaystyle\xrightarrow{\ \ \operatorname{div}\ \
}$ $\displaystyle Q_{I}$ $\displaystyle\xrightarrow{\qquad}$ $\displaystyle 0$
$\displaystyle\oplus_{\phantom{0}}$ $\displaystyle\oplus_{\phantom{0}}$
$\displaystyle\oplus_{\phantom{0}}$ (20c) $\displaystyle 0$
$\displaystyle\xrightarrow{\ \ \ \subset\ \ \ }$
$\displaystyle\tilde{\Sigma}_{E}$ $\displaystyle\xrightarrow{\ \
\operatorname{\mathbf{curl}}\ \ }$ $\displaystyle\tilde{\bm{V}}_{E}$
$\displaystyle\xrightarrow{\ \ \operatorname{div}\ \ }$
$\displaystyle\tilde{Q}_{E}$ $\displaystyle\xrightarrow{\qquad}$
$\displaystyle 0$

where the exterior spaces $\tilde{\Sigma}_{E}$, $\tilde{\bm{V}}_{E}$, and
$\tilde{Q}_{E}$ are given by Eq. 17.

######  Proof 3.9.

[6, Theorem 2.6] gives the exactness of Eq. 20a while Theorem 3.1 gives the
exactness of Eq. 20b. Moreover, the decomposition Eq. 18 and the exactness the
sequences Eqs. 20a and 20b imply that
$\dim\tilde{\Sigma}_{E}+\dim\tilde{Q}_{E}-\dim\tilde{\bm{V}}_{E}=0.$ Since
$\operatorname{\mathbf{curl}}\tilde{\Sigma}_{E}\subset\tilde{\bm{V}}_{E}$ and
$\operatorname{div}\tilde{\bm{V}}_{E}\subseteq\tilde{Q}_{E}$, we conclude that
the sequence Eq. 20c is exact using analogous arguments to those used in
Theorem 3.1.

The exactness of the final row in Eq. 20 gives the following analogue of
Corollary 3.3 for the exterior velocity space:

######  Corollary 3.10.

The exterior velocity space $\tilde{\bm{V}}_{E}$ admits the following
decomposition:
$\tilde{\bm{V}}_{E}=\operatorname{\mathbf{curl}}\tilde{\Sigma}_{E}\oplus\\{\bm{v}\in\bm{V}_{0}:\operatorname{div}\bm{v}\in\tilde{Q}_{E},\
a(\bm{v},\operatorname{\mathbf{curl}}\phi)=0,\ \forall\phi\in\Sigma_{0}\\}$.

Theorems 1.1 and 3.4 show that the mixed finite element pairs appearing in the
first two rows of Eq. 20 are uniformly inf-sup stable. The next result shows
that the boundary spaces are also stable with the same inf-sup constant as for
the full velocity and pressure spaces:

######  Theorem 3.11.

Let $\beta$ be the discrete inf-sup constant defined in Eq. 3. If
$q\in\tilde{Q}_{E}$, then there exists a $\bm{v}\in\tilde{\bm{V}}_{E}$ such
that $\operatorname{div}\bm{v}=q$ and

(21)
$\displaystyle|\bm{v}|_{\bm{H}^{1}(\Omega)}\leq\beta^{-1}\|q\|_{L^{2}(\Omega)}.$

Consequently, the spaces $\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$ are uniformly
inf-sup stable:

(22) $\displaystyle\inf_{0\neq
q\in\tilde{Q}_{E}}\sup_{\bm{0}\neq\bm{v}\in\tilde{\bm{V}}_{E}}\frac{b(\bm{v},q)}{|\bm{v}|_{\bm{H}^{1}(\Omega)}\|q\|_{L^{2}(\Omega)}}\geq\beta.$

######  Proof 3.12.

Let $q\in\tilde{Q}_{E}$ be given. By Theorem 1.1, there exists a
$\bm{w}\in\bm{V}_{0}$ such that $\operatorname{div}\bm{w}=q$ and
$\|\bm{w}\|_{\bm{H}^{1}(\Omega)}\leq\beta^{-1}\|q\|_{L^{2}(\Omega)}$, where
$\beta$ is independent of $h$ and $k$. According to Theorems 3.6 and 17b,
there exists functions $\bm{w}_{I}\in\bm{V}_{I}$,
$\tilde{\bm{w}}_{E}\in\tilde{\bm{V}}_{E}$ such that
$\bm{w}=\bm{w}_{I}+\tilde{\bm{w}}_{E}$.
$Q_{I}\ni\operatorname{div}\bm{w}_{I}=\operatorname{div}(\bm{w}-\tilde{\bm{w}}_{E})\in\tilde{Q}_{E}$
since $\operatorname{div}\bm{w}=q\in\tilde{Q}_{E}$, and so
$\operatorname{div}\bm{w}_{I}=0$. By the exact sequence property Eq. 13,
$\bm{w}_{I}=\operatorname{\mathbf{curl}}\phi_{I}$ for some
$\phi\in\Sigma_{I}$, and thus
$\bm{w}=\operatorname{\mathbf{curl}}\phi_{I}+\tilde{\bm{w}}_{E}$. Note that
this decomposition of $\bm{w}$ is $a(\cdot,\cdot)$ orthogonal by definition:
$a(\operatorname{\mathbf{curl}}\phi_{I},\tilde{\bm{w}}_{E})=0$ and

$\displaystyle|\bm{w}|_{H^{1}(\Omega)}^{2}=a(\operatorname{\mathbf{curl}}\phi_{I},\operatorname{\mathbf{curl}}\phi_{I})+a(\tilde{\bm{w}}_{E},\tilde{\bm{w}}_{E})=|\operatorname{\mathbf{curl}}\phi_{I}|_{H^{1}(\Omega)}^{2}+|\tilde{\bm{w}}_{E}|_{H^{1}(\Omega)}^{2}.$

Define $\bm{v}:=\tilde{\bm{w}}_{E}$. Then,
$\operatorname{div}\bm{v}=\operatorname{div}\tilde{\bm{w}}_{E}=\operatorname{div}(\tilde{\bm{w}}_{E}+\operatorname{\mathbf{curl}}\phi_{I})=\operatorname{div}\bm{w}=q$,
and
$|\bm{v}|_{H^{1}(\Omega)}\leq|\bm{w}|_{H^{1}(\Omega)}\leq\beta^{-1}\|q\|_{L^{2}(\Omega)}$.
Eqs. 22 and 21 follow at once.

##  4 Stokes Extension Operator

Let $\bm{V}:=V\times V$ denote the discrete velocity space in the absence of
essential boundary conditions and $Q_{I}^{\perp}$ be the orthogonal complement
of $Q_{I}$ in $Q$ with the corresponding projection $\tilde{\Pi}:Q\to
Q_{I}^{\perp}$,

(23) $\displaystyle(\tilde{\Pi}q,r)=(q,r),\ \forall r\in
Q_{I}^{\perp}:=\\{q\in Q:(q,r)=0\ \forall r\in Q_{I}\\}.$

It is worthwhile noting that Eq. 23 means that
$\tilde{Q}_{E}=Q_{I}^{\perp}\cap L^{2}_{0}(\Omega)$, so that the space
$Q_{I}^{\perp}$ corresponds to boundary degrees of freedom. Let
$K\in\mathcal{T}$. Then, thanks to Theorem 3.4, there exist
$\bm{u}_{S,K}\in\bm{\mathcal{P}}_{k}(K)$ and $p_{S,K}\in\mathcal{P}_{k-1}(K)$
satisfying

(24a) $\displaystyle a_{K}(\bm{u}_{S,K},\bm{v})+b_{K}(\bm{v},p_{S,K})$
$\displaystyle=0$ $\displaystyle\forall\bm{v}\in\bm{V}_{I}(K)$ (24b)
$\displaystyle b_{K}(\bm{u}_{S,K},q)$ $\displaystyle=0$ $\displaystyle\forall
q\in Q_{I}(K)$ (24c) $\displaystyle\bm{u}_{S,K}$ $\displaystyle=\bm{u}$
$\displaystyle\text{on }\partial K$ (24d) $\displaystyle p_{S,K}(\bm{a})$
$\displaystyle=p|_{K}(\bm{a})$ $\displaystyle\bm{a}\in\mathcal{V}_{K}$ (24e)
$\displaystyle\int_{K}p_{S,K}\ d\bm{x}$ $\displaystyle=\int_{K}p\ d\bm{x},$

where $a_{K}(\cdot,\cdot)$ and $b_{K}(\cdot,\cdot)$ denote the restrictions of
the bilinear forms to element $K$. We define the Stokes extension map
$\bm{V}\times Q\ni(\bm{u},p)\mapsto\mathscr{E}(\bm{u},p)=:(\bm{u}_{S},p_{S})$
by the rule $\bm{u}_{S}=\bm{u}_{S,K}$ and $p_{S}=p_{S,K}$ on each element
$K\in\mathcal{T}$.

The first result deals with the Stokes extension of a given velocity field
paired a zero pressure:

######  Theorem 4.1.

Let $\bm{\Pi}_{\bm{V}}:\bm{V}\to\bm{V}$, $\Pi_{Q}:\bm{V}\to Q$ be defined by
the rule

(25)
$\displaystyle\bm{V}\ni\bm{u}\mapsto(\bm{\Pi}_{\bm{V}}\bm{u},\Pi_{Q}\bm{u}):=\mathscr{E}(\bm{u},0).$

Then, $\Pi_{Q}\bm{u}\in Q_{I}$ and

(26)
$\displaystyle\|\bm{\Pi}_{\bm{V}}\bm{u}\|_{\bm{H}^{1}(K)}+\|\Pi_{Q}\bm{u}\|_{L^{2}(K)}\leq
C\|\bm{u}\|_{\bm{H}^{1/2}(\partial K)},\quad\forall K\in\mathcal{T},$

where $\|\cdot\|_{\bm{H}^{1/2}(\partial K)}$ is the usual trace norm and $C$
is independent of $k$ and $\bm{u}$. In particular, if
$\bm{u}\in\tilde{\bm{V}}_{E}$, then $\bm{\Pi}_{\bm{V}}\bm{u}=\bm{u}$.
Moreover, the following equivalence of semi-norms holds:

(27) $\displaystyle|\bm{u}|_{\bm{H}^{1/2}(\partial
K)}\leq|\bm{\Pi}_{\bm{V}}\bm{u}|_{\bm{H}^{1}(K)}\leq
C\beta^{-1}|\bm{u}|_{\bm{H}^{1/2}(\partial K)},\quad\forall K\in\mathcal{T},$

where $C$ is independent of $k$, $h_{K}$, $\beta$, and $\bm{u}$.

######  Proof 4.2.

Let $K\in\mathcal{T}$ and $\bm{u}\in\bm{V}$ be given. Conditions Eqs. 24d and
24e imply that $\Pi_{Q}\bm{u}\in Q_{I}$. Thanks to [9, Theorem 7.4], there
exists $\bm{w}\in\bm{\mathcal{P}}_{k}(K)$ such that

(28) $\displaystyle\bm{w}|_{\partial K}=\bm{u}|_{\partial
K}\quad\text{and}\quad\|\bm{w}\|_{\bm{H}^{1}(K)}\leq
C\|\bm{u}\|_{\bm{H}^{1/2}(\partial K)},$

with $C$ independent of $k$. In particular,
$\bm{\Pi}_{\bm{V}}\bm{u}-\bm{w}=\bm{u}_{I}$ where $\bm{u}_{I}\in\bm{V}_{I}(K)$
satisfies

$\displaystyle\begin{aligned}
a_{K}(\bm{u}_{I},\bm{v})+b_{K}(\bm{v},\Pi_{Q}\bm{u})&=-a_{K}(\bm{w},\bm{v})&&\forall\bm{v}\in\bm{V}_{I}(K)\\\
b_{K}(\bm{u}_{I},q)&=-b_{K}(\bm{w},q)&&\forall q\in Q_{I}(K).\end{aligned}$

Using [18, Corollary 4.1] and Theorem 3.4, we conclude that

$\displaystyle\|\bm{u}_{I}\|_{\bm{H}^{1}(K)}+\|\Pi_{Q}\bm{u}\|_{L^{2}(K)}\leq
C\|\bm{w}\|_{\bm{H}^{1}(K)}$

Equation 26 now follows from the triangle inequality and Eq. 28.

Now let $\bm{u}\in\tilde{\bm{V}}_{E}$. Then,
$\operatorname{div}\bm{\Pi}_{\bm{V}}\bm{u}\in\tilde{Q}_{E}$ by Eq. 24b and
$\bm{w}:=\bm{u}-\bm{\Pi}_{\bm{V}}\bm{u}\in\bm{V}_{I}$ by Eq. 24c. Moreover,
$\operatorname{div}\bm{w}\in Q_{I}\cap\tilde{Q}_{E}$ since
$\operatorname{div}\bm{u},\operatorname{div}\bm{\Pi}_{\bm{V}}\bm{u}\in\tilde{Q}_{E}$.
Thus, $\operatorname{div}\bm{w}=0$ and
$\bm{w}=\operatorname{\mathbf{curl}}\phi$ with $\phi\in\Sigma_{I}$ by the
exact sequence property Eq. 13. For any $\psi\in\Sigma_{I}$,

$\displaystyle
a(\operatorname{\mathbf{curl}}\phi,\operatorname{\mathbf{curl}}\psi)=a(\bm{u},\operatorname{\mathbf{curl}}\psi)-a(\bm{\Pi}_{\bm{V}}\bm{u},\operatorname{\mathbf{curl}}\psi)=0$

by the definition of $\tilde{\bm{V}}_{E}$ and choosing
$\bm{v}=\operatorname{\mathbf{curl}}\psi$ for $\psi\in\Sigma_{I}$ in Eq. 24a.
Since $a(\operatorname{\mathbf{curl}}\cdot,\operatorname{\mathbf{curl}}\cdot)$
is coercive on $\Sigma_{I}$, $\phi\equiv 0$, and
$\bm{u}=\bm{\Pi}_{\bm{V}}\bm{u}$.

The equivalence Eq. 27 is proved by arguing as in [12, Theorem 4.1].

The next result complements Theorem 4.1:

######  Theorem 4.3.

For $p\in Q$, $\mathscr{E}(\bm{0},p)=(\bm{0},\tilde{\Pi}p)$ where
$\tilde{\Pi}$ is defined in Eq. 23, and

(29) $\displaystyle\|\tilde{\Pi}p\|_{L^{2}(K)}\leq\|p\|_{L^{2}(K)}.$

In particular, if $p\in Q_{I}^{\perp}$, then
$\mathscr{E}(\bm{0},p)=(\bm{0},p)$.

######  Proof 4.4.

Let $p\in Q$ and consider the Stokes extension
$(\tilde{\bm{u}},\tilde{p}):=\mathscr{E}(\bm{0},p)$. Since $Q=Q_{I}\oplus
Q_{I}^{\perp}$, the pressure $\tilde{p}$ may be written in the form
$\tilde{p}=p_{I}+\tilde{\Pi}p$. In particular, $p_{I}\in Q_{I}$ satisfies

$\displaystyle
b_{K}(\bm{v},p_{I})=b_{K}(\bm{v},\tilde{p})-b_{K}(\bm{v},\tilde{\Pi}p)=b_{K}(\bm{v}_{K},\tilde{p}),\quad\forall\bm{v}\in\bm{V}_{I}(K),\
\forall K\in\mathcal{T},$

where we used the fact that $b_{K}(\bm{v},\tilde{\Pi}p)=0$ since
$\operatorname{div}\bm{V}_{I}(K)=Q_{I}(K)\perp Q_{I}^{\perp}$. Hence,

$\displaystyle\begin{aligned}
a_{K}(\tilde{\bm{u}},\bm{v})+b_{K}(\bm{v},p_{I})&=0&&\forall\bm{v}\in\bm{V}_{I}(K)\\\
b_{K}(\tilde{\bm{u}},q)&=0&&\forall q\in Q_{I}(K),\end{aligned}$

Equation 26 then gives $(\tilde{\bm{u}},p_{I})=\mathscr{E}(\bm{0},0)$; or,
equally well, $\tilde{\bm{u}}=\bm{0}$ and $\tilde{p}=\tilde{\Pi}p$. The
estimate Eq. 30 immediately follows since $\tilde{\Pi}$ is a projection. If
$p\in Q_{I}^{\perp}$, then
$\mathscr{E}(\bm{0},p)=(\bm{0},\tilde{\Pi}p)=(\bm{0},p)$.

Combining Theorems 4.1 and 4.3 leads to the following result:

######  Corollary 4.5.

The Stokes extension operator $\mathscr{E}(\cdot,\cdot)$ is linear and
continuous: For $K\in\mathcal{T}$,

(30) $\displaystyle\|\mathscr{E}(\bm{u},p)\|_{\bm{H}^{1}(K)\times
L^{2}(K)}\leq C\|\bm{u}\|_{\bm{H}^{1/2}(\partial
K)}+\|\tilde{\Pi}p\|_{L^{2}(K)}\quad\forall(\bm{u},p)\in\bm{V}\times Q,$

where $C$ is independent of $k$. Moreover, $\ker\mathscr{E}=\bm{V}_{I}\times
Q_{I}$ and
$\mathscr{E}(\bm{u},p)=(\bm{\Pi}_{\bm{V}}\bm{u},\Pi_{Q}\bm{u}+\tilde{\Pi}p)$.

######  Proof 4.6.

The linearity of $\mathscr{E}(\cdot,\cdot)$ is immediate from the definition
Eq. 24, and Eq. 30 then follows from Eqs. 26 and 29 using the triangle
inequality. A simple consequence of Eqs. 24c, 24d, and 24e is that
$\ker\mathscr{E}\subseteq\bm{V}_{I}\times Q_{I}$. Moreover, Eq. 30 gives that
$\bm{V}_{I}\times Q_{I}\subseteq\ker\mathscr{E}$. Thus,
$\ker\mathscr{E}=\bm{V}_{I}\times Q_{I}$.

##  5 Variational Form of the Schur Complement System

The results of the previous two sections are used to study the Schur
complement system Eq. 5. The first result relates the Schur complement matrix
$\bm{S}$ to the discrete Stokes extension map:

######  Lemma 5.1.

For all $(\bm{u},p),(\bm{v},q)\in\bm{V}_{0}\times Q$, the Stokes extension
satisfies

(31) $\displaystyle
a(\bm{u}_{S},\bm{v}_{S})+b(\bm{v}_{S},p_{S})+b(\bm{u}_{S},q_{S})=\begin{bmatrix}\vec{v}_{E}\\\
\vec{q}_{e}\end{bmatrix}^{T}\bm{S}\begin{bmatrix}\vec{u}_{E}\\\
\vec{p}_{e}\end{bmatrix}$

where $\bm{S}=\begin{bmatrix}\widetilde{\bm{A}}&\widetilde{\bm{B}}^{T}\\\
\widetilde{\bm{B}}&\widetilde{\bm{C}}\end{bmatrix}$. Consequently, the
following identities hold:

(32a) $\displaystyle\vec{u}_{E}^{T}\widetilde{\bm{A}}\vec{v}_{E}$
$\displaystyle=a(\bm{\Pi}_{\bm{V}}\bm{u},\bm{\Pi}_{\bm{V}}\bm{v})$
$\displaystyle\forall\bm{u},\bm{v}\in\bm{V}_{0},$ (32b)
$\displaystyle\vec{p}_{e}^{T}\widetilde{\bm{B}}\vec{u}_{E}$
$\displaystyle=b(\bm{\Pi}_{\bm{V}}\bm{u},\tilde{\Pi}p)$ $\displaystyle\forall
p\in Q,\ \bm{u}\in\bm{V}_{0},$ (32c) $\displaystyle\widetilde{\bm{C}}$
$\displaystyle=\bm{0}\quad\text{and}\quad\vec{g}_{e}^{*}=\vec{0},$

where $\bm{\Pi}_{\bm{V}}$ is defined in Eq. 25 and $\tilde{\Pi}$ is defined in
Eq. 23.

######  Proof 5.2.

Let $(\bm{u},p)\in\bm{V}_{0}\times Q$. The Stokes extension
$(\bm{u}_{S},p_{S})=\mathcal{E}(\bm{u},p)$ may be written as
$\bm{u}_{S}=\vec{\Phi}_{E}^{T}\vec{u}_{E}+\vec{\Phi}_{I}^{T}\vec{u}_{I}^{*}$
and
$p_{S}=\vec{\psi}_{e}^{T}\vec{p}_{e}+\vec{\psi}_{\iota}^{T}\vec{p}_{\iota}^{*}$
so that $\bm{u}=\vec{\Phi}_{E}^{T}\vec{u}_{E}+\vec{\Phi}_{I}^{T}\vec{u}_{I}$
and $p=\vec{\psi}_{e}^{T}\vec{p}_{e}+\vec{\psi}_{\iota}^{T}\vec{p}_{\iota}$
for suitable $\vec{u}_{E}$, $\vec{u}_{I}$, $\vec{p}_{e}$, and
$\vec{p}_{\iota}$. Thanks to Eqs. 24a and 24b, $\vec{u}_{I}^{*}$ and
$\vec{p}_{\iota}^{*}$ are given by

(33) $\displaystyle\begin{bmatrix}\vec{u}_{I}^{*}\\\
\vec{p}_{\iota}^{*}\end{bmatrix}=-\begin{bmatrix}\bm{A}_{II}&\bm{B}_{I\iota}\\\
\bm{B}_{\iota
I}&\bm{0}\end{bmatrix}^{-1}\begin{bmatrix}\bm{A}_{IE}&\bm{B}_{Ie}\\\
\bm{B}_{\iota E}&\bm{0}\end{bmatrix}\begin{bmatrix}\vec{u}_{E}\\\
\vec{p}_{e}\end{bmatrix}.$

Analogous relations hold replacing $(\bm{u},p)$ by
$(\bm{v},q)\in\bm{V}_{0}\times Q$. Now,

$\displaystyle
a(\bm{u}_{S},\bm{v}_{S})+b(\bm{v}_{S},p_{S})+b(\bm{u}_{S},q_{S})$
$\displaystyle\qquad=\begin{bmatrix}\begin{array}[]{c}\vec{v}_{E}\\\
\vec{q}_{e}\\\ \hline\cr\vec{v}_{I}^{*}\\\
\vec{q}_{\iota}^{*}\end{array}\end{bmatrix}^{T}\begin{bmatrix}\begin{array}[]{cc|cc}\bm{A}_{EE}&\bm{B}_{Ee}&\bm{A}_{EI}&\bm{B}_{E\iota}\\\
\bm{B}_{eE}&\bm{0}&\bm{B}_{eI}&\bm{0}\\\
\hline\cr\bm{A}_{IE}&\bm{B}_{Ie}&\bm{A}_{II}&\bm{B}_{I\iota}\\\ \bm{B}_{\iota
E}&\bm{0}&\bm{B}_{\iota
I}&\bm{0}\end{array}\end{bmatrix}\begin{bmatrix}\begin{array}[]{c}\vec{u}_{E}\\\
\vec{p}_{e}\\\ \hline\cr\vec{u}_{I}^{*}\\\
\vec{p}_{\iota}^{*}\end{array}\end{bmatrix}.$

and then Eq. 33, we obtain Eq. 31. Identities Eq. 32 are then obtained from
Eq. 31 as follows:

  1. (a)

Choose $p=q=0$. Then,
$(\bm{\Pi}_{\bm{V}}\bm{u}^{\dagger},\Pi_{Q}\bm{u}),(\bm{\Pi}_{\bm{V}}\bm{v},\Pi_{Q}\bm{v})\in\bm{V}\times
Q_{I}$ by Theorem 4.1, so
$b(\bm{\Pi}_{\bm{V}}\bm{v},\Pi_{Q}\bm{u})+b(\bm{\Pi}_{\bm{V}}\bm{u},\Pi_{Q}\bm{v})=0$
by Eq. 24b; Eq. 32a follows.

  2. (b)

Choose $p=0$ and $\bm{v}=\bm{0}$. By Theorem 4.3,
$\mathscr{E}(\bm{0},q)=(\bm{0},\tilde{\Pi}q)$, and Eq. 32b follows.

  3. (c)

Choose $\bm{u}=\bm{v}=\bm{0}$. By Theorem 4.3,
$\mathscr{E}(\bm{0},p)=(\bm{0},\tilde{\Pi}p)$,
$\mathscr{E}(\bm{0},q)=(\bm{0},\tilde{\Pi}q)$, and so
$\vec{q}_{e}^{T}\widetilde{\bm{C}}\vec{p}_{e}=0$. Furthermore,

$\displaystyle\vec{q}_{e}^{T}\vec{g}_{e}^{*}=\vec{q}_{e}^{T}\widetilde{\bm{B}}\vec{u}_{E}=b(\bm{\Pi}_{\bm{V}}\bm{u},\tilde{\Pi}q)=-b(\bm{u}-\bm{\Pi}_{\bm{V}}\bm{u},\tilde{\Pi}q),\quad\forall{q}\in{Q}$

by Eq. 2b. Since
$\operatorname{div}(\bm{u}-\bm{\Pi}_{\bm{V}}\bm{u})\in\operatorname{div}\bm{V}_{I}=Q_{I}\perp
Q_{I}^{\perp}$, $b(\bm{u}-\bm{\Pi}_{\bm{V}}\bm{u},\tilde{q})=0$, which
completes Eq. 32c.

The main result of this section relates the Schur complement problem Eq. 5 to
a Stokes problem posed on the boundary spaces
$\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$:

######  Theorem 5.3.

The Schur complement system Eq. 5, is equivalent to the following variational
problem: Find $(\bm{u},p)\in\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$ such that

(34a) $\displaystyle a(\bm{u},\bm{v})+b(\bm{v},p)$
$\displaystyle=(\bm{f},\bm{v})$
$\displaystyle\forall\bm{v}\in\tilde{\bm{V}}_{E}$ (34b) $\displaystyle
b(\bm{u},{q})$ $\displaystyle=0$ $\displaystyle\forall{q}\in\tilde{Q}_{E}.$

Moreover, the nonzero eigenvalues of the generalized eigenvalue problem
$\widetilde{\bm{B}}\widetilde{\bm{A}}^{-1}\widetilde{\bm{B}}^{T}\vec{q}_{e}$
$=\lambda\widetilde{\bm{M}}\vec{q}_{e}$ are contained in the interval
$[\beta^{2},1]$, where $\widetilde{\bm{M}}$ is the matrix associated with the
$L^{2}(\Omega)$-inner product on ${Q}_{I}^{\perp}$ and $\beta$ is the inf-sup
constant in Eq. 3. In particular, the nonzero eigenvalues $\lambda$ are
uniformly bounded away from zero in $h$ and $k$.

######  Proof 5.4.

Let $(\bm{u},p),(\bm{v},q)\in\tilde{\bm{V}}_{E}\times Q_{I}^{\perp}$.
Substituting the identities in Theorems 4.1 and 4.3 into Eq. 32 gives

$\displaystyle\begin{bmatrix}\vec{v}_{E}\\\
\vec{q}_{e}\end{bmatrix}^{T}\bm{S}\begin{bmatrix}\vec{u}_{E}\\\
\vec{p}_{e}\end{bmatrix}$
$\displaystyle=\vec{v}_{E}^{T}\widetilde{\bm{A}}\vec{u}_{E}^{T}+\vec{q}_{e}^{T}\widetilde{\bm{B}}\vec{u}_{E}+\vec{v}_{E}\widetilde{\bm{B}}^{T}\vec{p}_{e}$
$\displaystyle=a(\bm{\Pi}_{\bm{V}}\bm{u},\bm{\Pi}_{\bm{V}}\bm{v})+b(\bm{\Pi}_{\bm{V}}\bm{v},\tilde{\Pi}p)+b(\bm{\Pi}_{\bm{V}}\bm{u},\tilde{\Pi}q)$
$\displaystyle=a(\bm{u},\bm{v})+b(\bm{v},p)+b(\bm{u},q).$

Eq. 34 now follows from Eq. 32c on noting that
$\tilde{Q}_{E}=Q_{I}^{\perp}\cap L^{2}_{0}(\Omega)$. Arguing as in [14,
Theorem 3.22], the eigenvalue bound follows from the inf-sup condition Eq. 22.

##  6 Basis Functions

We first define a basis $\\{\phi_{i}\\}$ for the space of scalar-valued
functions $V$. The basis is constructed so that the exclusion of particular
functions gives a basis for $V_{0}=V\cap H^{1}_{0}(\Omega)$, which simplifies
both the enforcement of homogeneous boundary conditions and the implementation
of the preconditioner. A basis $\\{\bm{\Phi}_{i}\\}$ for the velocity space
$\bm{V}_{0}=V_{0}\times V_{0}$ is then obtained using functions of the form
$\phi_{j}\hat{\bm{e}}_{1}$ and $\phi_{j}\hat{\bm{e}}_{2}$. For the pressure
space, we only give a basis for $Q$ since the space $Q_{0}$ is not used in the
actual implementation.

###  6.1 Basis Functions on a Reference Triangle

We begin by defining basis functions for the pressure and velocity spaces on
the reference triangle $\hat{T}$ shown in Fig. 2(b).

####  6.1.1 Pressure Basis Functions

Let $\\{B^{k}_{\alpha}\\}_{\alpha\in\mathcal{I}}$ denote the Bernstein
polynomials [21]:

(35) $\displaystyle
B^{k}_{\alpha}=\frac{k!}{\alpha_{1}!\alpha_{2}!\alpha_{3}!}\lambda_{1}^{\alpha_{1}}\lambda_{2}^{\alpha_{2}}\lambda_{3}^{\alpha_{3}},$

where $\mathcal{I}=\\{\alpha\in\mathbb{Z}_{+}^{3}:|\alpha|=k\\}$ and
$\\{\lambda_{i},\ 1\leq i\leq 3\\}$, are the barycentric coordinates on the
reference triangle $\hat{T}$. The set
$\\{B_{\alpha}^{k}\\}_{\alpha\in\mathcal{I}}$ forms a basis for
$\mathcal{P}_{k}(\hat{T})$ [21]. Each Bernstein polynomial $B_{\alpha}^{k}$
can be identified with the domain point
$\bm{x}_{\alpha}=\frac{\alpha_{1}}{k}\hat{\bm{a}}_{1}+\frac{\alpha_{2}}{k}\hat{\bm{a}}_{2}+\frac{\alpha_{3}}{k}\hat{\bm{a}}_{3}$
on the reference triangle. Let
$\mathcal{I}_{0}=\\{\alpha\in\mathcal{I}:\alpha_{i}<k\\}$ denote the subset
corresponding to interior (non-vertex) points. Fix any
$\beta\in\mathcal{I}_{0}$; since all the Bernstein polynomials Eq. 35 share
the same average value, the set
$\\{B_{\alpha}^{k}-B_{\beta}^{k}\\}_{\alpha\in\mathcal{I}\setminus\\{\beta\\}}$
is a basis for $\mathcal{P}_{k}(\hat{T})\cap L^{2}_{0}(\hat{T})$. This set can
be partitioned into:

  1. (i)

Vertex functions: $\hat{\psi}_{i}:=B^{k}_{ke_{i}}-B^{k}_{\beta}$, $1\leq i\leq
3$, satisfying $\int_{\hat{T}}\hat{\psi}_{i}d\bm{x}=0$ and
$\hat{\psi}_{i}(\hat{\bm{a}}_{j})=\delta_{ij}$ for $1\leq i,j\leq 3$.

  2. (ii)

Interior functions: $\hat{\psi}_{\iota,\alpha}:=B^{k}_{\alpha}-B_{\beta}^{k}$,
$\alpha\in\mathcal{I}_{0}\setminus\\{\beta\\}$, satisfying
$\int_{\hat{T}}\hat{\psi}_{\iota,\alpha}d\bm{x}=0$ and
$\hat{\psi}_{\iota,\alpha}(\hat{\bm{a}}_{i})=0$, $1\leq i\leq 3$.

In order to obtain a basis for $\mathcal{P}_{k}(\hat{T})$, we supplement this
set with one additional function:

  1. (iii)

Average value function

(36) $\displaystyle\hat{\psi}_{\hat{T}}:=1-\sum_{i=1}^{3}\hat{\psi}_{i}.$

satisfying $|\hat{T}|^{-1}\int_{\hat{T}}\hat{\psi}_{\hat{T}}d\bm{x}=1$ and
$\hat{\psi}_{\hat{T}}(\bm{a}_{i})=0$, $1\leq i\leq 3$.

In summary, there are 3 vertex functions, one average value function and
$\frac{1}{2}(k+1)(k+2)-4$ interior functions which total
$\frac{1}{2}(k+1)(k+2)=\dim\mathcal{P}_{k}(\hat{T})$, and form a basis for the
pressure space $Q=\mathcal{P}_{k}(\hat{T})$ on the reference element.

$\bm{a}_{1}$$\bm{a}_{2}$$\bm{a}_{3}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\hat{\bm{t}}_{1}$$\hat{\bm{n}}_{1}$$\hat{\bm{t}}_{2}$$\hat{\bm{n}}_{2}$$\hat{\bm{t}}_{3}$$\hat{\bm{n}}_{3}$$K$
(a)

$\hat{\bm{a}}_{1}$$(0,0)$$\hat{\bm{a}}_{2}$$(1,0)$$\hat{\bm{a}}_{3}$$(0,1)$$\hat{\gamma}_{1}$$\hat{\gamma}_{2}$$\hat{\gamma}_{3}$$\hat{\bm{t}}_{1}$$\hat{\bm{n}}_{1}$$\hat{\bm{t}}_{2}$$\hat{\bm{n}}_{2}$$\hat{\bm{t}}_{3}$$\hat{\bm{n}}_{3}$$\hat{T}$
(b)

Figure 2: Notation for (a) general triangle $K$ and (b) reference triangle
$\hat{T}$.

####  6.1.2 Velocity Basis Functions

The construction of the basis functions for the velocity space $V$ is more
complicated owing to the higher continuity requirement. In particular, the
basis functions $\\{\hat{\phi}_{k}^{\beta}\\}$, $|\beta|=1$,
$k\in\\{1,2,3\\}$, associated with the derivative degrees of freedom at the
vertices should satisfy
$D^{\alpha}\hat{\phi}_{k}^{\beta}(\hat{\bm{a}}_{l})=\delta_{\alpha\beta}\delta_{kl}$,
$|\alpha|=1$, $l\in\\{1,2,3\\}$. In order to construct these functions, we
begin by considering the vector valued function given by

(37)
$\displaystyle\vec{J}_{1}:=\frac{\lambda_{1}^{2}}{P_{k-3}^{(3,3)}(-1)}\begin{bmatrix}\lambda_{2}P_{k-3}^{(3,3)}(\lambda_{2}-\lambda_{1})\\\
\lambda_{3}P_{k-3}^{(3,3)}(\lambda_{3}-\lambda_{1})\end{bmatrix},$

where ${P}_{k}^{(3,3)}$ is the Jacobi polynomial of degree $k$ [36]. The first
component of $\vec{J}_{1}$ vanishes on edge $\gamma_{2}$ and the gradient at
$\hat{\bm{a}}_{1}$ is given by $\begin{bmatrix}1&0\end{bmatrix}^{T}$, while
the second component vanishes on edge $\hat{\gamma}_{3}$ and has gradient
$\begin{bmatrix}0&1\end{bmatrix}^{T}$ at $\hat{\bm{a}}_{1}$. The factor
$\lambda_{1}^{2}$ means that both components of $\vec{J}_{1}$ and their
gradients vanish on the edge $\hat{\gamma}_{1}$. In summary, since
$x=\lambda_{2}$ and $y=\lambda_{3}$, we have

(38)
$\displaystyle\vec{J}_{1}(\hat{\bm{a}}_{k})=\vec{0}\quad\text{and}\quad\begin{bmatrix}\frac{\partial}{\partial
x}\\\ \frac{\partial}{\partial
y}\end{bmatrix}\vec{J}_{1}^{T}(\hat{\bm{a}}_{k})=\begin{bmatrix}\frac{\partial}{\partial\lambda_{2}}\\\
\frac{\partial}{\partial\lambda_{3}}\end{bmatrix}\vec{J}_{1}^{T}(\hat{\bm{a}}_{k})=\delta_{kl}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}.$

Defining $\vec{J}_{2}$ and $\vec{J}_{3}$ by cyclic permutations of the
indices, we conclude that $\vec{J}_{2}$ and $\vec{J}_{3}$ vanish at the
vertices and that, for $k\in\\{1,2,3\\}$,

(39) $\displaystyle\begin{bmatrix}\frac{\partial}{\partial\lambda_{3}}\\\
\frac{\partial}{\partial\lambda_{1}}\end{bmatrix}\vec{J}_{2}^{T}(\hat{\bm{a}}_{k})=\delta_{k2}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\quad\text{and}\quad\begin{bmatrix}\frac{\partial}{\partial\lambda_{1}}\\\
\frac{\partial}{\partial\lambda_{2}}\end{bmatrix}\vec{J}_{3}^{T}(\hat{\bm{a}}_{k})=\delta_{k3}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}.$

Substituting the identities

$\displaystyle\begin{bmatrix}\frac{\partial}{\partial\lambda_{3}}\\\
\frac{\partial}{\partial\lambda_{1}}\end{bmatrix}=\begin{bmatrix}-1&1\\\
-1&0\end{bmatrix}\begin{bmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial
y}\end{bmatrix}\quad\text{and}\quad\begin{bmatrix}\frac{\partial}{\partial\lambda_{1}}\\\
\frac{\partial}{\partial\lambda_{2}}\end{bmatrix}=\begin{bmatrix}0&-1\\\
1&-1\end{bmatrix}\begin{bmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\end{bmatrix}$

in Eq. 39 and rearranging gives

(40) $\displaystyle\begin{bmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\end{bmatrix}\left(\begin{bmatrix}-1&-1\\\
1&0\end{bmatrix}\vec{J}_{2}\right)^{T}(\hat{\bm{a}}_{k})$
$\displaystyle=\delta_{k2}\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ (41)
$\displaystyle\begin{bmatrix}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\end{bmatrix}\left(\begin{bmatrix}0&1\\\
-1&-1\end{bmatrix}\vec{J}_{3}\right)^{T}(\hat{\bm{a}}_{k})$
$\displaystyle=\delta_{k3}\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}.$

Armed with Eqs. 38, 40, and 41, we define the basis functions for the velocity
space as follows:

  1. (i)

$C^{0}$ vertex functions: $\hat{\phi}_{i}=\lambda_{i}^{2}(3-2\lambda_{i})$,
$\leq i\leq 3$, satisfying $\hat{\phi}_{i}(\hat{\bm{a}}_{j})=\delta_{ij}$,
$D\hat{\phi}_{i}(\hat{\bm{a}}_{j})=\bm{0}$, and
$\hat{\phi}_{i}|_{\hat{\gamma}_{i}}=0$ for $1\leq i,j\leq 3$.

  2. (ii)

$C^{1}$ vertex functions

$\displaystyle\begin{bmatrix}\hat{\phi}_{1}^{(1,0)}\\\
\hat{\phi}_{1}^{(0,1)}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\vec{J}_{1};$
$\displaystyle\begin{bmatrix}\hat{\phi}_{2}^{(1,0)}\\\
\hat{\phi}_{2}^{(0,1)}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}-1&-1\\\
1&0\end{bmatrix}\vec{J}_{2};$
$\displaystyle\begin{bmatrix}\hat{\phi}_{3}^{(1,0)}\\\
\hat{\phi}_{3}^{(0,1)}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}0&1\\\
-1&-1\end{bmatrix}\vec{J}_{3}$

which, thanks to Eqs. 38, 40, and 41, satisfy
$D^{\beta}\hat{\phi}_{i}^{\alpha}(\hat{\bm{a}}_{j})=\delta_{\alpha\beta}\delta_{ij}$
and $\hat{\phi}_{i}|_{\hat{\gamma}_{i}}=0$ for $1\leq i,j\leq 3$,
$|\alpha|=1$, $|\beta|\leq 1$.

  3. (iii)

Edge functions: Let $\hat{\gamma}$ be the edge connecting vertices
$\hat{\bm{a}}_{i}$ and $\hat{\bm{a}}_{j}$; then the basis functions associated
with the edge are defined by
$\hat{\phi}_{\hat{\gamma},l}=\lambda_{i}^{2}\lambda_{j}^{2}r_{l}(\lambda_{j}-\lambda_{i})$
where $\\{r_{l}\\}$ is any basis for $\mathcal{P}_{k-4}((-1,1))$. These
functions satisfy $D^{\alpha}\hat{\phi}_{\hat{\gamma},l}(\hat{\bm{a}}_{k})$
$=0$ for $|\alpha|\leq 1$, $1\leq k\leq 3$ and
$\hat{\phi}_{\hat{\gamma},l}|_{\hat{\gamma}^{\prime}}=\delta_{\hat{\gamma}\hat{\gamma^{\prime}}}$
for $\hat{\gamma}^{\prime}\in\mathcal{E}_{\hat{T}}$.

  4. (iv)

Interior functions: The basis functions associated with the element interior
are defined by $\hat{\phi}_{I,l}=\lambda_{1}\lambda_{2}\lambda_{3}s_{l}$ where
$\\{s_{l}\\}$ is any basis for $\mathcal{P}_{k-3}(\hat{T})$. These functions
satisfy $\hat{\phi}_{I,l}|_{\partial\hat{T}}=0$ and
$D\hat{\phi}_{I,l}(\hat{\bm{a}}_{k})=\bm{0}$ for $1\leq k\leq 3$.

It is easily seen that the above functions are linearly independent.
Furthermore, there are 3 functions per vertex,
$\dim\mathcal{P}_{k-4}((-1,1))=k-3$ functions per edge, and
$\dim\mathcal{P}_{k-3}(\hat{T})=\frac{1}{2}(k-2)(k-1)$ interior functions
which total $\frac{1}{2}(k+1)(k+2)=\dim\mathcal{P}_{k}(\hat{T})$. Hence, the
above functions also form a basis for $\mathcal{P}_{k}(\hat{T})$.

###  6.2 Basis Functions on a Mesh

We now define the global basis functions for the spaces $Q$ and $V$. The lower
continuity requirements imposed at corner vertices $\mathcal{V}_{C}$ means
that extra care must be taken when defining the global basis functions
associated with $\mathcal{V}_{C}$.

####  6.2.1 Pressure Basis Functions

The pressure space $Q$ requires $C^{0}$ continuity at all vertices except at
corner vertices, where the functions are allowed to be discontinuous. This
means that each element has its own degree of freedom at vertices
$\bm{a}\in\mathcal{V}_{C}$, whilst at the remaining vertices
$\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}$, all elements share a single
degree of freedom at the common vertex as shown in Fig. 3(a). Consequently,
any given vertex $\bm{a}\in\mathcal{V}$ is associated with either (a) a single
basis function supported on the patch $\mathcal{T}_{\bm{a}}$ if
$\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}$, or (b) a collection of basis
functions, each of which is supported on a single element
$K\in\mathcal{T}_{\bm{a}}$ if $\bm{a}\in\mathcal{V}_{C}$. The set of supports
of the pressure functions associated with a vertex $\bm{a}\in\mathcal{V}$ is
defined by

$\displaystyle\Omega_{\bm{a}}=\begin{cases}\\{\mathcal{T}_{\bm{a}}\\}&\bm{a}\in\mathcal{V}_{C}\\\
\\{K\in\mathcal{T}_{\bm{a}}\\}&\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}.\end{cases}$

That is, the cardinality of these sets is $|\Omega_{\bm{a}}|=1$ for noncorner
vertices (since there is only one vertex basis function associated to
$\bm{a}$) whilst $|\Omega_{\bm{a}}|\geq 2$ for corner vertices, thanks to the
assumption that the mesh is corner-split into at least two elements. The
corresponding global vertex functions
$\\{\psi_{\bm{a}}^{\omega}:\bm{a}\in\mathcal{V},\ \omega\in\Omega_{\bm{a}}\\}$
are defined to be pull-backs in the usual way:

(42) $\displaystyle\psi_{\bm{a}}^{\omega}$
$\displaystyle=\begin{cases}\hat{\psi}_{i}\circ\bm{F}_{K}^{-1}&\text{on
}K\subseteq\omega,\\\ 0&\text{otherwise},\end{cases}$ $\displaystyle\psi_{K}$
$\displaystyle=\begin{cases}\hat{\psi}_{\hat{T}}\circ\bm{F}_{K}^{-1}&\text{on
}K,\\\ 0&\text{otherwise},\end{cases}$

where $\hat{\bm{a}}_{i}=\bm{F}_{K}^{-1}(\bm{a})$.

Corner dofNoncorner dof (a)

Functional dofDerivative dof (b)

Figure 3: (a) The global pressure vertex degrees of freedom and (b) the global
velocity vertex degrees of freedom on a mesh of an example domain. Observe
that in (a) there are multiple pressure vertex degrees of freedom at corner
vertices but only a single degree of freedom at interior vertices and in (b)
there are three or more derivative degrees of freedom at corner vertices
(red), two derivative degrees of freedom aligned with the domain boundary at
noncorner boundary vertices (green), and two derivative degrees of freedom
aligned with the coordinate axes at interior vertices (blue).

The average value functions and interior functions are simpler. Each element
$K\in\mathcal{T}$ has a single function $\psi_{K}$, corresponding to the
average value over $K$, defined by Eq. 42. Similarly, each element
$K\in\mathcal{T}$ has $\frac{k}{2}(k+1)-4$ interior functions also defined to
be pull-backs.

####  6.2.2 Velocity Basis Functions

The velocity space $V$ imposes $C^{1}$ continuity at all vertices except at
corner vertices, where only $C^{0}$-continuity is required to ensure $V\subset
H^{1}(\Omega)$. This means that at corner vertices $\bm{a}\in\mathcal{V}_{C}$,
each element $K\in\mathcal{T}_{\bm{a}}$ has two degrees of freedom for the
gradient corresponding to the two tangential derivatives corresponding to the
two edges of $K$ that meet at $\bm{a}$. To enforce continuity between two
neighboring elements in $\mathcal{T}_{\bm{a}}$, the tangential derivative
corresponding to the common edge must be shared between the two elements. In
other words, each corner vertex $\bm{a}\in\mathcal{V}_{C}$ has one derivative
degree of freedom for each edge $\gamma\in\mathcal{E}_{\bm{a}}$. For the
remaining noncorner vertices $\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}$,
all elements in $\mathcal{T}_{\bm{a}}$ share two degrees of freedom at the
common vertex, corresponding to any two linearly independent directional
derivatives as in Fig. 3(b). Consequently, a given vertex
$\bm{a}\in\mathcal{V}$ is associated with either (a) two basis functions
supported on the patch $\mathcal{T}_{\bm{a}}$ if
$\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}$, or (b) a collection of basis
functions, each of which is associated to an edge
$\gamma\in\mathcal{E}_{\bm{a}}$ and supported on the pair of elements sharing
the common edge $\gamma$ if $\bm{a}\in\mathcal{V}_{C}$.

The set of unit vectors defining the directional derivative degrees of freedom
at a vertex $\bm{a}\in\mathcal{V}$ are chosen as follows:

(43) $\displaystyle
D_{\bm{a}}=\begin{cases}\\{\hat{\bm{e}}_{1},\hat{\bm{e}}_{2}\\}&\bm{a}\in\mathcal{V}_{I}\\\
\\{\hat{\bm{t}},\hat{\bm{n}}\\}&\bm{a}\in\mathcal{V}_{B}\\\
\\{\hat{\bm{t}}_{\gamma}:\gamma\in\mathcal{E}_{\bm{a}}\\}&\bm{a}\in\mathcal{V}_{C}\end{cases}$

where $\hat{\bm{t}}$ and $\hat{\bm{n}}$ are the unit tangent and normal
vectors at a noncorner boundary vertex $\bm{a}\in\mathcal{V}_{B}$ and
$\hat{\bm{t}}_{\gamma}$ denotes a unit tangent vector on an edge
$\gamma\in\mathcal{E}$ as illustrated in Fig. 3(b). For a given vertex
$\bm{a}\in\mathcal{V}$ and unit vector $\hat{\bm{\mu}}\in D_{\bm{a}}$, the
global basis function $\phi_{\bm{a}}^{\mu}$ has support

$\displaystyle\operatorname{supp}\phi_{\bm{a}}^{\mu}=\begin{cases}\\{K\in\mathcal{T}_{\bm{a}}:\exists\gamma\in\mathcal{E}_{K}\text{
with }\hat{\bm{t}}_{\gamma}=\pm\hat{\bm{\mu}}\\}&\bm{a}\in\mathcal{V}_{C},\\\
\mathcal{T}_{\bm{a}}&\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{C}.\end{cases}$

The global $C^{1}$ vertex functions come in pairs as follows: Given a
noncorner vertex $\bm{a}\in\mathcal{V}_{I}\cup\mathcal{V}_{B}$, let
$\hat{\bm{\mu}}_{1}$, $\hat{\bm{\mu}}_{2}$ be unit vectors such that
$D_{\bm{a}}=\\{\hat{\bm{\mu}}_{1},\hat{\bm{\mu}}_{2}\\}$ as in Eq. 43, and
define the basis functions by

(44) $\displaystyle\begin{bmatrix}\phi_{\bm{a}}^{\mu_{1}}\\\
\phi_{\bm{a}}^{\mu_{2}}\end{bmatrix}=\begin{bmatrix}\hat{\bm{\mu}}_{1}&\hat{\bm{\mu}}_{2}\end{bmatrix}^{-1}D\bm{F}_{K}\begin{bmatrix}\hat{\phi}_{i}^{(1,0)}\circ\bm{F}_{K}^{-1}\\\
\hat{\phi}_{i}^{(0,1)}\circ\bm{F}_{K}^{-1}\\\ \end{bmatrix}\quad\text{on
}K\in\mathcal{T}_{\bm{a}},$

where $\hat{\bm{a}}_{i}=\bm{F}_{K}^{-1}(\bm{a})$. The above construction
ensures that the basis functions are $C^{1}$ continuous at the vertex
$\bm{a}$: i.e.
$\hat{\bm{\mu}}_{i}\cdot\nabla\phi_{\bm{a}}^{\mu_{j}}(\bm{a})=\delta_{ij}$.
The case of a corner vertex $\bm{a}\in\mathcal{V}_{C}$ is more complicated
since, as mentioned above, each edge $\gamma\in\mathcal{E}_{\bm{a}}$
contributes one independent basis function at the vertex, also defined by the
expression Eq. 44, which is supported on the edge patch
$\\{K\in\mathcal{T}_{\bm{a}}:\gamma\in\mathcal{E}_{K}\\}$. The unit vectors
$\hat{\bm{\mu}}_{1},\hat{\bm{\mu}}_{2}$ in Eq. 44 associated with such an
element $K\in\mathcal{T}_{\bm{a}}$ are taken to be the pair of unit tangent
vectors on the two edges of $K$ having an endpoint at $\bm{a}$. This means
that the basis functions $\phi_{\bm{a}}^{\mu_{1}}$ and
$\phi_{\bm{a}}^{\mu_{2}}$ are the only $C^{1}$ vertex functions supported on
$K$.

The remaining $C^{0}$ vertex functions, edge functions, and interior functions
are again defined to be pull-backs of the corresponding functions on the
reference element in the usual way: i.e.
$\phi_{\bm{a}}=\hat{\phi}_{i}\circ\bm{F}_{K}^{-1}$ on
$K\in\mathcal{T}_{\bm{a}}$ and $\phi_{\bm{a}}=0$ otherwise, where
$\hat{\bm{a}}_{i}=\bm{F}_{K}^{-1}(\bm{a})$. Similarly, there are $k-3$ edge
functions per edge $\gamma\in\mathcal{E}$, supported on the patch of elements
containing that edge $\\{K\in\mathcal{T}:\gamma\in\mathcal{E}_{K}\\}$, and
there are $\frac{1}{2}(k-1)(k-2)$ interior functions per element
$K\in\mathcal{T}$.

####  6.2.3 Velocity Basis Functions with Homogeneous Boundary Conditions

The above construction gives a basis for $V$ in the absence of essential
boundary conditions. If nonhomogeneous essential boundary conditions are
imposed, then the values of the following basis functions will be constrained
by the boundary data:

  * •

the $C^{0}$ vertex function $\phi_{\bm{a}}$,
$\bm{a}\in\mathcal{V}\setminus\mathcal{V}_{I}$ at each vertex on the domain
boundary;

  * •

the $C^{1}$ vertex function at each noncorner boundary vertex corresponding to
the tangential derivative degree of freedom, i.e. $\phi_{\bm{a}}^{t}$ for
$\bm{a}\in\mathcal{V}_{B}$;

  * •

the pair of $C^{1}$ vertex functions at each corner boundary vertex
corresponding to the tangential derivatives along the domain boundary edges:
$\phi_{\bm{a}}^{t_{\gamma}}$, $\phi_{\bm{a}}^{t_{\gamma^{\prime}}}$ for
$\bm{a}\in\mathcal{V}_{C}$ where
$\gamma,\gamma^{\prime}\in\mathcal{E}_{\bm{a}}\cap\Gamma$; and

  * •

all $k-3$ edge functions for each edge on the domain boundary.

If homogeneous essential boundary conditions are imposed, then a basis for
$V_{0}=V\cap H^{1}_{0}(\Omega)$ is obtained by taking the following functions:

  * •

the $C^{0}$ vertex function at each interior vertex, i.e. $\phi_{\bm{a}}$,
$\bm{a}\in\mathcal{V}_{I}$;

  * •

the following $C^{1}$ vertex functions: $\phi_{\bm{a}}^{\mu}$,
$\bm{a}\in\mathcal{V}$, $\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}$ where

(45)
$\displaystyle\mathring{D}_{\bm{a}}=\begin{cases}\\{\hat{\bm{e}}_{1},\hat{\bm{e}}_{2}\\}&\bm{a}\in\mathcal{V}_{I}\\\
\\{\hat{\bm{n}}\\}&\bm{a}\in\mathcal{V}_{B}\\\
\\{\hat{\bm{t}}_{\gamma}:\gamma\in\mathcal{E}_{\bm{a}}\cap\mathcal{E}_{I}\\}&\bm{a}\in\mathcal{V}_{C}\end{cases}$

and $\mathcal{E}_{I}$ denotes the set of interior edges;

  * •

all $k-3$ edge functions on each interior edge; and

  * •

all interior functions on each element.

Condition Eq. 45 means that we keep both $C^{1}$ vertex functions for each
interior vertex, the $C^{1}$ vertex function associated with the the outward
normal of $\Gamma$ for each noncorner boundary vertex, and each $C^{1}$ vertex
function corresponding to an interior edge unit tangent vector at corner
vertices.

##  7 Constructing the Preconditioner Using Additive Schwarz Theory

In Section 2, we constructed the stiffness matrix for the Stokes problem Eq. 4
using bases for the spaces $\bm{V}_{0}$ and $Q$ and performed static
condensation to arrive at the Schur complement system Eq. 5. In Section 5, it
was shown that the algebraic Schur complement system Eq. 5 was related to the
mixed finite element problem Eq. 34 posed on the spaces
$\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$. The alert reader will have noticed a
slight discrepancy in the treatment of the average pressure mode over the
domain $\Omega$: in Sections 6.2 and 2, the average pressure modes were
included in the discretization (and it was pointed out that these modes span
the kernel of the Schur complement) whereas in Section 5, the pressure space
$\tilde{Q}_{E}=Q_{I}^{\perp}\cap L^{2}_{0}(\Omega)$ was used, which factors
out the singular mode. In order to construct a preconditioner in the form Eq.
7, we formulate an Additive Schwarz Method (ASM) over the spaces
$\tilde{\bm{V}}_{E}\times Q_{I}^{\perp}$ rather than the seemingly more
natural choice $\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$ suggested by Theorem
5.3.

###  7.1 Pressure ASM

We decompose the pressure space $Q_{I}^{\perp}$ as follows:

(46) $\displaystyle
Q_{I}^{\perp}=\bigoplus_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\omega\in\Omega_{\bm{a}}\end{subarray}}\tilde{Q}_{\bm{a},\omega}\oplus\bigoplus_{K\in\mathcal{T}}\tilde{Q}_{K},$

where (i) the vertex spaces
$\tilde{Q}_{\bm{a},\omega}:=\mathrm{span}\\{\tilde{\psi}_{\bm{a}}^{\omega}\\}$,
$\bm{a}\in\mathcal{V},\omega\in\Omega_{\bm{a}}$ with
$\tilde{\psi}_{\bm{a}}^{\omega}:=\tilde{\Pi}\psi_{\bm{a}}^{\omega}$ are
equipped with the inner product
$m_{\bm{a},\omega}(p,q):=|\omega|k^{-4}p(\bm{a})q(\bm{a})$, and (ii) the
element average spaces $\tilde{Q}_{K}:=\mathrm{span}\tilde{\psi}_{K}$,
$K\in\mathcal{T}$, with $\tilde{\psi}_{K}:=\tilde{\Pi}\psi_{K}$ are equipped
with the inner product

$\displaystyle m_{K}(p,q):=\frac{1}{|K|}\left(\int_{K}p\
d\bm{x}\right)\left(\int_{K}q\ d\bm{x}\right),\quad\forall
p,q\in\tilde{Q}_{K}.$

Applying the projection $\tilde{\Pi}$ to the formulae for $\psi_{K}$ Eqs. 36
and 42 gives

$\displaystyle\tilde{\psi}_{K}=\begin{cases}1-\sum\limits_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\omega\supseteq K\end{subarray}}\tilde{\psi}_{\bm{a}}^{\omega}&\text{on }K\\\
0&\text{otherwise},\end{cases}$

where the functions $\tilde{\psi}_{\bm{a}}^{\omega}$ are defined in (i) and we
use the fact that $\tilde{\Pi}$ preserves constants. The direct sum
decomposition Eq. 46 means that any $q\in Q_{I}^{\perp}$ may be uniquely
expressed in the form

$\displaystyle q=\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\omega\in\Omega_{\bm{a}}\end{subarray}}q_{\bm{a},\omega}+\sum_{K\in\mathcal{T}}q_{K},$

where

$\displaystyle q_{\bm{a},\omega}$
$\displaystyle=q|_{\omega}(\bm{a})\tilde{\psi}_{\bm{a}}^{\omega},\quad\bm{a}\in\mathcal{V},\
\omega\in\Omega_{\bm{a}},$ $\displaystyle q_{K}$
$\displaystyle=\left(\frac{1}{|K|}\int_{K}q\
d\bm{x}\right)\tilde{\psi}_{K},\quad K\in\mathcal{T}.$

The action of the associated ASM preconditioner on a residual $g\in
L^{2}(\Omega)$ is given by the solution $p\in Q_{I}^{\perp}$ of the
variational problem $\bar{m}(p,q)=(g,q)$ $\forall q\in Q_{I}^{\perp}$, where

(47)
$\displaystyle\bar{m}(p,q):=\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\omega\in\Omega_{\bm{a}}\end{subarray}}m_{\bm{a},\omega}(p_{\bm{a},\omega},q_{\bm{a},\omega})+\sum_{K\in\mathcal{T}}m_{K}(p_{K},q_{K}).$

The bilinear form $\bar{m}(\cdot,\cdot)$ gives rise to a matrix preconditioner
$\bar{\bm{M}}$ for the pressure space defined by

(48) $\displaystyle\vec{p}_{e}^{T}\bar{\bm{M}}\vec{q}_{e}$
$\displaystyle=\bar{m}(\tilde{\Pi}p,\tilde{\Pi}q)\qquad\forall p,q\in Q.$

###  7.2 Velocity ASM

We decompose the velocity space $\tilde{\bm{V}}_{E}$ as follows:

(49)
$\displaystyle\tilde{\bm{V}}_{E}=\tilde{\bm{V}}_{c}\oplus\bigoplus_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}\end{subarray}}\tilde{\bm{V}}_{\bm{a},\mu}\oplus\bigoplus_{\gamma\in\mathcal{E}_{I}}\tilde{\bm{V}}_{\gamma},$

where (i) the global $C^{0}$ vertex space
$\tilde{\bm{V}}_{c}:=\mathrm{span}\\{\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{1}),\
\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{2}):\bm{a}\in\mathcal{V}_{I}\\}$,
(ii) the $C^{1}$ vertex spaces
$\tilde{\bm{V}}_{\bm{a},\mu}:=\mathrm{span}\\{\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\bm{e}_{1}),\
\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\bm{e}_{2})\\}$, $\bm{a}\in\mathcal{V}$,
$\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}$, and (iii) the edge spaces
$\tilde{\bm{V}}_{\gamma}:=\mathrm{span}\\{\bm{\Pi}_{\bm{V}}(\phi_{\gamma,j}\bm{e}_{1}),\bm{\Pi}_{\bm{V}}(\phi_{\gamma,j}\bm{e}_{2}):1\leq
j\leq k-3\\}$, $\gamma\in\mathcal{E}_{I}$. Each of the velocity subspaces is
equipped with the inner product $a(\cdot,\cdot)$ restricted to the appropriate
space. The direct sum decomposition Eq. 49 means that any any
$\bm{u}\in\tilde{\bm{V}}_{E}$ may be uniquely expressed in the form

(50)
$\displaystyle\bm{u}=\bm{u}_{c}+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}\end{subarray}}\bm{u}_{\bm{a},\mu}+\sum_{\gamma\in\mathcal{E}_{I}}\bm{u}_{\gamma},$

where

$\displaystyle\bm{u}_{c}$
$\displaystyle=\sum_{\bm{a}\in\mathcal{V}_{I}}\sum_{i=1}^{2}(\bm{u}\cdot\hat{\bm{e}}_{i})(\bm{a})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i}),$
$\displaystyle\bm{u}_{\bm{a},\mu}$
$\displaystyle=\sum_{i=1}^{2}\partial_{\mu}(\bm{u}|_{K_{\mu}}\cdot\hat{\bm{e}}_{i})(\bm{a})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{i}),\quad\bm{a}\in\mathcal{V},\quad\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}},\quad
K_{\mu}\subseteq\operatorname{supp}\phi_{\bm{a}}^{\mu},$ and for each
$\gamma\in\mathcal{E}_{I}$, $\displaystyle\bm{u}_{\gamma}$
$\displaystyle=\begin{cases}\bm{u}-\bm{u}_{c}-\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}\end{subarray}}\bm{u}_{\bm{a},\mu}&\text{on
}\gamma\\\ \bm{0}&\text{on the remaining edges in $\mathcal{E}$.}\end{cases}$

The action of the associated ASM preconditioner on a residual
$\bm{f}\in\bm{L}^{2}(\Omega)$ is given by the solution
$\bm{u}\in\tilde{\bm{V}}_{E}$ of the variational problem
$\bar{a}(\bm{u},\bm{v})=(\bm{f},\bm{v})$ $\forall\bm{v}\in\tilde{\bm{V}}_{E}$,
where

(51)
$\displaystyle\bar{a}(\bm{u},\bm{v})=a(\bm{u}_{c},\bm{v}_{c})+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}\end{subarray}}a(\bm{u}_{\bm{a},\mu},\bm{v}_{\bm{a},\mu})+\sum_{\gamma\in\mathcal{E}_{I}}a(\bm{u}_{\gamma},\bm{v}_{\gamma}).$

The bilinear form $\bar{a}(\cdot,\cdot)$ gives rise to a matrix preconditioner
$\bar{\bm{A}}$ for the velocity space defined by

(52) $\displaystyle\vec{u}_{E}^{T}\bar{\bm{A}}\vec{v}_{E}$
$\displaystyle=\bar{a}(\bm{\Pi}_{\bm{V}}\bm{u},\bm{\Pi}_{\bm{V}}\bm{v}),\quad\forall\bm{u},\bm{v}\in\bm{V}_{0}.$

###  7.3 The Preconditioner and Main Result

######  Theorem 7.1.

Let $\bm{P}$ be defined as in Eq. 7 with $\bar{\bm{A}}$ given by Eq. 52 and
$\bar{\bm{M}}$ given by Eq. 48. Then, using $\bm{P}^{-1}$ as a preconditioner
for the MINRES method reduces the norm of the residual of MINRES by a factor
of at least $\frac{\sqrt{\sigma}-1}{\sqrt{\sigma}+1}$ every two iterations,
where $\sqrt{\sigma}\leq C(1+\log^{3}k)$ with $C$ independent of $k$ and $h$.

######  Proof 7.2.

Thanks to Theorem A.1 and the matrix correspondences Eqs. 48 and 52, there
holds

$\displaystyle\left[C_{2}\beta^{-2}(1+\log^{3}k)\right]^{-1}\bar{\bm{A}}\leq\widetilde{\bm{A}}\leq
C_{2}\bar{\bm{A}}\quad\text{and}\quad
C_{1}^{-1}\bar{\bm{M}}\leq\widetilde{\bm{M}}\leq C_{1}\bar{\bm{M}},$

where $\widetilde{\bm{M}}$ is the pressure mass matrix for the space
$Q_{I}^{\perp}$ and $\bm{A}\leq\bm{B}$ means that $\bm{B}-\bm{A}$ is positive
semidefinite. Additionally, the inf-sup condition for the spaces
$\tilde{\bm{V}}_{E}\times\tilde{Q}_{E}$ Eq. 22 and the boundedness of the
bilinear form $b(\cdot,\cdot)$ can be expressed in matrix form using Eq. 32b
and the same arguments in [14, Theorem 3.22] to arrive at

$\displaystyle{\beta}^{2}\leq\frac{\vec{q}_{e}^{T}\widetilde{\bm{B}}\widetilde{\bm{A}}^{-1}\widetilde{\bm{B}}^{T}\vec{q}_{e}}{\vec{q}_{e}^{T}\widetilde{\bm{M}}\vec{q}_{e}}\leq
1,\quad\forall\tilde{Q}_{E}\ni q=\vec{q}_{e}^{T}\vec{\psi}_{e},$

where $\beta$ is the discrete inf-sup constant in Eq. 3. Thus, Eq. 8 holds
with $\delta=\beta^{2}[C_{2}(1+\log^{3}k)]^{-1}$, $\Delta=C_{2}$,
$\theta=\beta^{2}C_{1}^{-1}$, and $\Delta=C_{1}$.

Let $\vec{r}_{n}$ denote the residual on the $n$-th iteration of MINRES with
the preconditioner $\bm{P}^{-1}$. Applying [14, Theorem 4.14] and using the
fact that the inf-sup constant $\beta$ is bounded below uniformly in $k$ and
$h$ gives

(53) $\displaystyle\|\vec{r}_{2n}\|_{\bm{P}^{-1}}\leq
2\left(\frac{\sqrt{\sigma}-1}{\sqrt{\sigma}+1}\right)^{n}\|\vec{r}_{0}\|_{\bm{P}^{-1}},$

where $\|\vec{r}\|_{\bm{P}^{-1}}^{2}:=\vec{r}^{T}\bm{P}^{-1}\vec{r}$ and
$\sqrt{\sigma}\leq C(1+\log^{3}k)$ with $C$ independent of $k$ and $h$. Since
all norms on finite dimension vector spaces are equivalent, Eq. 53 holds for
any choice of norm at the expense of replacing “2” by an appropriate constant
depending on the choice of norm, which completes the proof of Theorem 7.1.

Theorem 7.1 shows that the performance of the preconditioner deteriorates at
most as $\log^{3}k$ as the polynomial order is increased, but remains bounded
as the mesh is refined provided the shape regularity assumption Eq. 10 is
satisfied.

###  7.4 Implementation and Cost Analysis of the Preconditioner

To aid in the implementation and cost analysis of computing the actions of
$\bar{\bm{A}}^{-1}$ and $\bar{\bm{M}}^{-1}$, we assume, for convenience, the
interface degrees of freedom are ordered as follows:

  1. (i)

velocity $C^{0}$ vertex degrees of freedom,

  2. (ii)

velocity $C^{1}$ vertex degrees of freedom,

  3. (iii)

velocity edge degrees of freedom, grouped according to edge,

  4. (iv)

pressure vertex degrees of freedom,

  5. (v)

pressure average value degrees of freedom.

This ordering induces a block structure in the matrix $\widetilde{\bm{A}}$ in
which the diagonal subblocks are: $\widetilde{\bm{A}}_{c}$, corresponding to
the global interaction among all the global $C^{0}$ vertex functions;
$\widetilde{\bm{A}}_{\bm{a},\mu}$, the block-diagonal entry corresponding to
the $C^{1}$ vertex functions
$\\{\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{1},\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{2}\\}$;
whilst $\widetilde{\bm{A}}_{\gamma}$ corresponds to the interactions among the
edge functions associated to $\gamma$. The load vectors can be similarly split
into subvectors corresponding to the same groupings of degrees of freedom. The
block diagonal structure of $\bm{P}$ is then exploited to compute the action
of $\bm{P}^{-1}$ on a pair of vectors $\vec{f}$, $\vec{g}$ efficiently or in
parallel, as described in Algorithm 1.

Algorithm 1 Action of Preconditioner

1: $\widetilde{\bm{A}}$, $\vec{f}$, $\vec{g}$

2:function

3: $\vec{u}_{c}=\widetilde{\bm{A}}_{c}^{-1}\vec{f}_{c}$ $\triangleright$
Global velocity $C^{0}$ vertex function solve

4: for $\bm{a}\in\mathcal{V},\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}$, do
$\triangleright$ Block diagonal velocity $C^{1}$ vertex function solve

5: $u_{\bm{a},\mu}=\widetilde{\bm{A}}_{\bm{a},\mu}^{-1}\vec{f}_{\bm{a},\mu}$

6: end for

7: for $\gamma\in\mathcal{E}_{I}$ do $\triangleright$ Block diagonal velocity
edge solve

8: $\vec{u}_{\gamma}=\widetilde{\bm{A}}_{\gamma}^{-1}\vec{{f}}_{\gamma}$

9: end for

10: for $\bm{a}\in\mathcal{V}$, $\omega\in\Omega_{\bm{a}}$ do $\triangleright$
Diagonal pressure $C^{0}$ vertex function solve

11: $p_{\bm{a},\omega}=|\omega|^{-1}k^{4}{g}_{\bm{a},\omega}$

12: end for

13: for $K\in\mathcal{T}$ do $\triangleright$ Diagonal pressure average value
solve

14: $p_{K}=|K|^{-1}{g}_{K}$

15: end for

16: return $\vec{u}_{c}$, $(\vec{u}_{\bm{a},\mu})_{\bm{a},\mu}$,
$(\vec{u}_{\gamma})_{\gamma}$, $(p_{\bm{a},\omega})_{\bm{a},\omega}$,
$(p_{K})_{K}$ $\triangleright$ Return degrees of freedom

17:end function

The cost of computing the action of $\bm{P}^{-1}$ using Algorithm 1 comprises
of two parts: one-time setup costs and recurring costs associated with each
application of Algorithm 1. The setup cost is dominated by eliminating the
interior degrees of freedom on each element, which takes
$\mathcal{O}(|\mathcal{T}|k^{6})$ operations needed for the subassembly of the
Schur complement. The matrices $\bm{A}_{c}$, $\bm{A}_{\bm{a},\mu}$,
$\bm{a}\in\mathcal{V}$, $\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}$, and
$\bm{A}_{\gamma}$, $\gamma\in\mathcal{E}$, need only be factored once at a
cost of $\mathcal{O}(|\mathcal{V}|^{3}+|\mathcal{E}|k^{3})$ operations, giving
an overall setup cost of
$\mathcal{O}(|\mathcal{T}|k^{6}+|\mathcal{E}|k^{3}+|\mathcal{V}|^{3})$.

We now turn to the cost associated with each application of Algorithm 1. Line
2 of Algorithm 1 entails the solution of the linear system $\bm{A}_{c}$
involving all of the $C^{0}$ vertex functions which, thanks to the
prefactorisation of $\bm{A}_{c}$, costs $\mathcal{O}(|\mathcal{V}|^{2})$
operations per solve. Lines 3-5 require the solution of a 2x2 matrix on the
velocity $C^{1}$ vertex functions for each vertex $\bm{a}\in\mathcal{V}$ and
derivative degree of freedom $\hat{\bm{\mu}}\in\mathring{D}_{\bm{a}}$ at a
cost of $\mathcal{O}(|\mathcal{V}|)$ operations. Lines 6-8 entail a block
diagonal solve over each of the edges which, again thanks to the
prefactorisation of $\bm{A}_{\gamma}$, $\gamma\in\mathcal{E}$, can be applied
using $\mathcal{O}(|\mathcal{E}|k^{2})$ operations. Lines 9-11 require
$\mathcal{O}(|\mathcal{V}|)$ operations and lines 12-14 require
$\mathcal{O}(|\mathcal{T}|)$ operations by analogous arguments. In summary,
the overall cost per application of Algorithm 1 is
$\mathcal{O}(|\mathcal{E}|k^{2}+|\mathcal{V}|^{2}+|\mathcal{T}|)$, which is
comparable to nonoverlapping domain decomposition methods for second order
elliptic problems [37].

##  8 Numerical Examples

We illustrate the performance of the preconditioner described in Section 7 in
two numerical examples.

###  8.1 Moffatt Eddies

In the first example, we revisit the Moffatt problem [27] considered in [7],
in which the domain $\Omega$ is the wedge with a prescribed parabolic flow
profile on the top part of the boundary and no flow on the remainder of the
boundary:

$\displaystyle\bm{u}(x,0)=\begin{bmatrix}1-x^{2}\\\ 0\end{bmatrix},\ -1\leq
x\leq 1,\quad\text{and}\quad\bm{u}=\bm{0}\text{ on
}\Gamma\setminus(-1,1)\times\\{0\\}.$

The problem is approximated using a pure $p$-version finite element scheme on
the fixed mesh shown in Fig. 4(a). The results in [7] show that the $k=13$
solution resolves four to five eddies, equivalent to a $10^{13}$ range of
scales.

Let $\lambda_{min}^{\pm}$ and $\lambda_{max}^{\pm}$ denote the extremal
eigenvalues of $\bm{P}^{-1}\bm{S}$ so that

(54)
$\displaystyle\sigma(\bm{P}^{-1}\bm{S})\subseteq[-\lambda_{max}^{-},-\lambda_{min}^{-}]\cup\\{0\\}\cup[\lambda_{min}^{+},\lambda_{max}^{+}].$

According to Theorem 7.1, $\lambda_{max}^{\pm}\leq C$ and
$\lambda_{min}^{\pm}\geq C(1+\log^{3}k)^{-1}$ with constant $C$ independent of
$k$ and $h$. Fig. 4(b) displays the actual values of the extreme eigenvalues.
In agreement with theory, $\lambda_{max}^{\pm}$ is uniformly bounded in $k$
and $\lambda_{min}^{+}\geq C(1+\log^{3}k)^{-1}$. However, $\lambda_{min}^{-}$
appears to remain uniformly bounded in $k$, which would mean that, in
practice, the contraction factor in Theorem 7.1 is pessimistic. The residual
history for $k\in\\{4,7,10,13\\}$ for the preconditioned MINRES solver are
displayed in Fig. 4(c). The starting vector is taken to be
$\vec{x}+\vec{\epsilon}$, where $\vec{x}$ is the true solution of the Schur
complement system Eq. 5 and $\vec{\epsilon}$ is a random perturbation with
entries uniformly distributed in $(-1,1)$. Here, and in the remaining
examples, the relative residual is given by
$\sqrt{(\vec{r}^{T}\bm{P}^{-1}\vec{r})/(\vec{r}_{0}^{T}\bm{P}^{-1}\vec{r}_{0})}$,
where $\vec{r}_{0}$ is the initial residual vector, and MINRES is terminated
when the relative residual is smaller than $10^{-8}$. It is observed that, as
the polynomial order is raised, the iteration counts grow modestly consistent
with the results in Theorem 7.1.

(a)

(b)

(c)

Figure 4: (a) 18 element mesh, (b) extremal eigenvalues of
$\bm{P}^{-1}\bm{S}$, and (c) MINRES convergence history with
$k\in\\{4,7,10,13\\}$, stopping tolerance $10^{-8}$ for a sequence of random
initial iterates for Moffatt eddies problem. $1/\lambda_{min}^{+}$ grows as
$\log^{3}k$ while the other extreme eigenvalues remain bounded as the
polynomial degree $k$ is increased.

###  8.2 T-shaped Domain

In the next example, we consider the T-shaped domain example [2] where
$\bm{f}\equiv\bm{0}$ and boundary conditions are parabolic flow profile on the
leftmost and rightmost boundaries of the domain and no flow on the remainder
of the boundary:

$\displaystyle\bm{u}\left(\pm\frac{3}{2},y\right)$
$\displaystyle=\begin{bmatrix}y(1-y)\\\ 0\end{bmatrix},\ 0\leq y\leq
1,\quad\text{and}\quad\bm{u}=\bm{0}\text{ on
}\Gamma\setminus\left\\{\pm\frac{3}{2}\right\\}\times(0,1).$

The sequence of meshes is shown in Fig. 5, in which the elements are
geometrically graded and which were proved to give exponential convergence of
the finite element solution [7, §7.2]. The mesh in Fig. 5(a) consists of one
layer of elements around the re-entrant corner and the most bottom corners,
with a grading factor of $\sigma=0.08$. We then refine the mesh by
successively adding layers of elements such that the innermost layer of
elements has a diameter proportional to $\sigma^{n}$, where $n$ is the number
of refinements. For example, the mesh corresponding to three levels is shown
in Figs. 5(b) and 5(c). Observe that, once a mesh contains two or more layers,
the shape regularity constant $\kappa$ Eq. 10 changes from 0.1695 to 0.0829
due to the presence of “needle” elements near the corners. In particular,
several estimates in the analysis depend on $\kappa$, and thus we would expect
the performance of the preconditioner to be worse for $n\geq 2$ than for
$n=1$.

As with the previous example, the extremal eigenvalues Eq. 54, displayed in
Fig. 6, remain bounded independently of the number of levels of geometric
refinement, whilst $1/\lambda_{min}^{+}$ increases by a factor of roughly 10
after one level of refinement due to the change in shape regularity mentioned
above. This value is an order of magnitude greater than the value of
$1/\lambda_{min}^{+}$ observed for the Moffatt ($\kappa=0.1508$) example and
accounts for the increase of the resulting iteration counts observed in the
residual histories for $k\in\\{4,7,10,13\\}$ in Fig. 7. Thus, as one might
expect, the preconditioner $\bm{P}^{-1}$ is less effective on meshes
containing high aspect ratio elements owing to the fact that the inf-sup
constants and inequalities employed in Appendix A all depend on the shape
regularity constant appearing in Eq. 10. Nevertheless, similar to the behavior
observed in the previous example, for each fixed $n$, the iteration counts
grow modestly in $k$. For each fixed $k$, the iteration counts are bounded in
$n$, and remain virtually unchanged for $n\geq 3$.

(a)

(b)

(c)

Figure 5: (a) Mesh with $n=1$ layer of elements, (b) Mesh with $n=3$ layers of
elements, and (c) Zoom on re-entrant corner of mesh with $n=3$ layers of
elements.

(a)

(b)

(c)

Figure 6: Extremal eigenvalues of $\bm{P}^{-1}\bm{S}$ for the T-shape problem
with (a) $n=1$, (b) $n=2$, and (c) $n=3$ layers of geometrically graded
elements at the corners. All of the extreme eigenvalues are uniformly bounded
in $n$ for each fixed $k$. In addition, the introduction of small-angle
“needle” elements for $n\geq 2$ greatly increases $1/\lambda_{min}^{+}$.

(a)

(b)

(c)

Figure 7: MINRES convergence history with $k\in\\{4,7,10,13\\}$, stopping
tolerance $10^{-8}$ for a sequence of random initial iterates for the T-shape
problem with (a) $n=1$, (b) $n=2$, and (c) $n=3$ layers of geometrically
graded elements at the corners.

##  Appendix A Technical Lemmas

In this section, we establish a spectral equivalence of the ASM
preconditioners given in Section 7 to the inner products appearing in the
Stokes equations. The main result is the following theorem, which is an
immediate consequence of Lemmas A.6, A.8, A.10, and A.16 proved later in this
section:

######  Theorem A.1.

There exists positive constants $C_{1}$ and $C_{2}$, independent of $k$ and
$h$, such that

(55) $\displaystyle C_{1}^{-1}(p,p)\leq\bar{m}(p,p)\leq C_{1}(p,p)\quad\forall
p\in Q_{I}^{\perp},$

and

(56) $\displaystyle C_{2}^{-1}a(\bm{u},\bm{u})\leq\bar{a}(\bm{u},\bm{u})\leq
C_{2}\beta^{-2}(1+\log^{3}k)a(\bm{u},\bm{u})\quad\forall\bm{u}\in\tilde{\bm{V}}_{E},$

where $\bar{m}(\cdot,\cdot)$ is defined in Eq. 47, $\bar{a}(\cdot,\cdot)$ is
defined in Eq. 51, and $\beta$ is the inf-sup constant defined in Eq. 3.

###  A.1 Pressure ASM

We begin with the pressure ASM. The first lemma establishes a key estimate for
the norm of the pressure vertex functions:

######  Lemma A.2.

The pressure vertex functions functions $\tilde{\psi}_{\bm{a}}^{\omega}$,
$\bm{a}\in\mathcal{V}$, $\omega\in\Omega_{\bm{a}}$, satisfy

(57) $\displaystyle\|\tilde{\psi}_{\bm{a}}^{\omega}\|_{L^{2}(K)}\leq
Ch_{K}k^{-2}\quad\forall K\in\mathcal{T},$

with $C$ independent of $k$, $h_{K}$, and $\bm{a}$.

######  Proof A.3.

Let $\bm{a}\in\mathcal{V}$, $\omega\in\Omega_{\bm{a}}$. Define the function
$\chi\in Q$ by the rule $\chi=\chi_{K}$ on each element $K\in\mathcal{T}$
where $\chi_{K}$ is chosen as in [7, Lemma 4.1]. In particular,
$\chi_{K}\in\mathcal{P}_{k-1}(K)\cap L^{2}_{0}(K)$ satisfies (i)
$\chi_{K}\equiv 0$ if $K\nsubseteq\omega$, (ii)
$\chi_{K}(\bm{b})=\delta_{\bm{a}\bm{b}}$ for $\bm{b}\in\mathcal{V}_{K}$, and
(iii) $\|\chi_{K}\|_{L^{2}(K)}\leq Ch_{K}k^{-2}$ with $C$ independent of
$h_{K}$, $k$, and $\bm{a}$. By Eq. 24d, Eq. 24e, and Theorem 4.3,
$\tilde{\psi}_{\bm{a}}=\tilde{\Pi}\chi$, and
$\|\tilde{\psi}_{\bm{a}}^{\omega}\|_{L^{2}(K)}\leq\|\chi\|_{L^{2}(K)}\leq
Ch_{K}k^{-2}$ $\forall K\in\mathcal{T}$.

We now show that the inner products on the subspaces are coercive:

######  Lemma A.4.

There exists a positive constant $C$ independent of $k$ and $h$ such that

$\displaystyle(p,p)$ $\displaystyle\leq Cm_{\bm{a},\omega}(p,p)$
$\displaystyle\forall
p\in\tilde{Q}_{\bm{a},\omega},\quad\bm{a}\in\mathcal{V},\quad\omega\in\Omega_{\bm{a}},$
$\displaystyle(p,p)$ $\displaystyle\leq Cm_{K}(p,p)$ $\displaystyle\forall
p\in\tilde{Q}_{K},\quad K\in\mathcal{T}.$

######  Proof A.5.

Let $p\in\tilde{Q}_{\bm{a},\omega}$, $\bm{a}\in\mathcal{V}$,
$\omega\in\Omega_{\bm{a}}$. Then, $p=p(\bm{a})\tilde{\psi}_{\bm{a}}^{\omega}$,
and by Eq. 57 and shape regularity Eq. 10, there holds

$\displaystyle(p,p)\leq
C\sum_{K\in\mathcal{T}:K\subseteq\omega}h_{K}^{2}k^{-4}|p(\bm{a})|^{2}\leq
C|\omega|k^{-4}=m_{\bm{a},\omega}(p,p).$

Now let $p\in Q_{K}$, $K\in\mathcal{T}$. Then, $p=(|K|^{-1}\int_{K}p\
d\bm{x})\tilde{\psi}_{K}$ and since $\tilde{\psi}_{\bm{a}}^{\omega}\in
L^{2}_{0}(K)$,

$\displaystyle\int_{K}\tilde{\psi}_{K}^{2}\
d\bm{x}=\int_{K}\left\\{1+\left(\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\omega\supseteq
K\end{subarray}}\tilde{\psi}_{\bm{a}}^{\omega}\right)^{2}\right\\}\
d\bm{x}\leq|K|+Ch_{K}^{2}k^{-4}\leq C|K|.$

Thus, $(p,p)\leq Cm_{K}(p,p)$.

We are now able to establish the left-hand side of the equivalence Eq. 55:

######  Lemma A.6.

There exists a constant $C$ independent of $k$ and $h$ such that

(58) $\displaystyle(p,p)\leq C\bar{m}(p,p)\quad\forall p\in Q_{I}^{\perp}.$

######  Proof A.7.

Let $p\in Q_{I}^{\perp}$. By Cauchy-Schwarz, there holds

$\displaystyle(p,p)_{K}\leq
4\left[\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\ \omega\supseteq
K\end{subarray}}(p_{\bm{a},\omega},p_{\bm{a},\omega})_{K}+(p_{K},p_{K})_{K}\right]$

where $(p,q)_{K}:=\int_{K}pq\ d\bm{x}$. Eq. 58 now follows from Lemma A.4 and
summing over the elements.

The right-hand side of the equivalence Eq. 55 is covered by the next result:

######  Lemma A.8.

There exists a positive constant $C$ independent of $k$ and $h$ such that

(59) $\displaystyle\bar{m}(p,p)\leq C(p,p)\quad\forall p\in Q_{I}^{\perp}.$

######  Proof A.9.

By [4, Lemma 6.1], there holds

$\displaystyle|p|_{K}(\bm{a})|^{2}k^{-4}=|(p|_{K}\circ\bm{F}_{K})(\hat{\bm{a}})|^{2}k^{-4}\leq
C\|p\circ\bm{F}_{K}\|_{L^{2}(\hat{T})}^{2}\leq Ch_{K}^{2}\|p\|_{L^{2}(K)}^{2}$

with $\hat{\bm{a}}=\bm{F}_{K}^{-1}(\bm{a})$, and by shape regularity,

$\displaystyle
m_{\bm{a},\omega}(p_{\bm{a},\omega},p_{\bm{a},\omega})=|\omega|k^{-4}|p|_{{\omega}}(\bm{a})|^{2}\leq
C\sum_{K\in\mathcal{T}:K\subseteq\omega}\|p\|_{L^{2}(K)}^{2}.$

Summing over $\bm{a}\in\mathcal{V}$, $\omega\in\Omega_{\bm{a}}$ and again
using shape regularity to bound the overlap
$|\\{\omega:\exists\bm{a}\in\mathcal{V}:K\subseteq\omega\in\Omega_{\bm{a}}\\}|$
gives

$\displaystyle\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\omega\in\Omega_{\bm{a}}\end{subarray}}m_{\bm{a},\omega}(p_{\bm{a},\omega},p_{\bm{a},\omega})\leq
C\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}\\\
\omega\in\Omega_{\bm{a}}\end{subarray}}\sum_{K\in\mathcal{T}:K\subseteq\omega}\|p\|_{L^{2}(K)}^{2}\leq
C\|p\|_{L^{2}(\Omega)}^{2}.$

To bound the remaining $m_{K}(\cdot,\cdot)$ terms, we use Cauchy-Schwarz:

$\displaystyle\sum_{K\in\mathcal{T}}m_{K}(p_{K},p_{K})=\sum_{K\in\mathcal{T}}\frac{1}{|K|}\left(\int_{K}p\
d\bm{x}\right)^{2}\leq C\sum_{K\in\mathcal{T}}\|p\|_{L^{2}(K)}^{2}\leq
C\|p\|_{L^{2}(\Omega)}^{2},$

which completes the proof of Eq. 59.

###  A.2 Velocity ASM

We now turn to the velocity space, and start by extending the decomposition
Eq. 50 as follows. For $\bm{u}\in\tilde{\bm{V}}_{E}$, we define
$\bm{u}_{\gamma}\equiv\bm{0}$ for
$\gamma\in\mathcal{E}\setminus\mathcal{E}_{I}$ and
$\bm{u}_{\bm{a},\mu}\equiv\bm{0}$ for $\bm{a}\in\mathcal{V}$,
$\hat{\bm{\mu}}\in D_{\bm{a}}\setminus\mathring{D}_{\bm{a}}$. Since the inner
product on each of the subspace was taken to be $a(\cdot,\cdot)$, we
immediately obtain the left-hand side of the equivalence Eq. 56:

######  Lemma A.10.

For all $\bm{u}\in\tilde{\bm{V}}_{E}$, there holds

(60) $\displaystyle a(\bm{u},\bm{u})\leq 10\bar{a}(\bm{u},\bm{u}).$

######  Proof A.11.

First recall that there are exactly 2 directional derivative degrees of
freedom per velocity component per vertex on any given element, i.e. for
$K\in\mathcal{T}$,
$|\\{\hat{\bm{\mu}}:K\subseteq\operatorname{supp}\phi_{\bm{a}}^{\mu}\\}|=2$.
By Cauchy-Schwarz, there holds

$\displaystyle|\bm{u}|_{\bm{H}^{1}(K)}^{2}\leq
10\left\\{|\bm{u}_{c}|_{\bm{H}^{1}(K)}^{2}+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}|\bm{u}_{\bm{a},\mu}|_{\bm{H}^{1}(K)}^{2}+\sum_{\gamma\in\mathcal{E}_{K}}|\bm{u}_{\gamma}|_{\bm{H}^{1}(K)}^{2}\right\\},$
$\displaystyle\forall K\in\mathcal{T}.$

Equation 60 now follows by summing over the elements.

To prove the right-hand side of Eq. 56, we need to establish some properties
of the velocity vertex functions:

######  Lemma A.12.

The $C^{0}$ velocity vertex functions satisfy the following: For
$K\in\mathcal{T}$,

(61)
$\displaystyle\mathbb{R}^{2}\ni\bm{c}=\sum_{\bm{a}\in\mathcal{V}_{K}}\sum_{i=1}^{2}(\bm{c}\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\quad\text{on
}K$

and

(62)
$\displaystyle\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq C\quad\forall\bm{a}\in\mathcal{V},\ i=1,2,$

where $C$ depends only on the shape regularity parameter.

Moreover, the $C^{1}$ velocity vertex functions satisfy

(63)
$\displaystyle\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq C\|D\bm{F}_{K}\|_{L^{\infty}(\hat{T})}k^{-2}$
$\displaystyle\bm{a}\in\mathcal{V},\ \hat{\bm{\mu}}\in D_{\bm{a}},\ i=1,2,$

where $C$ depends only on the shape regularity parameter.

######  Proof A.13.

Let $K\in\mathcal{T}$. A simple computation reveals that
$1=(\lambda_{1}+\lambda_{2}+\lambda_{3})^{3}=\sum_{\bm{a}\in\mathcal{V}_{K}}\phi_{\bm{a}}+6\lambda_{1}\lambda_{2}\lambda_{3}$
on $K$, where $\\{\lambda_{i}:\ 1\leq i\leq 3\\}$ are the barycentric
coordinates on $K$, and hence, for any $\bm{c}\in\mathbb{R}^{2}$,

$\displaystyle\bm{c}=\sum_{\bm{a}\in\mathcal{V}}\sum_{i=1}^{2}(\bm{c}\cdot\hat{\bm{e}}_{i})(\phi_{\bm{a}}\hat{\bm{e}}_{i})+\underbrace{6\bm{c}\sum_{K\in\mathcal{T}}\lambda_{1}\lambda_{2}\lambda_{3}}_{\in\bm{V}_{I}}.$

Applying $\bm{\Pi}_{\bm{V}}$ to both sides of this identity and noting that
Theorem 4.1 gives $\bm{\Pi}_{\bm{V}}:\bm{V}_{I}\to\\{\bm{0}\\}$, we obtain

$\displaystyle\bm{\Pi}_{\bm{V}}\bm{c}=\sum_{\bm{a}\in\mathcal{V}}\sum_{i=1}^{2}(\bm{c}\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i}).$

Finally, Theorem 4.1 implies that $\mathscr{E}(\bm{c},0)=(\bm{c},0)$ and Eq.
61 follows at once.

Now let $K\in\mathcal{T}$ and $i\in\\{1,2\\}$. Clearly Eq. 62 holds if
$\bm{a}\notin\mathcal{V}_{K}$ since $\phi_{\bm{a}}\hat{\bm{e}}_{i}=\bm{0}$.
Otherwise, if $\bm{a}\in\mathcal{V}_{K}$, we apply a scaling argument in
conjunction with Eq. 26 to arrive at

$\displaystyle\frac{1}{|K|^{2}}\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\|_{\bm{L}^{2}(K)}^{2}+|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})|_{\bm{H}^{1}(K)}^{2}\leq
C\left\\{|\phi_{\bm{a}}|_{H^{1/2}(\partial K)}^{2}+\frac{1}{|\partial
K|}\|\phi_{\bm{a}}\|_{L^{2}(\partial K)}^{2}\right\\},$

where $C$ is a positive constant independent of $k$ and $h_{K}$. Thus,

$\displaystyle\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq C\|\hat{\phi}_{j}\|_{H^{1/2}(\partial\hat{T})}\leq
C\|\hat{\phi}_{j}\|_{H^{1}(\hat{T})}\leq C,$

where $\hat{\bm{a}}_{j}=\bm{F}_{K}^{-1}(\bm{a})$. For
$K\subseteq\operatorname{supp}\phi_{\bm{a}}^{\mu}$, we argue similarly and use
Eq. 44 to obtain

$\displaystyle\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq
C\left\|\begin{bmatrix}\hat{\bm{\mu}}&\hat{\bm{\xi}}\end{bmatrix}^{-1}\right\|\|D\bm{F}_{K}\|_{L^{\infty}(\hat{T})}\left\|\begin{bmatrix}\hat{\phi}_{j}^{(1,0)}\\\
\hat{\phi}_{j}^{(0,1)}\end{bmatrix}\right\|_{\bm{H}^{1/2}(\partial\hat{T})},$

where $\hat{\bm{\mu}}\neq\hat{\bm{\xi}}\in D_{\bm{a}}$ is chosen such that
$\operatorname{supp}\phi_{\bm{a}}^{\xi}\supseteq K$,
$\hat{\bm{a}}_{j}=\bm{F}_{K}^{-1}(\bm{a})$, and $\|\cdot\|$ is any matrix
norm. By the definition of $D_{\bm{a}}$ Eq. 43 and shape regularity,
$\left\|\begin{bmatrix}\hat{\bm{\mu}}&\hat{\bm{\xi}}\end{bmatrix}^{-1}\right\|$
is uniformly bounded by a constant depending only on $\kappa$ Eq. 10. Since
$\hat{\phi}_{j}^{(1,0)}(\hat{\bm{a}})=\hat{\phi}_{j}^{(0,1)}(\hat{\bm{a}})={0}$
for $\hat{\bm{a}}\in\mathcal{V}_{\hat{T}}$ by the construction Eq. 37, there
holds

$\displaystyle\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}^{\mu}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq
C\|D\bm{F}_{K}\|_{L^{\infty}(\hat{T})}\|J\|_{H^{1/2}_{00}(I)}$

where $I=(-1,1)$, $H^{1/2}_{00}(I)$ is the usual Sobolev space (defined as,
e.g. [23]), and

$\displaystyle
J(t)=\frac{1}{P^{(3,3)}_{k-3}(-1)}\left(\frac{1+t}{2}\right)\left(\frac{1-t}{2}\right)^{2}P^{(3,3)}_{k-3}(t).$

Thanks to [5, Lemma B.1], $\|J\|_{L^{2}(I)}\leq Ck^{-3}$ with $C$ independent
of $k$. Using interpolation, and the inverse estimate
$\|J^{\prime}\|_{L^{2}(I)}\leq Ck^{2}\|J\|_{L^{2}(I)}$ [10, Lemma 5.4], we
obtain $\|J\|_{H^{1/2}_{00}(I)}\leq
C\|J\|_{L^{2}(I)}^{1/2}\|J\|_{H^{1}(I)}^{1/2}\leq Ck^{-2}$, which completes
the proof of Eq. 63.

We now use the properties of the vertex functions to prove element-wise
stability of the subspace decomposition Eq. 50:

######  Lemma A.14.

For $\bm{u}\in\tilde{\bm{V}}_{E}$ and $K\in\mathcal{T}$, there holds

(64)
$\displaystyle|\bm{u}_{c}|_{\bm{H}^{1}(K)}^{2}+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}|\bm{u}_{\bm{a},\mu}|_{\bm{H}^{1}(K)}^{2}+\sum_{\gamma\in\mathcal{E}_{K}}|\bm{u}_{\gamma}|_{\bm{H}^{1}(K)}^{2}\leq
C\beta^{-2}(1+\log^{3}k)|\bm{u}|_{\bm{H}^{1}(K)}^{2},$

where $C$ is independent of $k$, $h_{K}$ and $\bm{u}$.

######  Proof A.15.

Let $\bm{u}\in\tilde{\bm{V}}_{E}$ and $K\in\mathcal{T}$. For any
$\bm{c}\in\mathbb{R}^{2}$, we have the decomposition

$\displaystyle\bm{u}-\bm{c}=\bm{u}_{c}-\sum_{\bm{a}\in\mathcal{V}_{K}}\sum_{i=1}^{2}(\bm{c}\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}\bm{u}_{\bm{a},\mu}+\sum_{\gamma\in\mathcal{E}_{K}}\bm{u}_{\gamma}\quad\text{on
}K$

thanks to Eq. 61. Thus,

$\displaystyle\hat{\bm{u}}-\bm{c}=\hat{\bm{u}}_{c}-\sum_{{\bm{a}}\in\mathcal{V}_{K}}\sum_{i=1}^{2}(\bm{c}\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}\hat{\bm{u}}_{\bm{a},\mu}+\sum_{\gamma\in\mathcal{E}_{K}}\hat{\bm{u}}_{\gamma}\quad\text{on
}\hat{T},$

where $\hat{\bm{u}}=\bm{u}\circ\bm{F}_{K}$,
$\hat{\bm{u}}_{c}=\bm{u}_{c}\circ\bm{F}_{K}$, etc. We first bound the energy
of $\hat{\bm{u}}_{c}$. Since

$\displaystyle\hat{\bm{u}}_{c}$
$\displaystyle=\sum_{\bm{a}\in\mathcal{V}_{K}}\sum_{i=1}^{2}(\bm{u}(\bm{a})\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}$
$\displaystyle=\sum_{\hat{\bm{a}}\in\mathcal{V}_{\hat{T}}}\sum_{i=1}^{2}(\hat{\bm{u}}(\hat{\bm{a}})\cdot\hat{\bm{e}}_{i})\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\quad\text{on
}\hat{T},$

where $\bm{a}=\bm{F}_{K}(\hat{\bm{a}})$, we use [9, Corollary 6.3] and Eq. 62
to obtain

$\displaystyle\|\hat{\bm{u}}_{c}-\bm{c}\|_{\bm{H}^{1}(\hat{T})}^{2}$
$\displaystyle\leq\sum_{\hat{\bm{a}}\in\mathcal{V}_{\hat{T}}}|\hat{\bm{u}}(\hat{\bm{a}})-\bm{c}|^{2}\sum_{i=1}^{2}\|\bm{\Pi}_{\bm{V}}(\phi_{\bm{a}}\hat{\bm{e}}_{i})\circ\bm{F}_{K}\|_{\bm{H}^{1}(\hat{T})}^{2}$
(65) $\displaystyle\leq C(1+\log
k)\|\hat{\bm{u}}-\bm{c}\|_{\bm{H}^{1}(\hat{T})}^{2}.$

We now bound the vertex derivative contribution. For
$\bm{a}\in\mathcal{V}_{K}$, we note that
$\partial_{\mu}(\bm{u}\cdot\hat{\bm{e}}_{i})(\bm{a})=\hat{\bm{\mu}}^{T}D\bm{F}_{K}^{-T}D(\bm{u}\cdot\hat{\bm{e}}_{i}\circ\bm{F}_{K})(\hat{\bm{a}})$,
$i=1,2$, where $\hat{\bm{a}}=\bm{F}_{K}^{-1}({\bm{a}})$. Applying [4, Lemma
6.1] to $D(\bm{u}\circ\bm{F}_{K})$, and using Eq. 63 and shape regularity
gives

(66) $\displaystyle\|\hat{\bm{u}}_{\bm{a},\mu}\|_{\bm{H}^{1}(\hat{T})}^{2}$
$\displaystyle\leq
C\|D\bm{F}_{K}^{-T}\|_{L^{\infty}(K)}^{2}\cdot\|D\bm{F}_{K}\|_{L^{\infty}(\hat{T})}^{2}\cdot|D\hat{\bm{u}}(\hat{\bm{a}})|^{2}k^{-4}\leq
C|\hat{\bm{u}}|_{\bm{H}^{1}(\hat{T})}^{2}.$

Now, we define

$\displaystyle\bm{u}^{\\#}:=(\bm{u}-\bm{c})-(\bm{u}_{c}-\bm{c})-\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}\bm{u}_{\bm{a},\mu}\quad\text{on
}K.$

Then, $D^{\alpha}\bm{u}^{\\#}(\bm{a})=\bm{0}$ for $\bm{a}\in\mathcal{V}_{K}$,
$|\alpha|\leq 1$, and thanks to Eqs. 65 and 66,
$\hat{\bm{u}}^{\\#}:=\bm{u}^{\\#}\circ\bm{F}_{K}$ may be estimated as follows:

(67) $\displaystyle\|\hat{\bm{u}}^{\\#}\|_{\bm{H}^{1}(\hat{T})}^{2}\leq
C(1+\log k)\|\hat{\bm{u}}-\bm{c}\|_{\bm{H}^{1}(\hat{T})}^{2}$

Let $\gamma\in\mathcal{E}_{K}$. Equation 27, shape regularity Eq. 10, and the
trace theorem give

(68) $\displaystyle|\hat{\bm{u}}_{\gamma}|_{\bm{H}^{1}(\hat{T})}$
$\displaystyle\leq C\beta^{-1}|\bm{u}_{\gamma}|_{\bm{H}^{1/2}(\partial K)}\leq
C\beta^{-1}|\hat{\bm{u}}_{\gamma}|_{\bm{H}^{1/2}(\partial\hat{T})}\leq
C\beta^{-1}\|\hat{\bm{u}}^{\\#}\|_{\bm{H}_{00}^{1/2}(\hat{\gamma})},$

where $\hat{\gamma}=\bm{F}_{K}^{-1}(\gamma)$. Thanks to [9, Theorem 6.5] and
the trace theorem, we have the estimate

(69)
$\displaystyle\|\hat{\bm{u}}^{\\#}\|_{\bm{H}^{1/2}_{00}(\hat{\gamma})}\leq
C(1+\log k)\|\hat{\bm{u}}^{\\#}\|_{\bm{H}^{1/2}(\hat{\gamma})}\leq C(1+\log
k)\|\hat{\bm{u}}^{\\#}\|_{\bm{H}^{1}(\hat{T})}.$

Using Eqs. 69, 67, and 68 gives

(70) $\displaystyle|\hat{\bm{u}}_{\gamma}|_{\bm{H}^{1}(\hat{T})}^{2}\leq
C\beta^{-2}(1+\log^{2}k)\|\hat{\bm{u}}^{\\#}\|_{\bm{H}^{1}(\hat{T})}^{2}\leq
C\beta^{-2}(1+\log^{3}k)\|\hat{\bm{u}}-\bm{c}\|_{\bm{H}^{1}(\hat{T})}^{2},$

Combining Eqs. 65, 66, and 70 leads to

$\displaystyle|\hat{\bm{u}}_{c}|_{\bm{H}^{1}(\hat{T})}^{2}+\sum_{\begin{subarray}{c}{\bm{a}}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}|\hat{\bm{u}}_{\bm{a},\mu}|_{\bm{H}^{1}(\hat{T})}^{2}$
$\displaystyle+\sum_{\gamma\in\mathcal{E}_{K}}|\hat{\bm{u}}_{\gamma}|_{\bm{H}^{1}(\hat{T})}^{2}$
$\displaystyle\qquad\leq
C\beta^{-2}(1+\log^{3}k)\|\hat{\bm{u}}-\bm{c}\|_{\bm{H}^{1}(\hat{T})}^{2},$

where we used that
$|\hat{\bm{u}}_{c}|_{\bm{H}^{1}(\hat{T})}=|\hat{\bm{u}}_{c}-\bm{c}|_{\bm{H}^{1}(\hat{T})}$.
Taking the infimum over all $\bm{c}\in\mathbb{R}^{2}$ and applying the
quotient norm equivalence [28, Theorem 7.2] gives

$\displaystyle|\hat{\bm{u}}_{c}|_{\bm{H}^{1}(\hat{T})}^{2}+\sum_{\begin{subarray}{c}\bm{a}\in\mathcal{V}_{K}\\\
\hat{\bm{\mu}}:K\in\operatorname{supp}\phi_{\bm{a}}^{\mu}\end{subarray}}|\hat{\bm{u}}_{\bm{a},\mu}|_{\bm{H}^{1}(\hat{T})}^{2}+\sum_{\gamma\in\mathcal{E}_{K}}|\hat{\bm{u}}_{\gamma}|_{\bm{H}^{1}(\hat{T})}^{2}\leq
C\beta^{-2}(1+\log^{3}k)|\hat{\bm{u}}|_{\bm{H}^{1}(\hat{T})}^{2}.$

Equation 64 now follows from shape regularity Eq. 10.

Summing Eq. 64 over the elements leads to the following:

######  Lemma A.16.

There exists a constant $C$ independent of $k$ and $h$ such that

(71) $\displaystyle\bar{a}(\bm{u},\bm{u})\leq
C\beta^{-2}(1+\log^{3}k)a(\bm{u},\bm{u})\quad\forall\bm{u}\in\tilde{\bm{V}}_{E}.$

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