# A priori and a posteriori error analysis of the lowest-order NCVEM for
second-order linear indefinite elliptic problems

Carsten Carstensen, Rekha Khot and Amiya K. Pani22footnotemark: 2 Department
of Mathematics, Humboldt-Universität zu Berlin, 10099 Berlin, Germany. Email:
<EMAIL_ADDRESS>of Mathematics, Indian Institute of Technology
Bombay, Powai, Mumbai, 400076. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The nonconforming virtual element method (NCVEM) for the approximation of the
weak solution to a general linear second-order non-selfadjoint indefinite
elliptic PDE in a polygonal domain $\Omega$ is analyzed under reduced elliptic
regularity. The main tool in the a priori error analysis is the connection
between the nonconforming virtual element space and the Sobolev space
$H^{1}_{0}(\Omega)$ by a right-inverse $J$ of the interpolation operator
$I_{h}$. The stability of the discrete solution allows for the proof of
existence of a unique discrete solution, of a discrete inf-sup estimate and,
consequently, for optimal error estimates in the $H^{1}$ and $L^{2}$ norms.
The explicit residual-based a posteriori error estimate for the NCVEM is
reliable and efficient up to the oscillation terms. Numerical experiments on
different types of polygonal meshes illustrate the robustness of an error
estimator and support the improved convergence rate of an adaptive mesh-
refinement in comparison to the uniform mesh-refinement.

Keywords: second-order linear indefinite elliptic problems, virtual elements,
nonconforming,

polytopes, enrichment, stability, a priori error estimates, a residual-based a
posteriori error

estimate, adaptive mesh-refinement.

AMS subject classifications: 65N12, 65N15, 65N30, 65N50.

##  1 Introduction

The nonconforming virtual element method approximates the weak solution $u\in
H^{1}_{0}(\Omega)$ to the second-order linear elliptic boundary value problem

$\displaystyle{\cal L}u:=-\text{div}(\textbf{A}\nabla u+\textbf{b}u)+\gamma
u=f\quad\mbox{in}\quad\Omega$ (1.1)

for a given $f\in L^{2}(\Omega)$ in a bounded polygonal Lipschitz domain
$\Omega\subset{\mathbb{R}}^{2}$ subject to homogeneous Dirichlet boundary
conditions.

###  1.1 General introduction

The virtual element method (VEM) introduced in [4] is one of the well-received
polygonal methods for approximating the solutions to partial differential
equations (PDEs) in the continuation of the mimetic finite difference method
[7]. This method is becoming increasingly popular [1, 6, 5, 16, 17, 3] for its
ability to deal with fairly general polygonal/polyhedral meshes. On the
account of its versatility in shape of polygonal domains, the local finite-
dimensional space (the space of shape functions) comprises non-polynomial
functions. The novelty of this approach lies in the fact that it does not
demand for the explicit construction of non-polynomial functions and the
knowledge of degrees of freedom along with suitable projections onto
polynomials is sufficient to implement the method.

Recently, Beirão da Veiga et al. discuss a conforming VEM for the indefinite
problem (1.1) in [6]. Cangiani et al. [17] develop a nonconforming VEM under
the additional condition

$\displaystyle 0\leq\gamma-\frac{1}{2}\text{div}(\textbf{b}),$ (1.2)

which makes the bilinear form coercive and significantly simplifies the
analysis. The two papers [6, 17] prove a priori error estimates for a solution
$u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ in a convex domain $\Omega$. The a
priori error analysis for the nonconforming VEM in [17] can be extended to the
case when the exact solution $u\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$
with $\sigma>1/2$ as it is based on traces. This paper shows it for all
$\sigma>0$ and circumvents any trace inequality. Huang et al. [31] discuss a
priori error analysis of the nonconforming VEM applied to Poisson and
Biharmonic problems for $\sigma>0$. An a posteriori error estimate in [16]
explores the conforming VEM for (1.1) under the assumption (1.2). There are a
few contributions [16, 9, 34] on residual-based a posteriori error control for
the conforming VEM. This paper presents a priori and a posteriori error
estimates for the nonconforming VEM without (1.2), but under the assumption
that the Fredholm operator ${\cal L}$ is injective.

###  1.2 Assumptions on (1.1)

This paper solely imposes the following assumptions (A1)-(A3) on the
coefficients $\textbf{A},\textbf{b},\gamma$ and the operator ${\cal L}$ in
(1.1) with $f\in L^{2}(\Omega)$.

  1. (A1)

The coefficients $\textbf{A}_{jk},\textbf{b}_{j},\gamma$ for $j,k=1,2$ are
piecewise Lipschitz continuous functions. For any decomposition $\cal{T}$
(admissible in the sense of Subsection $2.1$) and any polygonal domain
$P\in\cal{T}$, the coefficients $\textbf{A},\textbf{b},\gamma$ are bounded
pointwise a.e. by
$\|\textbf{A}\|_{\infty},\|\textbf{b}\|_{\infty},\|\gamma\|_{\infty}$ and
their piecewise first derivatives by
$|\textbf{A}|_{1,\infty},|\textbf{b}|_{1,\infty},|\gamma|_{1,\infty}$.

  2. (A2)

There exist positive constants $a_{0}$ and $a_{1}$ such that, for a.e.
$x\in\Omega$, $\textbf{A}(x)$ is SPD and

$\displaystyle
a_{0}|\xi|^{2}\leq\sum_{j,k=1}^{2}\textbf{A}_{jk}(x)\xi_{j}\xi_{k}\leq
a_{1}|\xi|^{2}\quad\text{for all}\;\xi\in{\mathbb{R}}^{2}.$ (1.3)

  3. (A3)

The linear operator ${\cal L}:H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ is
injective, i.e., zero is not an eigenvalue of ${\cal L}$ .

Since the bounded linear operator ${\cal L}$ is a Fredholm operator [30, p.
321], (A3) implies that ${\cal L}$ is bijective with bounded inverse ${\cal
L}^{-1}:H^{-1}(\Omega)\to H^{1}_{0}(\Omega)$. The Fredholm theory also entails
the existence of a unique solution to the adjoint problem, that is, for every
$g\in L^{2}(\Omega)$, there exists a unique solution $\Phi\in
H^{1}_{0}(\Omega)$ to

$\displaystyle{\cal
L}^{*}\Phi:=-\text{div}(\textbf{A}\nabla\Phi)+\textbf{b}\cdot\nabla\Phi+\gamma\Phi=g.$
(1.4)

The bounded polygonal Lipschitz domain $\Omega$, the homogeneous Dirichlet
boundary conditions, and (A1)-(A2) lead to some $0<\sigma\leq 1$ and positive
constants $C_{\text{reg}}$ and $C^{*}_{\text{reg}}$ (depending only on
$\sigma,\Omega$ and coefficients of ${\cal L}$) such that, for any $f,g\in
L^{2}(\Omega)$, the unique solution $u$ to (1.1) and the unique solution
$\Phi$ to (1.4) belong to $H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ and
satisfy

$\displaystyle\|u\|_{1+\sigma,\Omega}\leq
C_{\text{reg}}\|f\|_{L^{2}(\Omega)}\;\text{
and}\;\;\|\Phi\|_{1+\sigma,\Omega}\leq
C^{*}_{\text{reg}}\|g\|_{L^{2}(\Omega)}.$ (1.5)

(The restriction $\sigma\leq 1$ is for convenience owing to the limitation to
first-order convergence of the scheme.)

###  1.3 Weak formulation

Given the coefficients $\textbf{A},\textbf{b},\gamma$ with (A1)-(A2), define,
for all $u,v\in V:=H^{1}_{0}(\Omega)$,

$a(u,v):=(\textbf{A}\nabla u,\nabla v)_{L^{2}(\Omega)},\hskip
14.22636ptb(u,v):=(u,\textbf{b}\cdot\nabla v)_{L^{2}(\Omega)},\hskip
14.22636ptc(u,v):=(\gamma u,v)_{L^{2}(\Omega)}$ (1.6)

and

$B(u,v):=a(u,v)+b(u,v)+c(u,v)$ (1.7)

(with piecewise versions $a_{\mathrm{pw}},b_{\mathrm{pw}},c_{\mathrm{pw}}$ and
$B_{\mathrm{pw}}$ for $\nabla$ replaced by the piecewise gradient
$\nabla_{\mathrm{pw}}$ and local contributions $a^{P},b^{P},c^{P}$ defined in
Subsection 3.1 throughout this paper). The weak formulation of the problem
(1.1) seeks $u\in V$ such that

$B(u,v)=(f,v)\quad\text{for all}\;v\in V.$ (1.8)

Assumptions (A1)-(A3) imply that the bilinear form $B(\cdot,\cdot)$ is
continuous and satisfies an inf-sup condition [11]

$\displaystyle 0<\beta_{0}:=\inf_{0\neq v\in V}\sup_{0\neq w\in
V}\frac{B(v,w)}{\|v\|_{1,\Omega}\|w\|_{1,\Omega}}.$ (1.9)

###  1.4 Main results and outline

Section $2$ introduces the VEM and guides the reader to the first-order
nonconforming VEM on polygonal meshes. It explains the continuity of the
interpolation operator and related error estimates in detail. Section $3$
starts with the discrete bilinear forms and their properties, followed by some
preliminary estimates for the consistency error and the nonconformity error.
The nonconformity error uses a new conforming companion operator resulting in
the well-posedness of the discrete problem for sufficiently fine meshes.
Section $4$ proves the discrete inf-sup estimate and optimal a priori error
estimates. Section $5$ discusses both reliability and efficiency of an
explicit residual-based a posteriori error estimator. Numerical experiments in
Section $6$ for three computational benchmarks illustrate the performance of
an error estimator and show the improved convergence rate in adaptive mesh-
refinement.

###  1.5 Notation

Throughout this paper, standard notation applies to Lebesgue and Sobolev
spaces $H^{m}$ with norm $\|\cdot\|_{m,\cal{D}}$ (resp. seminorm
$|\cdot|_{m,\cal{D}}$) for $m>0$, while $(\cdot,\cdot)_{L^{2}({\cal D})}$ and
$\|\cdot\|_{L^{2}({\cal D})}$ denote the $L^{2}$ scalar product and $L^{2}$
norm on a domain ${\cal D}$. The space $C^{0}(\cal D)$ consists of all
continuous functions vanishing on the boundary of a domain ${\cal D}$. The
dual space of $H^{1}_{0}(\Omega)$ is denoted by $H^{-1}(\Omega)$ with dual
norm $\|\cdot\|_{-1}$. An inequality $A\lesssim B$ abbreviates $A\leq CB$ for
a generic constant $C$, that may depend on the coefficients of ${\cal L}$, the
universal constants $\sigma$, $\rho$ (from (M2) below), but that is
independent of the mesh-size. Let $\mathcal{P}_{k}({\cal D})$ denote the set
of polynomials of degree at most $k\in\mathbb{N}_{0}$ defined on a domain
${\cal D}$ and let $\Pi_{k}$ denote the piecewise $L^{2}$ projection on
$\mathcal{P}_{k}({\cal T})$ for any admissible partition
$\mathcal{T}\in\mathbb{T}$ (hidden in the notation $\Pi_{k}$). The notation
$H^{s}(P):=H^{s}(\text{int}P)$ for a compact polygonal domain $P$ means the
Sobolev space $H^{s}$ [30] defined in the interior $\text{int}(P)$ of $P$
throughout this paper. The outward normal derivative is denoted by
$\frac{\partial\;\bullet}{\partial\textbf{n}_{P}}=\textbf{n}_{P}\cdot\nabla\bullet$
for the exterior unit normal vector $\textbf{n}_{P}$ along the boundary
$\partial P$ of the domain $P$.

##  2 First-order virtual element method on a polygonal mesh

This section describes class of admissible partitions of $\Omega$ into
polygonal domains and the lowest-order nonconforming virtual element method
for the problem (1.1) [17, 3].

###  2.1 Polygonal meshes

A polygonal domain $P$ in this paper is a non-void compact simply-connected
set $P$ with polygonal boundary $\partial P$ so that $\text{int}(P)$ is a
Lipschitz domain. The polygonal boundary $\partial P$ is a simple closed
polygon described by a finite sequence of distinct points. The set ${\cal
N}(\partial P)=\\{z_{1},z_{2},\dots,z_{J}\\}$ of nodes of a polygon $P$ is
enumerated with $z_{J+1}:=z_{1}$ such that
$E(j):=\text{conv}\\{z_{j},z_{j+1}\\}$ defines an edge and all $J$ edges cover
the boundary $\partial P=E(1)\cup\dots\cup E(J)$ with an intersection
$E(j)\cap E(j+1)=\\{z_{j+1}\\}$ for $j=1,\dots,J-1$ and $E(J)\cap
E(1)={z_{1}}$ with $\text{dist}(E(j),E(k))>0$ for all distinct indices $j\neq
k$.

Let $\mathbb{T}$ be a family of partitions of $\overline{\Omega}$ into
polygonal domains, which satisfies the conditions (M1)-(M2) with a universal
positive constant $\rho$.

  1. (M1)

Admissibility. Any two distinct polygonal domains $P$ and $P^{\prime}$ in
$\mathcal{T}\in\mathbb{T}$ are disjoint or share a finite number of edges or
vertices.

Figure 2.1:

  2. (M2)

Mesh regularity. Every polygonal domain $P$ of diameter $h_{P}$ is star-shaped
with respect to every point of a ball of radius greater than equal to $\rho
h_{P}$ and every edge $E$ of $P$ has a length $|E|$ greater than equal to
$\rho h_{P}$.

Here and throughout this paper, $h_{\mathcal{T}}|_{P}:=h_{P}$ denotes the
piecewise constant mesh-size and
$\mathbb{T}(\delta):=\\{\mathcal{T}\in\mathbb{T}:h_{\text{max}}\leq\delta\leq
1\\}$ with the maximum diameter $h_{\text{max}}$ of the polygonal domains in
$\mathcal{T}$ denotes the subclass of partitions of $\overline{\Omega}$ into
polygonal domains of maximal mesh-size $\leq\delta$. Let $|P|$ denote the area
of polygonal domain $P$ and $|E|$ denote the length of an edge $E$. With a
fixed orientation to a polygonal domain $P$, assign the outer unit normal
$\textbf{n}_{P}$ along the boundary $\partial P$ and
$\textbf{n}_{E}:=\textbf{n}_{P}|_{E}$ for an edge $E$ of $P$. Let
$\mathcal{E}$ (resp. $\widehat{\mathcal{E}}$) denote the set of edges $E$ of
$\mathcal{T}$ (resp. of $\widehat{{\cal T}}$) and $\mathcal{E}(P)$ denote the
set of edges of polygonal domain $P\in\mathcal{T}$. For a polygonal domain
$P$, define

$\displaystyle\text{mid}(P):=\frac{1}{|P|}\int_{P}x\,dx\quad\text{and}\quad\text{mid}(\partial
P):=\frac{1}{|\partial P|}\int_{\partial P}x\,ds.$

Let $\mathcal{P}_{k}({\cal T}):=\\{v\in L^{2}(\Omega):\forall
P\in\mathcal{T}\quad v|_{P}\in\mathcal{P}_{k}(P)\\}$ for $k\in\mathbb{N}_{0}$
and $\Pi_{k}$ denote the piecewise $L^{2}$ projection onto
$\mathcal{P}_{k}({\cal T})$. The notation $\Pi_{k}$ hides its dependence on
$\mathcal{T}$ and also assume $\Pi_{k}$ applies componentwise to vectors.
Given a decomposition ${\cal T}\in\mathbb{T}$ of $\Omega$ and a function $f\in
L^{2}(\Omega)$, its oscillation reads

$\displaystyle\mathrm{osc}_{k}(f,P):=\|h_{P}(1-\Pi_{k})f\|_{L^{2}(P)}\quad\text{and}\quad\mathrm{osc}_{k}(f,{\cal
T}):=\left(\sum_{P\in{\cal
T}}\|h_{P}(1-\Pi_{k})f\|_{L^{2}(P)}^{2}\right)^{\displaystyle\nicefrac{{1}}{{2}}}$

with $\mathrm{osc}(f,\bullet):=\mathrm{osc}_{0}(f,\bullet)$.

######  Remark 1 (consequence of mesh regularity assumption).

There exists an interior node $c$ in the sub-triangulation $\widehat{{\cal
T}}(P):=\\{T(E)=\text{conv}(c,E):E\in\mathcal{E}(P)\\}$ of a polygonal domain
$P$ with $h_{T(E)}\leq h_{P}\leq C_{\text{sr}}h_{T(E)}$ as illustrated in
Figure 2.2. Each polygonal domain $P$ can be divided into triangles so that
the resulting sub-triangulation $\widehat{{\cal T}}|_{P}:=\widehat{{\cal
T}}(P)$ of $\mathcal{T}$ is shape-regular. The minimum angle in the sub-
triangulation solely depends on $\rho$ [13, Sec. 2.1].

(a)

(b)

Figure 2.2: (a) Polygon $P$ and (b) its sub-triangulation $\widehat{{\cal
T}}(P)$.

######  Lemma 2.1 (Poincaré-Friedrichs inequality).

There exists a positive constant $C_{\mathrm{PF}}$, that depends solely on
$\rho$, such that

$\displaystyle\|f\|_{L^{2}(P)}\leq C_{\mathrm{PF}}h_{P}|f|_{1,P}$ (2.1)

holds for any $f\in H^{1}(P)$ with $\sum_{j\in J}\int_{E(j)}f\,ds=0$ for a
nonempty subset $J\subseteq\\{1,\dots,m\\}$ of indices in the notation
$\partial P=E(1)\cup\dots\cup E(m)$ of Figure 2.2. The constant
$C_{\mathrm{PF}}$ depends exclusively on the number $m:=|\mathcal{E}(P)|$ of
the edges in the polygonal domain $P$ and the quotient of the maximal area
divided by the minimal area of a triangle in the triangulation $\widehat{{\cal
T}}(P)$.

Some comments on $C_{\mathrm{PF}}$ for anisotropic meshes are in order before
the proof gives an explicit expression for $C_{\mathrm{PF}}$.

######  Example 2.1.

Consider a rectangle $P$ with a large aspect ratio divided into four congruent
sub-triangles all with vertex $c=\text{mid}(P)$. Then, $m=4$ and the quotient
of the maximal area divided by the minimal area of a triangle in the criss-
cross triangulation $\widehat{{\cal T}}(P)$ is one. Hence $C_{\mathrm{PF}}\leq
1.4231$ (from the proof below) is independent of the aspect ratio of $P$.

###### Proof of Lemma 2.1.

The case $J=\\{1,\dots,m\\}$ with $f\in H^{1}(P)$ and $\int_{\partial
P}f\,ds=0$ is well-known cf. e.g. [13, Sec. 2.1.5], and follows from the
Bramble-Hilbert lemma [14, Lemma 4.3.8] and the trace inequality [13, Sec.
2.1.1]. The remaining part of the proof shows the inequality (2.1) for the
case $J\subseteq\\{1,\dots,m\\}$. The polygonal domain $P$ and its
triangulation $\widehat{{\cal T}}(P)$ from Figure 2.2 has the center $c$ and
the nodes $z_{1},\dots,z_{m}$ for the $m:=|\mathcal{E}(P)|=|\widehat{{\cal
T}}(P)|$ edges $E(1),\dots,E(m)$ and the triangles $T(1),\dots,T(m)$ with
$T(j)=T(E(j))=\text{conv}\\{c,E(j)\\}=\text{conv}\\{c,z_{j},z_{j+1}\\}$ for
$j=1,\dots,m$. Here and throughout this proof, all indices are understood
modulo $m$, e.g., $z_{0}=z_{m}$. The proof uses the trace identity

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(x-c)\cdot\nabla f(x)\,dx$ (2.2)

for $f\in H^{1}(P)$ as in the lemma. This follows from an integration by parts
and the observation that $(x-c)\cdot\textbf{n}_{F}=0$ on
$F\in\mathcal{E}(T(j))\backslash E(j)$ and the height
$(x-c)\cdot\textbf{n}_{E(j)}=\frac{2|T(j)|}{|E(j)}$ of the edge $E(j)$ in the
triangle $T(j)$, for $x\in E(j)$; cf. [24, Lemma 2.1] or [25, Lemma 2.6] for
the remaining details. Another version of the trace identity (2.2) concerns
$\text{conv}\\{z_{j},c\\}=:F(j)=\partial T(j-1)\cap\partial T(j)$ and reads

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{F(j)}f\,ds$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j-1)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j-1)}(x-z_{j-1})\cdot\nabla f(x)\,dx$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j+1})\cdot\nabla f(x)\,dx$ (2.3)

in $T(j-1)$ and $T(j)$. The three trace identities in (2.2)-(2.3) are
rewritten with the following abbreviations, for $j=1,\dots m$,

$\displaystyle x_{j}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds,\quad
f_{j}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx,\quad
a_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(x-c)\cdot\nabla f(x)\,dx,$ $\displaystyle
b_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j})\cdot\nabla f(x)\,dx,\quad
c_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j+1})\cdot\nabla f(x)\,dx.$

Let $t_{\text{min}}=\min_{T\in\widehat{{\cal T}}(P)}|T|$ and
$t_{\text{max}}=\max_{T\in\widehat{{\cal T}}(P)}|T|$ abbreviate the minimal
and maximal area of a triangle in $\widehat{{\cal T}}(P)$ and let
$\widehat{\Pi}_{0}f\in\mathcal{P}_{0}(\widehat{{\cal T}}(P))$ denote the
piecewise integral means of $f$ with respect to the triangulation
$\widehat{{\cal T}}(P)$. The Poincaré inequality in a triangle with the
constant $C_{\text{P}}:=1/j_{1,1}$ and the first positive root $j_{1,1}\approx
3.8317$ of the Bessel function $J_{1}$ from [24, Thm. 2.1] allows for

$\displaystyle\|f-\widehat{\Pi}_{0}f\|_{L^{2}(T(j))}\leq
C_{\text{P}}h_{T(j)}|f|_{1,T(j)}\quad\text{for}\;j=1,\dots,m.$

Hence $\|f-\widehat{\Pi}_{0}f\|_{L^{2}(P)}\leq C_{\text{P}}h_{P}|f|_{1,P}$.
This and the Pythagoras theorem (with
$f-\widehat{\Pi}_{0}f\perp\mathcal{P}_{0}(\widehat{{\cal T}}(P))$ in
$L^{2}(P)$) show

$\displaystyle\|f\|^{2}_{L^{2}(P)}=\|\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P))}+\|f-\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P))}\leq\|\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P))}+C_{\text{P}}^{2}h_{P}^{2}|f|^{2}_{1,P}.$
(2.4)

It remains to bound the term $\|\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P))}$. The
assumption on $f$ reads $\sum_{j\in J}\int_{E(j)}f\,ds=\sum_{j\in
J}|E(j)|x_{j}=0$ for a subset $J\subset\\{1,\dots,m\\}$ so that
$0\in\text{conv}\\{|E(1)|x_{1},\dots,|E(m)|x_{m}\\}$. It follows
$0\in\text{conv}\\{x_{1},\dots,x_{m}\\}$ and it is known that this implies

$\displaystyle\sum_{k=1}^{m}x_{k}^{2}\leq{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}$
(2.5)

for a constant ${\cal{M}}=\frac{1}{2(1-\cos(\pi/m))}$ that depends exclusively
on $m$ [25, Lemma 4.2]. Recall (2.2) in the form $x_{j}=f_{j}+a_{j}$ to deduce
from a triangle inequality and (2.5) that

$\displaystyle\frac{1}{2}\sum_{j=1}^{m}f_{j}^{2}\leq\sum_{k=1}^{m}x_{k}^{2}+\sum_{\ell=1}^{m}a_{\ell}^{2}\leq{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}+\sum_{\ell=1}^{m}a_{\ell}^{2}.$

This shows that

$\displaystyle
t_{\text{max}}^{-1}\|\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P)}=t_{\text{max}}^{-1}\sum_{j=1}^{m}|T(j)|f_{j}^{2}\leq\sum_{j=1}^{m}f_{j}^{2}\leq
2{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}+2\sum_{\ell=1}^{m}a_{\ell}^{2}.$

Recall (2.2)-(2.3) in the form $f_{j}-f_{j-1}=b_{j-1}-c_{j}$ and
$x_{j}-x_{j-1}=f_{j}-f_{j-1}+a_{j}-a_{j-1}=b_{j-1}-a_{j-1}+a_{j}-c_{j}$ for
all $j=1,\dots,m$. This and the Cauchy-Schwarz inequality imply the first two
estimates in

$\displaystyle 2|x_{j}-x_{j-1}|$
$\displaystyle=\bigg{|}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j-1)}(c-z_{j-1})\cdot\nabla
f(x)\,dx+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T(j)}(z_{j+1}-c)\cdot\nabla f(x)\,dx\bigg{|}$
$\displaystyle\leq\max\\{|c-z_{j-1}|,|c-z_{j+1}|\\}\Big{(}|T(j-1)|^{-1/2}|f|_{1,T(j-1)}+|T(j)|^{-1/2}|f|_{1,T(j)}\Big{)}$
$\displaystyle\leq h_{P}t_{\text{min}}^{-1/2}|f|_{1,T(j-1)\cup T(j)}$

with the definition of $h_{P}$ and $t_{\text{min}}$ in the end. The inequality
$\int_{T(j)}|x-c|^{2}\;dx\leq\frac{1}{2}h^{2}_{T(j)}|T(j)|$ [25, Lemma 2.7]
and the Cauchy-Schwarz inequality show, for $j=1,\dots,m$, that

$\displaystyle|a_{j}|\leq 2^{-3/2}h_{T(j)}|T(j)|^{-1/2}|f|_{1,|T(j)|}\leq
2^{-3/2}h_{P}t_{\text{min}}^{-1/2}|f|_{1,|T(j)|}.$

The combination of the previous three displayed estimates result in

$\displaystyle
4h_{P}^{-2}(t_{\text{min}}/t_{\text{max}})\|\widehat{\Pi}_{0}f\|^{2}_{L^{2}(P)}\leq
2{\cal{M}}\sum_{k=1}^{m}|f|^{2}_{T(k-1)\cup
T(k)}+\sum_{\ell=1}^{m}|f|^{2}_{1,T(\ell)}=(4{\cal{M}}+1)|f|^{2}_{1,P}.$

This and (2.4) conclude the proof with the constant
$C_{\mathrm{PF}}^{2}=({\cal{M}}+1/4)(t_{\text{max}}/t_{\text{min}})+C_{\text{P}}^{2}$.
∎

In the nonconforming VEM, the finite-dimensional space $V_{h}$ is a subset of
the piecewise Sobolev space

$H^{1}(\mathcal{T}):=\\{v\in L^{2}(\Omega):\forall P\in\mathcal{T}\quad
v|_{P}\in H^{1}(P)\\}\equiv\prod_{P\in\mathcal{T}}H^{1}(P).$

The piecewise $H^{1}$ seminorm (piecewise with respect to $\mathcal{T}$ hidden
in the notation for brevity) reads

$|v_{h}|_{1,\text{pw}}:=\bigg{(}\sum_{P\in\mathcal{T}}|v_{h}|_{1,P}^{2}\bigg{)}^{1/2}\quad\text{for
any}\;v_{h}\in H^{1}(\mathcal{T}).$

###  2.2 Local virtual element space

The first nonconforming virtual element space [3] is a subspace of harmonic
functions with edgewise constant Neumann boundary values on each polygon. The
extended nonconforming virtual element space [1, 17] reads

$\displaystyle\widehat{V}_{h}(P):=\begin{rcases}\begin{dcases}v_{h}\in
H^{1}(P):&\Delta v_{h}\in\mathcal{P}_{1}(P)\quad\text{and}\quad\forall
E\in\mathcal{E}(P)\quad{\frac{\partial
v_{h}}{\partial\textbf{n}_{P}}}\Big{|}_{E}\in\mathcal{P}_{0}(E)\end{dcases}\end{rcases}.$
(2.6)

######  Definition 2.2 (Ritz projection).

Let $\Pi^{\nabla}_{1}$ be the Ritz projection from $H^{1}(P)$ onto the affine
functions $\mathcal{P}_{1}(P)$ in the $H^{1}$ seminorm defined, for $v_{h}\in
H^{1}(P)$, by

$\displaystyle(\nabla\Pi^{\nabla}_{1}v_{h}-\nabla
v_{h},\nabla\chi)_{L^{2}(P)}=0\quad\text{for
all}\;\chi\in\mathcal{P}_{1}(P)\quad\text{and}\quad\int_{\partial
P}\Pi^{\nabla}_{1}v_{h}\,ds=\int_{\partial P}v_{h}\,ds.$ (2.7)

######  Remark 2 (integral mean).

For $P\in\mathcal{T}$ and $f\in H^{1}(P)$,
$\nabla\Pi^{\nabla}_{1}f=\Pi_{0}\nabla f$. (This follows from (2.7.a) and the
definition of the $L^{2}$ projection operator $\Pi_{0}$ (acting componentwise)
onto the piecewise constants $\mathcal{P}_{0}(P;\mathbb{R}^{2})$.)

######  Remark 3 (representation of $\Pi^{\nabla}_{1}$).

For $P\in\mathcal{T}$ and $f\in H^{1}(P)$, the Ritz projection
$\Pi^{\nabla}_{1}f$ reads

$\displaystyle(\Pi^{\nabla}_{1}f)(x)=\frac{1}{|P|}\Big{(}\int_{\partial
P}f\textbf{n}_{P}\,ds\Big{)}\cdot\Big{(}x-\text{mid}(\partial
P)\Big{)}+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\partial P}f\,ds\quad\text{for}\;x\in P.$ (2.8)

(The proof of (2.8) consists in the verification of (2.7): The equation
(2.7.a) follows from Remark 2 with an integration by parts. The equation
(2.7.b) follows from the definition of $\text{mid}(\partial P)$ as the
barycenter of $\partial P$. ∎)

The enhanced virtual element spaces [1, 17] are designed with a computable
$L^{2}$ projection $\Pi_{1}$ onto $\mathcal{P}_{1}(\mathcal{T})$. The
resulting local discrete space under consideration throughout this paper reads

$\displaystyle
V_{h}(P):=\begin{rcases}\begin{dcases}v_{h}\in\widehat{V}_{h}(P):v_{h}-\Pi^{\nabla}_{1}v_{h}\perp\mathcal{P}_{1}(P)\quad\text{in}\;L^{2}(P)\end{dcases}\end{rcases}.$
(2.9)

The point in the selection of $V_{h}(P)$ is that the Ritz projection
$\Pi^{\nabla}_{1}v_{h}$ coincides with the $L^{2}$ projection $\Pi_{1}v_{h}$
for all $v_{h}\in V_{h}(P)$. The degrees of freedom on $P$ are given by

$\displaystyle\text{dof}_{E}(v)=\frac{1}{|E|}\int_{E}v\,ds\quad\textrm{for
all}\;E\in\mathcal{E}(P)\;\text{and}\;v\in V_{h}(P).$ (2.10)

######  Proposition 2.3.

$(a)$ The vector space $\widehat{V}_{h}(P)$ from (2.6) is of dimension
$3+|\mathcal{E}(P)|$. $(b)$ $V_{h}(P)$ from (2.9) is of dimension
$|\mathcal{E}(P)|$ and the triplet
$(P,V_{h}(P),\text{dof}_{E}:E\in\mathcal{E}(P))$ is a finite element in the
sense of Ciarlet [28].

###### Proof.

Let $E(1),\dots,E(m)$ be an enumeration of the edges $\mathcal{E}(P)$ of the
polygonal domain $P$ in a consecutive way as depicted in Figure 2.2.a and
define
$W(P):=\mathcal{P}_{1}(P)\times\mathcal{P}_{0}(E{(1)})\times\dots\times\mathcal{P}_{0}(E{(m)})$.
Recall $\widehat{V}_{h}(P)$ from (2.6) and identify the quotient space
$\widehat{V}_{h}(P)/\mathbb{R}\equiv\left\\{f\in\widehat{V}_{h}(P):\right.\\\
\left.\int_{\partial P}f\,ds=0\right\\}$ with all functions in
$\widehat{V}_{h}(P)$ having zero integral over the boundary $\partial P$ of
$P$. Since the space $\widehat{V}_{h}(P)$ consists of functions with an affine
Laplacian and edgewise constant Neumann data, the map

$\displaystyle S:\widehat{V}_{h}(P)/\mathbb{R}\to W(P),\quad\quad
f\mapsto\left(-\Delta f,\frac{\partial
f}{\partial\textbf{n}_{P}}\Big{|}_{E{(1)}},\dots,\frac{\partial
f}{\partial\textbf{n}_{P}}\Big{|}_{E{(m)}}\right)$

is well-defined and linear. The compatibility conditions for the existence of
a solution of a Laplacian problem with Neumann data show that the image of $S$
is equal to

$\displaystyle\mathcal{R}(S)=\left\\{(f_{1},g_{1},\dots,g_{m})\in
W(P):\int_{P}f_{1}dx+\sum_{j=1}^{m}g_{j}|E(j)|=0\right\\}.$

(The proof of this identity assumes the compatible data
$(f_{1},g_{1},\dots,g_{m})$ from the set on the right-hand side and solves the
Neumann problem with a unique solution $\widehat{u}$ in
$\widehat{V}_{h}(P)/\mathbb{R}$ and $S\widehat{u}=(f_{1},g_{1},\dots,g_{m})$.)
It is known that the Neumann problem has a unique solution up to an additive
constant and so $S$ is a bijection and the dimension $m+2$ of
$\widehat{V}_{h}(P)/\mathbb{R}$ is that of $\mathcal{R}(S)$. In particular,
dimension of $\widehat{V}_{h}(P)$ is $m+3$. This proves $(a)$.
Let $\Lambda_{0},\Lambda_{1},\Lambda_{2}:H^{1}(P)\to\mathbb{R}$ be linear
functionals

$\displaystyle\Lambda_{0}f:=\Pi_{0}f,\quad\Lambda_{j}f:={\cal
M}_{j}((\Pi^{\nabla}_{1}-\Pi_{1})f)$

with ${\cal M}_{j}f:=\Pi_{0}((x_{j}-c_{j})f)$ for $j=1,2$ and $f\in H^{1}(P)$
that determines an affine function $p_{1}\in\mathcal{P}_{1}(P)$ such that
$(P,\mathcal{P}_{1}(P),(\Lambda_{0},\Lambda_{1},\Lambda_{2}))$ is a finite
element in the sense of Ciarlet. For any edge $E(j)\in\mathcal{E}(P)$, define
$\Lambda_{j+2}f=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds$ as integral mean of the traces of $f$
in $H^{1}(P)$ on $E(j)$. It is elementary to see that
$\Lambda_{0},\dots,\Lambda_{m+2}$ are linearly independent: If $f$ in
$\widehat{V}_{h}(P)$ belongs to the kernel of all the linear functionals, then
$\Pi^{\nabla}_{1}f=0$ from (2.8) with $\Lambda_{j}f=0$ for each
$j=3,\dots,2+m$. Since the functionals $\Lambda_{j}f=0$ for $j=1,2$,
$(x_{j}-c_{j})(\Pi^{\nabla}_{1}-\Pi_{1})f=0$ and $\Pi^{\nabla}_{1}f=0$ imply
$\Pi_{1}f=0$. An integration by parts leads to

$\displaystyle\|\nabla f\|_{L^{2}(P)}^{2}=(-\Delta
f,f)_{L^{2}(P)}+\Big{(}f,\frac{\partial
f}{\partial\textbf{n}_{P}}\Big{)}_{L^{2}(\partial P)}=0.$

This and $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\partial P}f\,ds=0$ show $f\equiv 0$.
Consequently, the intersection $\cap_{j=0}^{m+2}\text{Ker}(\Lambda_{j})$ of
all kernels Ker$(\Lambda_{0}),\dots,\text{Ker}(\Lambda)_{m+2}$ is trivial and
so that the functionals $\Lambda_{0},\dots,\Lambda_{m+2}$ are linearly
independent. Since the number of the linear functionals is equal to the
dimension of $\widehat{V}_{h}(P)$,
$(P,\widehat{V}_{h}(P),\\{\Lambda_{0},\dots,\Lambda_{m+2}\\})$ is a finite
element in the sense of Ciarlet and there exists a nodal basis
$\psi_{0},\dots,\psi_{m+2}$ of $\widehat{V}_{h}(P)$ with

$\displaystyle\Lambda_{j}(\psi_{k})=\delta_{jk}\quad\text{for
all}\;j,k=0,\dots,m+2.$

The linearly independent functions $\psi_{3},\dots,\psi_{m+2}$ belong to
$V_{h}(P)$ and so dim$(V_{h}(P))\geq m$. Since
$V_{h}(P)\subset\widehat{V}_{h}(P)$ and three linearly independent conditions
$(1-\Pi^{\nabla}_{1})v_{h}\perp\mathcal{P}_{1}(P)$ in $L^{2}(P)$ are imposed
on $\widehat{V}_{h}(P)$ to define $V_{h}(P)$, dim$(V_{h}(P))\leq m$. This
shows that dim$(V_{h}(P))=m$ and hence, the linear functionals
$\text{dof}_{E}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}\bullet\,ds$ for $E\in\mathcal{E}(P)$ form a
dual basis of $V_{h}(P)$. This concludes the proof of $(b)$. ∎

######  Remark 4 (stability of $L^{2}$ projection).

The $L^{2}$ projection $\Pi_{k}$ for $k=0,1$ is $H^{1}$ and $L^{2}$ stable in
$V_{h}(P)$, in the sense that any $v_{h}$ in $V_{h}(P)$ satisfies

$\displaystyle\|\Pi_{k}v_{h}\|_{L^{2}(P)}\leq\|v_{h}\|_{L^{2}(P)}\;\text{and}\;\|\nabla(\Pi_{k}v_{h})\|_{L^{2}(P)}\leq\|\nabla
v_{h}\|_{L^{2}(P)}.$ (2.11)

(The first inequality follows from the definition of $\Pi_{k}$. The
orthogonality in (2.9) and the definition of $\Pi_{1}$ imply that the Ritz
projection $\Pi^{\nabla}_{1}$ and the $L^{2}$ projection $\Pi_{1}$ coincide on
the space $V_{h}(P)$ for $P\in\mathcal{T}$. This with the definition of the
Ritz projection $\Pi^{\nabla}_{1}$ verifies the second inequality. ∎)

######  Definition 2.4 (Fractional order Sobolev space [14]).

Let $\alpha:=(\alpha_{1},\alpha_{2})$ denote a multi-index with
$\alpha_{j}\in\mathbb{N}_{0}$ for $j=1,2$ and
$|\alpha|:=\alpha_{1}+\alpha_{2}.$ For a real number $m$ with $0<m<1$, define

$\displaystyle H^{1+m}(\omega):=\left\\{v\in
H^{1}(\omega):\frac{|v^{\alpha}(x)-v^{\alpha}(y)|}{|x-y|^{(1+m)}}\in
L^{2}(\omega\times\omega)\quad\text{for all}\;|\alpha|=1\right\\}$

with $v^{\alpha}$ as the partial derivative of $v$ of order $\alpha$. Define
the seminorm $|\cdot|_{1+m}$ and Sobolev-Slobodeckij norm $\|\cdot\|_{1+m}$ by

$\displaystyle|v|_{1+m,\omega}^{2}=\sum_{|\alpha|=1}\int_{\omega}\int_{\omega}\frac{{|v^{\alpha}(x)-v^{\alpha}(y)|}^{2}}{|x-y|^{2(1+m)}}\,dx\,dy\quad\text{and}\quad\|v\|_{1+m,\omega}^{2}=\|v\|^{2}_{1,\omega}+|v|_{1+m,\omega}^{2}.$

######  Proposition 2.5 (approximation by polynomials [29, Thm. 6.1]).

Under the assumption (M2), there exists a positive constant $C_{\mathrm{apx}}$
(depending on $\rho$ and on the polynomial degree $k$) such that, for every
$v\in H^{m}(P)$, the $L^{2}$ projection $\Pi_{k}(P)$ on $\mathcal{P}_{k}$ for
$k\in\mathbb{N}_{0}$ satisfies

$\displaystyle\|v-\Pi_{k}v\|_{L^{2}(P)}+h_{P}|v-\Pi_{k}v|_{1,P}\leq
C_{\mathrm{apx}}h_{P}^{m}|v|_{m,P}\quad\text{for}\;1\leq m\leq k+1.$ (2.12)

###  2.3 Global virtual element space

Define the global nonconforming virtual element space, for any
$\mathcal{T}\in\mathbb{T}$, by

$\displaystyle V_{h}:=\left\\{v_{h}\in H^{1}(\mathcal{T}):\forall
P\in\mathcal{T}\quad v_{h}|_{P}\in V_{h}(P)\quad\text{and}\quad\forall
E\in\mathcal{E}\quad\int_{E}[v_{h}]_{E}\,ds=0\right\\}.$ (2.13)

Let $[\cdot]_{E}$ denote the jump across an edge $E\in\mathcal{E}$: For two
neighboring polygonal domains $P^{+}$ and $P^{-}$ sharing a common edge
$E\in\mathcal{E}(P^{+})\cap\mathcal{E}(P^{-})$,
$[v_{h}]_{E}:=v_{h|P^{+}}-v_{h|P^{-}}$, where $P^{+}$ denote the adjoint
polygonal domain with $\textbf{n}_{P^{+}|E}=\textbf{n}_{E}$ and $P^{-}$ denote
the polygonal domain with $\textbf{n}_{P^{-}|E}=-\textbf{n}_{E}$. If
$E\subset\partial\Omega$ is a boundary edge, then $[v_{h}]_{E}:=v_{h}|_{E}$.

######  Example 2.2.

If each polygonal domain $P$ is a triangle, then the finite-dimensional space
$V_{h}$ coincides with CR-FEM space. (Since the dimension of the vector space
$V_{h}(P)$ is three and $\mathcal{P}_{1}(P)\subset V_{h}(P)$,
$V_{h}(P)=\mathcal{P}_{1}(P)$ for $P\in\mathcal{T}$.)

######  Lemma 2.6.

There exists a universal constant $C_{\mathrm{F}}$ (that depends only on
$\rho$ from (M2)) such that, for all ${\cal T}\in\mathbb{T}$, any $v_{h}\in
V_{h}$ from (2.13) satisfies

$\displaystyle\|v_{h}\|_{L^{2}(\Omega)}\leq
C_{\mathrm{F}}|v_{h}|_{1,\mathrm{pw}}.$ (2.14)

###### Proof.

Recall from Remark 1 that $\widehat{{\cal T}}$ is a shape regular sub-
triangulation of $\mathcal{T}$ into triangles. Since $V_{h}\subset
H^{1}(\widehat{{\cal T}})$ and the Friedrichs’ inequality holds for all
functions in $H^{1}(\widehat{{\cal T}})$ [14, Thm. 10.6.16], there exists a
positive constant $C_{\text{F}}$ such that the (first) inequality holds in

$\displaystyle\|v_{h}\|_{L^{2}(\Omega)}\leq
C_{\text{F}}\left(\sum_{T\in\widehat{{\cal T}}}\|\nabla
v_{h}\|_{L^{2}(T)}^{2}\right)^{1/2}=C_{\text{F}}|v_{h}|_{1,\mathrm{pw}}.$

The (second) equality follows for $v_{h}\in H^{1}(P)$ with $P\in\mathcal{T}$.
∎

Lemma 2.6 implies that the seminorm $|\cdot|_{1,\mathrm{pw}}$ is equivalent to
the norm
$\|\cdot\|_{1,\mathrm{pw}}:=\|\cdot\|^{2}_{L^{2}(\Omega)}+|\cdot|^{2}_{1,\mathrm{pw}}$
in $V_{h}$ with mesh-size independent equivalence constants.

###  2.4 Interpolation

######  Definition 2.7 (interpolation operator).

Let $(\psi_{E}:E\in\mathcal{E})$ be the nodal basis of $V_{h}$ defined by
$\text{dof}_{E}(\psi_{E})=1$ and $\text{dof}_{F}(\psi_{E})=0$ for all other
edges $F\in\mathcal{E}\setminus\\{E\\}$. The global interpolation operator
$I_{h}:H^{1}_{0}(\Omega)\to V_{h}$ reads

$\displaystyle
I_{h}v:=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v\,ds\Big{)}\psi_{E}\quad\text{for}\;v\in V.$

Since a Sobolev function $v\in V$ has traces and the jumps $[v]_{E}$ vanish
across any edge $E\in\mathcal{E}$, the interpolation operator $I_{h}$ is well-
defined. Recall $\rho$ from (M2), $C_{\mathrm{PF}}$ from Lemma 2.1, and
$C_{\text{apx}}$ from Proposition 2.5.

######  Theorem 2.8 (interpolation error).

  1. $\left(a\right)$

There exists a positive constant $C_{\mathrm{Itn}}$ (depending on $\rho$) such
that any $v\in H^{1}(P)$ and its interpolation $I_{h}v\in V_{h}(P)$ satisfy

$\displaystyle\|\nabla I_{h}v\|_{L^{2}(P)}\leq C_{\mathrm{Itn}}\|\nabla
v\|_{L^{2}(P)}.$

  2. $\left(b\right)$

Any $P\in\mathcal{T}\in\mathbb{T}$ and $v\in H^{1}(P)$ satisfy
$|v-I_{h}v|_{1,P}\leq(1+C_{\mathrm{Itn}})\|(1-\Pi_{0})\nabla v\|_{L^{2}(P)}$
and

$\displaystyle
h_{P}^{-1}\|(1-\Pi_{1}I_{h})v\|_{L^{2}(P)}+|(1-\Pi_{1}I_{h})v|_{1,P}\leq(1+C_{\mathrm{PF}})\|(1-\Pi_{0})\nabla
v\|_{L^{2}(P)}.$

  3. $\left(c\right)$

The positive constant
$C_{\mathrm{I}}:=C_{\mathrm{apx}}(1+C_{\mathrm{Itn}})(1+C_{\mathrm{PF}})$, any
$0<\sigma\leq 1$, and any $v\in H^{1+\sigma}(P)$ with the local interpolation
$I_{h}v|_{P}\in V_{h}(P)$ satisfy

$\displaystyle\|v-I_{h}v\|_{L^{2}(P)}+h_{P}|v-I_{h}v|_{1,P}\leq
C_{\mathrm{I}}h^{1+\sigma}_{P}|v|_{1+\sigma,P}.$ (2.15)

###### Proof of $(a)$.

The boundedness of the interpolation operator in $V_{h}(P)$ is mentioned in
[17] with a soft proof in its appendix. The subsequent analysis aims at a
clarification that $C_{\text{I}}$ depends exclusively on the parameter $\rho$
in (M2). The elementary arguments apply to more general situations in
particular to 3D. Given $I_{h}v\in V_{h}(P)$, $q_{1}:=-\Delta
I_{h}v\in\mathcal{P}_{1}(P)$ is affine and $\int_{E}(v-I_{h}v)\,ds=0$. Since
$\frac{\partial I_{h}v}{\partial\textbf{n}_{P}}$ is edgewise constant, this
shows $\int_{E}{\frac{\partial
I_{h}v}{\partial\textbf{n}_{P}}}|_{E}(v-I_{h}v)\,ds=0$ for all
$E\in\mathcal{E}(P)$ and so $\big{\langle}\frac{\partial
I_{h}v}{\partial\textbf{n}_{P}},v-I_{h}v\big{\rangle}_{\partial P}=0$. An
integration by parts leads to

$\displaystyle(\nabla
I_{h}v,\nabla(I_{h}v-v))_{L^{2}(P)}=(q_{1},I_{h}v-v)_{L^{2}(P)}=(q_{1},\Pi^{\nabla}_{1}I_{h}v-v)_{L^{2}(P)}$

with $q_{1}\in\mathcal{P}_{1}(P)$ and $\Pi_{1}v_{h}=\Pi^{\nabla}_{1}v_{h}$ for
$v_{h}\in V_{h}(P)$ in the last step. Consequently,

$\displaystyle\|\nabla I_{h}v\|_{L^{2}(P)}^{2}$ $\displaystyle=(\nabla
I_{h}v,\nabla(I_{h}v-v))_{L^{2}(P)}+(\nabla I_{h}v,\nabla v)_{L^{2}(P)}$
$\displaystyle=(q_{1},\Pi^{\nabla}_{1}I_{h}v-v)_{L^{2}(P)}+(\nabla
I_{h}v,\nabla v)_{L^{2}(P)}$
$\displaystyle\leq\|q_{1}\|_{L^{2}(P)}\|v-\Pi^{\nabla}_{1}I_{h}v\|_{L^{2}(P)}+\|\nabla
I_{h}v\|_{L^{2}(P)}\|\nabla v\|_{L^{2}(P)}$ (2.16)

with the Cauchy inequality in the last step. Remark 2 and 3 on the Ritz
projection, and the definition of $I_{h}$ show

$\displaystyle\Pi_{0}\nabla v=\nabla\Pi^{\nabla}_{1}v=|P|^{-1}\int_{\partial
P}v\,\textbf{n}_{P}\,ds=|P|^{-1}\int_{\partial
P}I_{h}v\textbf{n}_{P}\,ds=\Pi_{0}\nabla I_{h}v=\nabla\Pi^{\nabla}_{1}I_{h}v.$
(2.17)

The function $f:=v-\Pi^{\nabla}_{1}I_{h}v\in H^{1}(P)$ satisfies
$\int_{\partial P}f\,ds=\int_{\partial P}(v-I_{h}v)\,ds=0$ and the Poincaré-
Friedrichs inequality from Lemma 2.1.a shows

$\displaystyle\|v-\Pi^{\nabla}_{1}I_{h}v\|_{L^{2}(P)}\leq
C_{\text{PF}}h_{P}\|\nabla(v-\Pi^{\nabla}_{1}I_{h}v)\|_{L^{2}(P)}=C_{\text{PF}}h_{P}\|(1-\Pi_{0})\nabla
v\|_{L^{2}(P)}$ (2.18)

with (2.17) in the last step. Let $\phi_{c}\in S^{1}_{0}(\widehat{{\cal
T}}(P)):=\\{w\in C^{0}(P):w|_{T(E)}\in\mathcal{P}_{1}(T(E))\quad\text{for
all}\;E\in\mathcal{E}(P)\\}$ denote the piecewise linear nodal basis function
of the interior node $c$ with respect to the triangulation $\widehat{{\cal
T}}(P)=\\{T(E):E\in\mathcal{E}(P)\\}$ (cf. Figure 2.2.b for an illustration of
$\widehat{{\cal T}}(P)$). An inverse estimate

$\displaystyle\|f_{1}\|_{L^{2}(T(E))}\leq
C_{1}\|\phi_{c}^{1/2}f_{1}\|_{L^{2}(T(E))}\quad\text{for
all}\;f_{1}\in\mathcal{P}_{1}(\widehat{{\cal T}}(P))$

on the triangle $T(E):=\text{conv}(E\cup\\{c\\})$ holds with the universal
constant $C_{1}$. A constructive proof computes the mass matrices for $T$ with
and without the weight $\phi_{c}$ to verify that the universal constant
$C_{1}$ does not depend on the shape of the triangle $T(E)$. This implies

$\displaystyle
C_{1}^{-1}\|q_{1}\|_{L^{2}(P)}^{2}\leq(\phi_{c}q_{1},q_{1})_{L^{2}(P)}=(-\Delta
I_{h}v,\phi_{c}q_{1})=(\nabla I_{h}v,\nabla(\phi_{c}q_{1}))_{L^{2}(P)}$ (2.19)

with an integration by parts for $\phi_{c}q_{1}\in H^{1}_{0}(P)$ and $I_{h}v$
in the last step. The mesh-size independent constant $C_{2}$ in the standard
inverse estimate

$\displaystyle h_{T(E)}\|\nabla q_{2}\|_{L^{2}(T(E))}\leq
C_{2}\|q_{2}\|_{L^{2}(T(E))}\quad\text{for
all}\;q_{2}\in\mathcal{P}_{2}(T(E))$

depends merely on the angles in the triangle $T(E),E\in\mathcal{E}(P),$ and so
exclusively on $\rho$. With $C^{-1}_{\text{sr}}h_{P}\leq h_{T(E)}$ from Remark
1, this shows

$\displaystyle
C_{2}^{-1}C_{\text{sr}}^{-1}h_{P}\|\nabla\phi_{c}q_{1}\|_{L^{2}(P)}\leq\|\phi_{c}q_{1}\|_{L^{2}(P)}\leq\|q_{1}\|_{L^{2}(P)}.$

This and (2.19) lead to

$\displaystyle\|q_{1}\|_{L^{2}(P)}\leq
C_{1}C_{2}C_{\text{sr}}h_{P}^{-1}\|\nabla I_{h}v\|_{L^{2}(P)}.$ (2.20)

The combination with (2.16)-(2.18) proves

$\displaystyle\|\nabla I_{h}v\|_{L^{2}(P)}^{2}$
$\displaystyle\leq(C_{1}C_{2}C_{\text{sr}}C_{\text{PF}}\|(1-\Pi_{0})\nabla
v\|_{L^{2}(P)}+\|\nabla v\|_{L^{2}(P)})\|\nabla I_{h}v\|_{L^{2}(P)}$
$\displaystyle\leq(1+C_{1}C_{2}C_{\text{sr}}C_{\text{PF}})\|\nabla
v\|_{L^{2}(P)}\|\nabla I_{h}v\|_{L^{2}(P)}.\qed$

###### Proof of $(b)$.

The identity (2.17) reads $\Pi_{0}\nabla(1-I_{h})v=0$ and the triangle
inequality results in

$\displaystyle|v-I_{h}v|_{1,P}$
$\displaystyle=\|(1-\Pi_{0})\nabla(1-I_{h})v\|_{L^{2}(p)}$
$\displaystyle\leq\|(1-\Pi_{0})\nabla v\|_{L^{2}(P)}+\|(1-\Pi_{0})\nabla
I_{h}v\|_{L^{2}(P)}.$ (2.21)

Since $I_{h}$ is the identity in $\mathcal{P}_{1}(P)$, it follows
$(1-\Pi_{0})\nabla I_{h}v=(1-\Pi_{0})\nabla I_{h}(v-\Pi^{\nabla}_{1}v).$ This
and the boundedness of the interpolation operator $I_{h}$ lead to

$\displaystyle\|(1-\Pi_{0})\nabla I_{h}v\|_{L^{2}(P)}$
$\displaystyle\leq\|\nabla I_{h}(1-\Pi^{\nabla}_{1})v\|_{L^{2}(P)}$
$\displaystyle\leq
C_{\mathrm{Itn}}\|\nabla(1-\Pi^{\nabla}_{1})v\|_{L^{2}(P)}=C_{\mathrm{Itn}}\|(1-\Pi_{0})\nabla
v\|_{L^{2}(P)}$ (2.22)

with Remark 2 in the last step. The combination of (2.21) and (2.22) proves
the first part of $(b)$.
The identity $|(1-\Pi_{1}I_{h})v|_{1,P}=\|(1-\Pi_{0})\nabla v\|_{L^{2}(P)}$
follows from (2.17). Since $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$ and
$\int_{\partial P}v\,ds=\int_{\partial P}I_{h}v\,ds=\int_{\partial
P}\Pi^{\nabla}_{1}I_{h}v\,ds$, the Poincaré-Friedrichs inequality

$\|(1-\Pi_{1}I_{h})v\|_{L^{2}(P)}\leq
C_{\text{PF}}h_{P}|(1-\Pi_{1}I_{h})v|_{1,P}$

follows from Lemma 2.1.a. This concludes the proof of $(b)$. ∎

###### Proof of $(c)$.

This is an immediate consequence of the part $(b)$ with (2.12) and the
Poincaré-Friedrichs inequality for $v-I_{h}v$ (from above) in Lemma 2.1.a. ∎

##  3 Preliminary estimates

This subsection formulates the discrete problem along with the properties of
the discrete bilinear form such as boundedness and a G$\mathring{a}$rding-type
inequality.

###  3.1 The discrete problem

Denote the restriction of the bilinear forms $a(\cdot,\cdot),\hskip
2.84526ptb(\cdot,\cdot)$ and $c(\cdot,\cdot)$ on a polygonal domain
$P\in\mathcal{T}$ by $a^{P}(\cdot,\cdot),\hskip 2.84526ptb^{P}(\cdot,\cdot)$
and $c^{P}(\cdot,\cdot)$. The corresponding local discrete bilinear forms are
defined for $u_{h},v_{h}\in V_{h}(P)$ by

$\displaystyle a_{h}^{P}(u_{h},v_{h})$
$\displaystyle:=(\textbf{A}\nabla\Pi_{1}u_{h},\nabla\Pi_{1}v_{h})_{L^{2}(P)}+S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h}),$
(3.1) $\displaystyle b_{h}^{P}(u_{h},v_{h})$ $\displaystyle:=\
(\Pi_{1}u_{h},\textbf{b}\cdot\nabla\Pi_{1}v_{h})_{L^{2}(P)},$ (3.2)
$\displaystyle c_{h}^{P}(u_{h},v_{h})$
$\displaystyle:=(\gamma\Pi_{1}u_{h},\Pi_{1}v_{h})_{L^{2}(P)},$ (3.3)
$\displaystyle B_{h}^{P}(u_{h},v_{h})$
$\displaystyle:=a_{h}^{P}(u_{h},v_{h})+b_{h}^{P}(u_{h},v_{h})+c_{h}^{P}(u_{h},v_{h}).$
(3.4)

Choose the stability term $S^{P}(u_{h},v_{h})$ as a symmetric positive
definite bilinear form on $V_{h}(P)\times V_{h}(P)$ for a positive constant
$C_{s}$ independent of $P$ and $h_{P}$ satisfying

$C_{s}^{-1}a^{P}(v_{h},v_{h})\leq S^{P}(v_{h},v_{h})\leq
C_{s}a^{P}(v_{h},v_{h})\quad\text{for all}\hskip 2.84526ptv_{h}\in
V_{h}(P)\hskip 2.84526pt\text{with}\hskip 2.84526pt\Pi_{1}v_{h}=0.$ (3.5)

For some positive constant approximation $\overline{\textbf{A}}_{P}$ of A over
$P$ and the number $N_{P}:=|\mathcal{E}(P)|$ of the degrees of freedom (2.10)
of $V_{h}(P)$, a standard example of a stabilization term from [4],[36, Sec.
4.3] with a scaling coefficient $\overline{\textbf{A}}_{P}$ reads

$\displaystyle
S^{P}(v_{h},w_{h}):=\overline{\textbf{A}}_{P}\sum_{r=1}^{N_{P}}\text{dof}_{r}(v_{h})\text{dof}_{r}(w_{h})\quad\text{for
all}\;v_{h},w_{h}\in V_{h}.$ (3.6)

Note that an approximation $\overline{\textbf{A}}_{P}$ is a positive real
number (not a matrix) and can be chosen as $\sqrt{a_{0}a_{1}}$ with the
positive constants $a_{0}$ and $a_{1}$ from (A2). For $f\in L^{2}(\Omega)$ and
$v_{h}\in V_{h}$, define the right-hand side functional $f_{h}$ on $V_{h}$ by

$\displaystyle(f_{h},v_{h})_{L^{2}(P)}$
$\displaystyle:=(f,\Pi_{1}v_{h})_{L^{2}(P)}.$ (3.7)

The sum over all the polygonal domains $P\in\mathcal{T}$ reads

$\displaystyle a_{h}(u_{h},v_{h})$
$\displaystyle:=\sum_{P\in\mathcal{T}}a_{h}^{P}(u_{h},v_{h}),\hskip
14.22636ptb_{h}(u_{h},v_{h}):=\sum_{P\in\mathcal{T}}b_{h}^{P}(u_{h},v_{h}),$
$\displaystyle c_{h}(u_{h},v_{h})$
$\displaystyle:=\sum_{P\in\mathcal{T}}c_{h}^{P}(u_{h},v_{h}),\hskip
14.22636pts_{h}(u_{h},v_{h}):=\sum_{P\in\mathcal{T}}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h}),$
$\displaystyle B_{h}(u_{h},v_{h})$
$\displaystyle:=\sum_{P\in\mathcal{T}}B_{h}^{P}(u_{h},v_{h}),\hskip
14.22636pt(f_{h},v_{h}):=\sum_{P\in\mathcal{T}}(f_{h},v_{h})_{P}\quad\text{for
all}\;u_{h},v_{h}\in V_{h}.$

The discrete problem seeks $u_{h}\in V_{h}$ such that

$\displaystyle B_{h}(u_{h},v_{h})=(f_{h},v_{h})\quad\text{for all}\;v_{h}\in
V_{h}.$ (3.8)

######  Remark 5 (polygonal mesh with small edges).

The conditions (M1)-(M2) are well established and apply throughout the paper.
The sub-triangulation $\widehat{{\cal T}}$ may not be shape-regular without
the edge condition $|E|\geq\rho h_{P}$ for an edge $E\in\mathcal{T}(P)$ and
$P\in\mathcal{T}$, but satisfies the maximal angle condition and the arguments
employed in the proof of [8, Lemma 6.3] can be applied to show (2.20) in
Theorem 2.8.a. For more general star-shaped polygon domains with short edges,
the recent anisotropic analysis [8, 15, 18] indicates that the stabilization
term has to be modified as well to avoid a logarithmic factor in the optimal
error estimates.

###  3.2 Properties of the discrete bilinear form

The following proposition provides two main properties of the discrete
bilinear form $B_{h}$.

######  Proposition 3.1.

There exist positive universal constants $M,\alpha$ and a universal
nonnegative constant $\beta$ depending on the coefficients
$\textbf{A},\textbf{b},\gamma$ such that

  1. $\left(a\right)$

Boundedness: $|B_{h}(u_{h},v_{h})|\leq
M|u_{h}|_{1,\mathrm{pw}}|v_{h}|_{1,\mathrm{pw}}\quad\text{for all}\hskip
2.84526ptu_{h},v_{h}\in V_{h}.$

  2. $\left(b\right)$

G$\mathring{a}$rding-type inequality:
$\alpha|v_{h}|^{2}_{1,\mathrm{pw}}-\beta\|v_{h}\|^{2}_{L^{2}(\Omega)}\leq
B_{h}(v_{h},v_{h})\quad\text{for all}\hskip 2.84526ptv_{h}\in V_{h}.$

###### Proof of $\left(a\right)$.

The upper bound of the coefficients from the assumption (A1), the Cauchy-
Schwarz inequality, the stability (2.11) of $\Pi_{1}$, and the definition
(3.5) of the stabilization term imply the boundedness of $B_{h}$ with
$M:=(1+C_{s})\|\textbf{A}\|_{\infty}+C_{\mathrm{F}}\|\textbf{b}\|_{\infty}+C_{\mathrm{F}}^{2}\|\gamma\|_{\infty}$.
The details of the proof follow as in [6, Lemma 5.2] with the constant
$C_{\mathrm{F}}$ from Lemma 2.6. ∎

###### Proof of $\left(b\right)$.

The first step shows that $a_{h}(\cdot,\cdot)$ is coercive. For $v_{h}\in
V_{h}(P)$, $\Pi_{1}v_{h}=\Pi^{\nabla}_{1}v_{h}$ and
$\nabla\Pi_{1}v_{h}\perp\nabla(v_{h}-\Pi^{\nabla}_{1}v_{h})$ in
$L^{2}(P;\mathbb{R}^{2})$. This orthogonality, the assumption (A2), and the
definition of the stability term (3.5) with the constant $C_{s}^{-1}\leq 1$
imply for $\alpha_{0}=a_{0}C_{s}^{-1}$ that

$\displaystyle\alpha_{0}|v_{h}|_{1,\mathrm{pw}}^{2}\leq
a_{0}\|\nabla_{\mathrm{pw}}\Pi_{1}v_{h}\|^{2}_{L^{2}(\Omega)}+a_{0}C_{s}^{-1}\|\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\quad\leq\left(\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}v_{h},\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+C_{s}^{-1}(\textbf{A}\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h},\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h}\right)_{L^{2}(\Omega)}$
$\displaystyle\quad\leq(\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}v_{h},\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})v_{h},(1-\Pi_{1})v_{h})=a_{h}(v_{h},v_{h}).$
(3.9)

The Cauchy-Schwarz inequality, (2.11), and the Young inequality lead to

$\displaystyle|b_{h}(v_{h},v_{h})+c_{h}(v_{h},v_{h})|$
$\displaystyle\leq\|\textbf{b}\|_{\infty}\|\Pi_{1}v_{h}\|_{L^{2}(\Omega)}\|\nabla_{\mathrm{pw}}\Pi_{1}v_{h}\|_{L^{2}(\Omega)}+\|\gamma\|_{\infty}\|\Pi_{1}v_{h}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\|\textbf{b}\|_{\infty}\|v_{h}\|_{L^{2}(\Omega)}|v_{h}|_{1,\mathrm{pw}}+\|\gamma\|_{\infty}\|v_{h}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\frac{\|\textbf{b}\|^{2}_{\infty}}{2\alpha_{0}}\|v_{h}\|_{L^{2}(\Omega)}^{2}+\frac{\alpha_{0}}{2}|v_{h}|^{2}_{1,\mathrm{pw}}+\|\gamma\|_{\infty}\|v_{h}\|_{L^{2}(\Omega)}^{2}.$
(3.10)

The combination of (3.9)-(3.10) proves

$\displaystyle\frac{\alpha_{0}}{2}|v_{h}|^{2}_{1,\mathrm{pw}}-\left(\frac{\|\textbf{b}\|^{2}_{\infty}}{2\alpha_{0}}+\|\gamma\|_{\infty}\right)\|v_{h}\|^{2}_{L^{2}(\Omega)}\leq
B_{h}(v_{h},v_{h}).$

This concludes the proof of $\left(b\right)$ with
$\alpha=\frac{\alpha_{0}}{2}$ and
$\beta=\frac{\|\textbf{b}\|^{2}_{\infty}}{2\alpha_{0}}+\|\gamma\|_{\infty}$. ∎

######  Remark 6 ($\|\cdot\|_{h}\approx|\cdot|_{1,\mathrm{pw}}$).

The discrete space $V_{h}$ of the nonconforming VEM is endowed with the
natural norm $\|\cdot\|_{h}:=a_{h}(\cdot,\cdot)^{1/2}$ induced by the scalar
product $a_{h}$. The boundedness of $a_{h}$ is proven in $(a)$, while (3.9)
shows the converse estimate in the equivalence
$\|\cdot\|_{h}\approx|\cdot|_{1,\mathrm{pw}}$ in $V_{h}$, namely

$\alpha_{0}|v_{h}|^{2}_{1,\mathrm{pw}}\leq
a_{h}(v_{h},v_{h})\leq\|\textbf{A}\|_{\infty}(1+C_{s})|v_{h}|_{1,\mathrm{pw}}^{2}\quad\text{for
all}\;v_{h}\in V_{h}.$

###  3.3 Consistency error

This subsection discusses the consistency error between the continuous
bilinear form $B$ and the corresponding discrete bilinear form $B_{h}$. Recall
the definition $B^{P}(\cdot,\cdot)\equiv
a^{P}(\cdot,\cdot)+b^{P}(\cdot,\cdot)+c^{P}(\cdot,\cdot)$ and
$B_{h}^{P}(\cdot,\cdot)\equiv
a_{h}^{P}(\cdot,\cdot)+b_{h}^{P}(\cdot,\cdot)+c_{h}^{P}(\cdot,\cdot)$ for a
polygonal domain $P\in\mathcal{T}$ from Subsection 2.1.

######  Lemma 3.2 (consistency).

$(a)$ There exists a positive constant $C_{\text{cst}}$ (depending only on
$\rho$) such that any $v\in H^{1}(\Omega)$ and $w_{h}\in V_{h}$ satisfy

$\displaystyle B^{P}(\Pi_{1}v,w_{h})-B_{h}^{P}(\Pi_{1}v,w_{h})\leq
C_{\mathrm{cst}}\,h_{P}\|v\|_{1,P}|w_{h}|_{1,P}\quad\text{for
all}\;P\in\mathcal{T}.$ (3.11)

$(b)$ Any $f\in L^{2}(\Omega)$ and $f_{h}:=\Pi_{1}f$ satisfy

$\displaystyle\|f-f_{h}\|_{V_{h}^{*}}:=\sup_{0\neq v_{h}\in
V_{h}}\frac{(f-f_{h},v_{h})}{\|v_{h}\|_{1,\mathrm{pw}}}\leq
C_{\mathrm{PF}}\,\mathrm{osc}_{1}(f,\mathcal{T}).$ (3.12)

###### Proof.

Observe that $S^{P}((1-\Pi_{1})\Pi_{1}v,(1-\Pi_{1})w_{h})=0$ follows from
$(1-\Pi_{1})\Pi_{1}v=0$. The definition of $B^{P}$ and $B_{h}^{P}$ show

$\displaystyle
B^{P}(\Pi_{1}v,w_{h})-B_{h}^{P}(\Pi_{1}v,w_{h})=:T_{1}+T_{2}+T_{3}.$ (3.13)

The term $T_{1}$ in (3.13) is defined as the difference of the contributions
from $a^{P}$ and $a^{P}_{h}$. Their definitions prove the equality (at the end
of the first line below) and the definition of $\Pi_{1}$ prove the next
equality in

$\displaystyle T_{1}$
$\displaystyle:=a^{P}(\Pi_{1}v,w_{h})-a_{h}^{p}(\Pi_{1}v,w_{h})=(\textbf{A}\nabla\Pi_{1}v,\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}$
$\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\nabla\Pi_{1}v),\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}\leq
h_{P}|\textbf{A}|_{1,\infty}|v|_{1,P}|w_{h}|_{1,P}.$

The last inequality follows from the Cauchy-Schwarz inequality, the Lipschitz
continuity of A, and the stabilities
$\|\nabla\Pi_{1}v_{h}\|_{L^{2}(P)}\leq\|\nabla v_{h}\|_{L^{2}(P)}$ and
$\|\nabla(1-\Pi_{1})w_{h}\|_{L^{2}(P)}\leq\|\nabla w_{h}\|_{L^{2}(P)}$ from
Remark 4. Similar arguments apply to $T_{2}$ from the differences of $b^{P}$
and $b^{P}_{h}$, and $T_{3}$ from those of $c^{P}$ and $c_{h}^{P}$ in (3.13).
This leads to

$\displaystyle T_{2}$
$\displaystyle:=b^{P}(\Pi_{1}v,w_{h})-b_{h}^{P}(\Pi_{1}v,w_{h})$
$\displaystyle=((\textbf{b}-\Pi_{0}\textbf{b})\Pi_{1}v,\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}+((\Pi_{0}\textbf{b})(1-\Pi_{0})(\Pi_{1}v),\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}$
$\displaystyle\leq(|\textbf{b}|_{1,\infty}+C_{\mathrm{apx}}\|\textbf{b}\|_{\infty})h_{P}\|v\|_{1,P}|w_{h}|_{1,P},$
$\displaystyle T_{3}$
$\displaystyle:=c^{P}(\Pi_{1}v,w_{h})-c_{h}^{P}(\Pi_{1}v,w_{h})=(\gamma\Pi_{1}v,(1-\Pi_{1})w_{h})_{L^{2}(P)}\leq
C_{\mathrm{PF}}\,\|\gamma\|_{\infty}h_{P}\|v\|_{L^{2}(P)}|w_{h}|_{1,P}.$

The inequality for the last step in $T_{2}$ follows from the Cauchy-Schwarz
inequality, the Lipschitz continuity of b, the estimate
$\|(1-\Pi_{0})\Pi_{1}v\|_{L^{2}(P)}\leq\|(1-\Pi_{0})v\|_{L^{2}(P)}\leq
C_{\text{apx}}h_{P}|v|_{1,P}$ from (2.12), and the above stabilities
$\|\nabla\Pi_{1}v_{h}\|_{L^{2}(P)}\leq\|\nabla v_{h}\|_{L^{2}(P)}$ and
$\|\nabla(1-\Pi_{1})w_{h}\|_{L^{2}(P)}\leq\|\nabla w_{h}\|_{L^{2}(P)}$. The
inequality for the last step in $T_{3}$ follows from the Cauchy-Schwarz
inequality, $\|\Pi_{1}v\|_{L^{2}(P)}$ $\leq\|v\|_{L^{2}(P)}$ from (2.11) and
the Poincaŕe-Friedrichs inequality in Lemma 2.1.a for $w_{h}-\Pi_{1}w_{h}$
with $\int_{\partial P}(w_{h}-\Pi_{1}w_{h})\,ds=0$ from
$\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$. The combination of the above estimates
shows (3.11). The proof of (3.12) adapts the arguments in the above analysis
of $T_{3}$ and the definition of $\mathrm{osc}_{1}(f,\mathcal{T})$ in
Subsection 2.1 for the proof of

$\displaystyle(f-f_{h},w_{h})_{L^{2}(P)}=(f-\Pi_{1}f,w_{h}-\Pi_{1}w_{h})_{L^{2}(P)}\leq
C_{\text{PF}}|w_{h}|_{1,P}\,\mathrm{osc}_{1}(f,P).$

This concludes the proof. ∎

###  3.4 Nonconformity error

Enrichment operators play a vital role in the analysis of nonconforming finite
element methods [12]. For any $v_{h}\in V_{h},$ the objective is to find a
corresponding function $Jv_{h}\in H_{0}^{1}(\Omega)$. The idea is to map the
VEM nonconforming space into the Crouzeix-Raviart finite element space

$\displaystyle\text{CR}_{0}^{1}(\widehat{{\cal T}}):=\\{v\in{\cal
P}_{1}(\widehat{{\cal T}}):\hskip 2.84526pt$
$\displaystyle\forall\;E\in\widehat{{\cal E}}\quad v\hskip 2.84526pt\text{is
continuous at mid}(E)\quad\text{and}$ $\displaystyle\forall\;E\in{\cal
E}(\partial\Omega)\quad v(\text{mid}(E))=0\\}$

with respect to the shape-regular triangulation $\widehat{{\cal T}}$ from
Remark 1. Let $\psi_{E}$ be the edge-oriented basis functions of
CR${}_{0}^{1}(\widehat{{\cal T}})$ with $\psi_{E}(\text{mid}\hskip
2.84526ptE)=1$ and $\psi_{E}(\text{mid}\hskip 2.84526ptF)=0$ for all other
edges $F\in\widehat{{\cal E}}\setminus\\{E\\}.$ Define the interpolation
operator $I_{\text{CR}}:V_{h}\to\text{CR}_{0}^{1}(\widehat{{\cal T}})$, for
$v_{h}\in V_{h}$, by

$\displaystyle I_{\text{CR}}v_{h}=\sum_{F\in\widehat{{\cal
E}}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{F}v_{h}\,ds\right)\psi_{F}.$ (3.14)

The definition of $V_{h}$ implies $\int_{F}[v_{h}]\,ds=0$ for $v_{h}\in V_{h}$
and for all $F\in\mathcal{E}$. Since $v_{h}|_{P}\in H^{1}(P)$, it follows
$\int_{F}[v_{h}]\,ds=0$ for all $F\in\widehat{{\cal E}}\setminus\mathcal{E}$.
This shows $\int_{F}v_{h|T^{\pm}}\,ds$ is unique for all edges $F=\partial
T^{+}\cap\partial T^{-}\in\widehat{{\cal E}}$ and, consequently,
$I_{\text{CR}}v_{h}$ is well-defined (independent of the choice of traces
selected in the evaluation of $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{F}v_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{F}v_{h}|_{T^{+}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{F}v_{h}|_{T^{-}}\,ds$). The approximation property
of $I_{\text{CR}}$ on each $T\in\widehat{{\cal T}}$ reads

$\displaystyle
h_{T}^{-1}\|v_{h}-I_{\text{CR}}v_{h}\|_{L^{2}(T)}+|v_{h}-I_{\text{CR}}v_{h}|_{1,T}\leq
2|v_{h}|_{1,T}$ (3.15)

(cf. [23, Thm 2.1] or [21, Thm 4] for explicit constants). Define an
enrichment operator $E_{h}:\text{CR}_{0}^{1}(\widehat{{\cal T}})\to
H_{0}^{1}(\Omega)$ by averaging the function values at each interior vertex
$z$, that is,

$\displaystyle E_{h}v_{\text{CR}}(z)=\frac{1}{|\widehat{{\cal
T}}(z)|}\sum_{T\in\widehat{{\cal T}}(z)}{v_{\text{CR}}}|_{T}(z)$ (3.16)

and zero on boundary vertices. In (3.16) the set $\widehat{{\cal
T}}(z):=\\{T\in\widehat{{\cal T}}\hskip 2.84526pt|\hskip 2.84526ptz\in T\\}$
of neighboring triangles has the cardinality $|\widehat{{\cal T}}(z)|\geq 3$.

The following lemma describes the construction of a modified companion
operator $J:V_{h}\to H_{0}^{1}(\Omega)$, which is a right-inverse of the
interpolation operator $I_{h}$ from Definition 2.7.

######  Lemma 3.3 (conforming companion operator).

There exists a linear map $J:V_{h}\to H^{1}_{0}(\Omega)$ and a universal
constant $C_{\mathrm{J}}\lesssim 1$ such that any $v_{h}\in V_{h}$ satisfies
$I_{h}Jv_{h}=v_{h}$ and

  1. (a)

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds$ for any edge $E\in\widehat{{\cal
E}},$

  2. (b)

$\displaystyle\nabla_{\mathrm{pw}}(v_{h}-Jv_{h})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$
in $L^{2}(\Omega;\mathbb{R}^{2}),$

  3. (c)

$\displaystyle v_{h}-Jv_{h}\perp\mathcal{P}_{1}(\mathcal{T})$ in
$L^{2}(\Omega),$

  4. (d)

$\|h_{\mathcal{T}}^{-1}(v_{h}-Jv_{h})\|_{L^{2}(\Omega)}+|v_{h}-Jv_{h}|_{1,\mathrm{pw}}\leq
C_{\mathrm{J}}|v_{h}|_{1,\mathrm{pw}}.$

###### Design of $J$ in Lemma 3.3.

∎ Given $v_{h}\in V_{h}$, let
$v_{\text{CR}}:=I_{\text{CR}}v_{h}\in\text{CR}^{1}_{0}(\widehat{{\cal T}})$.
There exists an operator $J^{\prime}:\text{CR}_{0}^{1}(\widehat{{\cal T}})\to
H_{0}^{1}(\Omega)$ from [22, Prop. 2.3] such that any
$v_{\text{CR}}\in\text{CR}_{0}^{1}(\widehat{{\cal T}})$ satisfies

  1. (a’)

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}J^{\prime}v_{CR}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v_{CR}\,ds$ for any edge $E\in\widehat{{\cal
E}},$

  2. (b’)

$\displaystyle\int_{P}\nabla_{\mathrm{pw}}(v_{\text{CR}}-J^{\prime}v_{\text{CR}})\,dx=0$
for all $P\in\mathcal{T}$,

  3. (c’)

$\displaystyle\|h_{\widehat{{\cal
T}}}^{-1}(v_{\text{CR}}-J^{\prime}v_{\text{CR}})\|_{L^{2}(\Omega)}+|v_{\text{CR}}-J^{\prime}v_{\text{CR}}|_{1,\mathrm{pw}}\leq
C_{\mathrm{J^{\prime}}}\min_{v\in
H^{1}_{0}(\Omega)}|v_{\text{CR}}-v|_{1,\mathrm{pw}}$

with a universal constant $C_{\mathrm{J^{\prime}}}$ from [25]. Set
$v:=J^{\prime}I_{\text{CR}}v_{h}\in V:=H^{1}_{0}(\Omega)$. Recall that
$\widehat{{\cal T}}(P)$ is a shape-regular triangulation of $P$ into a finite
number of triangles. For each $T\in\widehat{{\cal T}}(P)$, let $b_{T}\in
W_{0}^{1,\infty}(T)$ denote the cubic bubble-function
$27\lambda_{1}\lambda_{2}\lambda_{3}$ for the barycentric co-ordinates
$\lambda_{1},\lambda_{2},\lambda_{3}\in\mathcal{P}_{1}(T)$ of $T$ with
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{T}b_{T}\,dx=9/20$ and $\|\nabla
b_{T}\|_{L^{2}(T)}\lesssim h_{T}^{-1}|T|^{1/2}\approx 1.$ Let $b_{T}$ be
extended by zero outside $T$ and, for $P\in\mathcal{T}$, define

$\displaystyle b_{P}:=\frac{20}{9}\sum_{T\in\widehat{{\cal T}}(P)}b_{T}\in
W_{0}^{1,\infty}(P)\subset W_{0}^{1,\infty}(\Omega)$ (3.17)

with $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{P}b_{P}\,dx=1$ and $\|\nabla
b_{P}\|_{L^{2}(P)}\lesssim h_{P}^{-1}|P|^{1/2}\approx 1$. Let
$v_{P}\in\mathcal{P}_{1}(\mathcal{T})$ be the Riesz representation of the
linear functional $\mathcal{P}_{1}(\mathcal{T})\to\mathbb{R}$ defined by
$w_{1}\mapsto(v_{h}-v,w_{1})_{L^{2}(\Omega)}$ for
$w_{1}\in\mathcal{P}_{1}(\mathcal{T})$ in the Hilbert space
$\mathcal{P}_{1}(\mathcal{T})$ endowed with the weighted $L^{2}$ scalar
product $(b_{P}\bullet,\bullet)_{L^{2}(P)}$. Hence $v_{P}$ exists uniquely and
satisfies $\Pi_{1}(v_{h}-v)=\Pi_{1}(b_{P}v_{P})$. Given the bubble-functions
$(b_{P}:P\in\mathcal{T})$ from (3.17) and the above functions
$(v_{P}:P\in\mathcal{T})$ for $v_{h}\in V_{h}$, define

$\displaystyle Jv_{h}:=v+\sum_{P\in\mathcal{T}}v_{P}b_{P}\in V.$ (3.18)

###### Proof of (a).

Since $b_{P}$ vanishes at any $x\in E\in\mathcal{E}$, it follows for any
$E\in\widehat{\mathcal{E}}$ that

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}J^{\prime}v_{\text{CR}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v_{\text{CR}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds,$

where the definition of $v=J^{\prime}v_{\text{CR}}$, (a’), and
$v_{\text{CR}}=I_{\text{CR}}v_{h}$ lead to the second, third, and fourth
equality. This proves (a). ∎

###### Proof of (b).

An integration by parts and (a) show, for all $v_{h}\in V_{h}$ with $Jv_{h}$
from (3.18), that

$\displaystyle\int_{P}\nabla Jv_{h}\,dx=\int_{\partial
P}Jv_{h}\textbf{n}_{P}\,ds=\sum_{E\in\mathcal{E}(P)}\Big{(}\int_{E}Jv_{h}\textbf{n}_{E}\,ds\Big{)}=\sum_{E\in\mathcal{E}(P)}\Big{(}\int_{E}v_{h}\textbf{n}_{E}\,ds\Big{)}=\int_{P}\nabla
v_{h}\,dx.$

Since this holds for all $P\in\mathcal{T}$, it proves (b). ∎

###### Proof of (c).

This is $\Pi_{1}v_{h}=\Pi_{1}Jv_{h}$ and guaranteed by the design of $J$ in
(3.18). ∎

###### Proof of (d).

This relies on the definition of $J$ in (3.18) and $J^{\prime}$ with (c’).
Since (a) allows for $\int_{\partial P}(v_{h}-Jv_{h})\,ds=0$, the Poincaré-
Friedrichs inequality from Lemma 2.1.a implies

$\displaystyle h_{P}^{-1}\|v_{h}-Jv_{h}\|_{L^{2}(P)}\leq
C_{\text{PF}}|v_{h}-Jv_{h}|_{1,P}.$

Hence it remains to prove
$|v_{h}-Jv_{h}|_{1,\mathrm{pw}}\lesssim|v_{h}|_{1,\mathrm{pw}}.$ Triangle
inequalities with $v_{h},Jv_{h},v=J^{\prime}v_{\text{CR}}$ and
$v_{\text{CR}}=I_{\text{CR}}v_{h}$ show the first and second inequality in

$\displaystyle|v_{h}-Jv_{h}|_{1,\mathrm{pw}}-|v-Jv_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq|v-v_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq|v_{h}-I_{\text{CR}}v_{h}|_{1,\mathrm{pw}}+|v_{\text{CR}}-J^{\prime}v_{\text{CR}}|_{1,\mathrm{pw}}\leq(1+C_{\mathrm{J^{\prime}}})|v_{h}|_{1,\mathrm{pw}}$
(3.19)

with (b’) for
$|v_{\text{CR}}|_{1,\mathrm{pw}}=\|\Pi_{0}\nabla_{\mathrm{pw}}v_{h}\|_{L^{2}(\Omega)}\leq\|\nabla_{\mathrm{pw}}v_{h}\|_{L^{2}(\Omega)}=|v_{h}|_{1,\mathrm{pw}}$
in the last step. The equivalence of norms in the finite-dimensional space
$\mathcal{P}_{1}(P)$ assures the existence of a positive constant $C_{b}$,
independent of $h_{P}$, such that any $\chi\in\mathcal{P}_{1}(P)$ satisfies
the inverse inequalities

$\displaystyle C_{b}^{-1}\|\chi\|^{2}_{L^{2}(P)}\leq$
$\displaystyle(b_{P},\chi^{2})_{L^{2}(P)}\leq C_{b}\|\chi\|^{2}_{L^{2}(P)},$
(3.20) $\displaystyle
C_{b}^{-1}\|\chi\|_{L^{2}(P)}\leq\|b_{P}\chi\|_{L^{2}(P)}$
$\displaystyle+h_{P}\|\nabla(b_{P}\chi)\|_{L^{2}(P)}\leq
C_{b}\|\chi\|_{L^{2}(P)}.$ (3.21)

These estimates are completely standard on shape-regular triangles [2, p. 27]
or [37]; so they hold on each $T\in\widehat{{\cal T}}$ and, by definition of
$b_{P}$, their sum is (3.20)-(3.21). The analysis of the term
$|v-Jv_{h}|_{1,\mathrm{pw}}$ starts with one $P\in\mathcal{T}$ and (3.18) for

$\displaystyle|v-Jv_{h}|_{1,P}=|v_{P}b_{P}|_{1,P}\leq
C_{b}h_{P}^{-1}\|v_{P}\|_{L^{2}(P)}$ (3.22)

with (3.21) in the last step. The estimate (3.20) leads to the first
inequality in

$\displaystyle
C_{b}^{-1}\|v_{P}\|^{2}_{L^{2}(P)}\leq(b_{P}v_{P},v_{P})_{L^{2}(P)}=(v_{h}-v,v_{P})_{L^{2}(P)}\leq\|v_{h}-v\|_{L^{2}(P)}\|v_{P}\|_{L^{2}(P)}.$

The equality results from $\Pi_{1}(v_{h}-v)=\Pi_{1}(v_{P}b_{P})$ and
$v_{P}\in\mathcal{P}_{1}(\mathcal{T})$, while the last step is the Cauchy-
Schwarz inequality. Consequently, $\|v_{P}\|_{L^{2}(P)}\leq
C_{b}\|v_{h}-v\|_{L^{2}(P)}$. This and (3.22) show

$\displaystyle|v-Jv_{h}|_{1,\mathrm{pw}}\leq
C_{b}^{2}\|h^{-1}_{\mathcal{T}}(v-v_{h})\|_{L^{2}(\Omega)}\leq
C_{b}^{2}C_{\mathrm{PF}}|v-v_{h}|_{1,\mathrm{pw}}$

with $\int_{\partial P}(v-v_{h})\,ds=0$ from $(a)$ and hence the Poincaré-
Friedrichs inequality for $v-v_{h}$ from Lemma 2.1.a in the last step. Recall
$|v-v_{h}|_{1,\mathrm{pw}}\lesssim|v_{h}|_{1,\mathrm{pw}}$ from (3.19) to
conclude $|v-Jv_{h}|_{1,\mathrm{pw}}\lesssim|v_{h}|_{1,\mathrm{pw}}$ from the
previous displayed inequality. This concludes the proof of (d). ∎

###### Proof of $I_{h}J=\text{id}\;\text{ in}\;V_{h}$.

Definition 2.7 and Lemma 3.3.a show, for all $v_{h}\in V_{h}$, that

$\displaystyle
I_{h}Jv_{h}=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds\Big{)}\psi_{E}=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds\Big{)}\psi_{E}=v_{h}.$

This concludes the proof of Lemma 3.3. ∎

Since $V_{h}$ is not a subset of $H^{1}_{0}(\Omega)$ in general, the
substitution of discrete function $v_{h}$ in the weak formulation leads to a
nonconformity error.

######  Lemma 3.4 (nonconformity error).

There exist positive universal constants $C_{\mathrm{NC}},C^{*}_{\mathrm{NC}}$
(depending on the coefficients $\textbf{A},\textbf{b}$ and the universal
constants $\rho,\sigma$) such that all $f,g\in L^{2}(\Omega)$ and all
$\mathcal{T}\in\mathbb{T}(\delta)$ (with the assumption
$h_{\text{max}}\leq\delta\leq 1$) satisfy $(a)$ and $(b)$.
$(a)$ The solution $u\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ to
$(\ref{1})$ satisfies

$\displaystyle\sup_{0\neq v_{h}\in
V_{h}}\frac{|B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})_{L^{2}(\Omega)}|}{\|v_{h}\|_{1,\mathrm{pw}}}\leq
C_{\mathrm{NC}}h_{\text{max}}^{\sigma}\|f\|_{L^{2}(\Omega)}.$ (3.23)

$(b)$ The solution $\Phi\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ to the
dual problem $(\ref{5})$ satisfies

$\displaystyle\sup_{0\neq v_{h}\in
V_{h}}\frac{|B_{\mathrm{pw}}(v_{h},\Phi)-(g,v_{h})_{L^{2}(\Omega)}|}{\|v_{h}\|_{1,\mathrm{pw}}}\leq
C^{*}_{\mathrm{NC}}h_{\text{max}}^{\sigma}\|g\|_{L^{2}(\Omega)}.$ (3.24)

###### Proof of $(a)$.

Given $v_{h}\in V_{h}$, define $Jv_{h}\in V$ and the piecewise averages
$\overline{\textbf{A}}:=\Pi_{0}(\textbf{A}),\overline{\textbf{b}}:=\Pi_{0}(\textbf{b})$,
and $\overline{\gamma}:=\Pi_{0}(\gamma)$ of the coefficients
$\textbf{A},\textbf{b}$, and $\gamma$. The choice of test function
$v:=Jv_{h}\in V$ in the weak formulation (1.8) having extra properties
provides the terms with oscillations in the further analysis. Abbreviate
$\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u$. The weak formulation (1.8),
Lemma 3.3.b-c, and the Cauchy-Schwarz inequality reveal that

$\displaystyle
B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})_{L^{2}(\Omega)}=B_{\mathrm{pw}}(u,v_{h}-Jv_{h})-(f,v_{h}-Jv_{h})_{L^{2}(\Omega)}$
$\displaystyle\leq\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\|_{L^{2}(\Omega)}\|\nabla_{\mathrm{pw}}(1-J)v_{h}\|_{L^{2}(\Omega)}+\|h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma
u)\|_{L^{2}(\Omega)}\|h_{\mathcal{T}}^{-1}(1-J)v_{h}\|_{L^{2}(\Omega)}.$
(3.25)

The first term on the right-hand side of (3.25) involves the factor

$\displaystyle\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|\textbf{A}\nabla u-\Pi_{0}(\textbf{A}\nabla
u)\|_{L^{2}(\Omega)}+\|\textbf{b}u-\Pi_{0}(\textbf{b}u)\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|(\textbf{A}-\overline{\textbf{A}})\nabla
u+\overline{\textbf{A}}(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}+\|(\textbf{b}-\overline{\textbf{b}})u+\overline{\textbf{b}}(1-\Pi_{0})u\|_{L^{2}(\Omega)}$
$\displaystyle\leq\Big{(}h_{\text{max}}(|\textbf{A}|_{1,\infty}+|\textbf{b}|_{1,\infty})+C_{\text{apx}}(h_{\text{max}}^{\sigma}\|\textbf{A}\|_{\infty}+h_{\text{max}}\|\textbf{b}\|_{\infty})\Big{)}\;\|u\|_{1+\sigma,\Omega}.$

The last inequality follows from the Lipschitz continuity of the coefficients
A and b, and the estimate (2.12). Lemma 3.3.d leads to the estimates
$\|\nabla_{\mathrm{pw}}(1-J)v_{h}\|_{L^{2}(\Omega)}\leq
C_{J}|v_{h}|_{1,\mathrm{pw}}$ and

$\displaystyle\|h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma
u)\|_{L^{2}(\Omega)}\|h_{\mathcal{T}}^{-1}(1-J)v_{h}\|_{L^{2}(\Omega)}\leq\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})C_{J}|v_{h}|_{1,\mathrm{pw}}.$

The substitution of the previous estimates in (3.25) with
$h_{\mathrm{max}}\leq 1$ (from $\delta\leq 1$ by assumption) and the
regularity (1.5) show

$\displaystyle B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})\leq
C_{\mathrm{NC}}h_{\text{max}}^{\sigma}\|f\|_{L^{2}(\Omega)}\|v_{h}\|_{1,\mathrm{pw}}$

with
$C_{\mathrm{NC}}:=C_{J}\Big{(}(|\textbf{A}|_{1,\infty}+|\textbf{b}|_{1,\infty}+C_{\text{apx}}(\|\textbf{A}\|_{\infty}+\|\textbf{b}\|_{\infty})+\|\gamma\|_{\infty})C_{\text{reg}}+1\Big{)}$.
This concludes the proof of Lemma 3.4.a. ∎

###### Proof of $(b)$.

The solution $\Phi\in V$ to (1.4) satisfies $B(v,\Phi)=(g,v)_{L^{2}(\Omega)}$
for all $v\in V.$ This implies

$B_{\mathrm{pw}}(v_{h},\Phi)-(g,v_{h})_{L^{2}(\Omega)}=B_{\mathrm{pw}}(v_{h}-Jv_{h},\Phi)-(g,v_{h}-Jv_{h})_{L^{2}(\Omega)}.$

The arguments in the proof of $(a)$ lead to the bound (3.24) with

$C^{*}_{\mathrm{NC}}:=C_{J}\Big{(}(|\textbf{A}|_{1,\infty}+C_{\text{apx}}\|\textbf{A}\|_{\infty}+\|\textbf{b}\|_{\infty}+\|\gamma\|_{\infty})C^{*}_{\text{reg}}+1\Big{)}.$

The remaining analogous details are omitted in the proof of Lemma 3.4.b for
brevity. ∎

##  4 A priori error analysis

This section focuses on the stability, existence, and uniqueness of the
discrete solution $u_{h}$. The a priori error analysis uses the discrete inf-
sup condition.

###  4.1 Existence and uniqueness of the discrete solution

######  Theorem 4.1 (stability).

There exist positive constants $\delta\leq 1$ and $C_{\mathrm{stab}}$
(depending on $\alpha,\beta,\sigma,\rho,$ and $C_{\mathrm{F}}$) such that, for
all $\mathcal{T}\in\mathbb{T}(\delta)$ and for all $f\in L^{2}(\Omega)$, the
discrete problem (3.8) has a unique solution $u_{h}\in V_{h}$ and

$\displaystyle|u_{h}|_{1,\mathrm{pw}}\leq
C_{\mathrm{stab}}\|f_{h}\|_{V_{h}^{*}}.$

###### Proof.

In the first part of the proof, suppose there exists some solution $u_{h}\in
V_{h}$ to the discrete problem (3.8) for some $f\in L^{2}(\Omega)$. (This is
certainly true for all $f\equiv 0\equiv u_{h}$, but will be discussed for all
those pairs at the end of the proof and shall lead to the uniqueness of
discrete solutions.) Since $u_{h}$ satisfies a G$\mathring{a}$rding-type
inequality in Proposition 3.1.b,

$\displaystyle\alpha|u_{h}|_{1,\mathrm{pw}}^{2}$
$\displaystyle\leq\beta\|u_{h}\|^{2}_{L^{2}(\Omega)}+B_{h}(u_{h},u_{h})=\beta\|u_{h}\|^{2}_{L^{2}(\Omega)}+(f_{h},u_{h})_{L^{2}(\Omega)}.$

This, (2.14), and the definition of the dual norm in (3.12) lead to

$\displaystyle\alpha|u_{h}|_{1,\mathrm{pw}}\leq\beta
C_{\text{F}}\|u_{h}\|_{L^{2}(\Omega)}+\|f_{h}\|_{V_{h}^{*}}.$ (4.1)

Given $g:=u_{h}\in L^{2}(\Omega)$, let $\Phi\in V\cap H^{1+\sigma}(\Omega)$
solve the dual problem ${\cal L}^{*}\Phi=g$ and let $I_{h}\Phi\in V_{h}$ be
the interpolation of $\Phi$ from Subsection 2.4. Elementary algebra shows

$\displaystyle\|u_{h}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=\Big{(}(g,u_{h})_{L^{2}(\Omega)}-B_{\mathrm{pw}}(u_{h},\Phi)\Big{)}+B_{\mathrm{pw}}(u_{h},\Phi-
I_{h}\Phi)$
$\displaystyle\quad+\Big{(}B_{\mathrm{pw}}(u_{h},I_{h}\Phi)-B_{h}(u_{h},I_{h}\Phi)\Big{)}+(f_{h},I_{h}\Phi)_{L^{2}(\Omega)}.$
(4.2)

Rewrite a part of the third term corresponding to diffusion on the right-hand
side of (4.2) as

$\displaystyle
a^{P}(u_{h},I_{h}\Phi)-a_{h}^{P}(u_{h},I_{h}\Phi)=(\textbf{A}\nabla
u_{h},\nabla(1-\Pi_{1})I_{h}\Phi)_{L^{2}(P)}$
$\displaystyle+(\nabla(1-\Pi_{1})u_{h},(\textbf{A}-\Pi_{0}\textbf{A})(\nabla\Pi_{1}I_{h}\Phi))_{L^{2}(P)}-S^{P}\big{(}(1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Phi\big{)}.$

The Cauchy-Schwarz inequality in the semi-scalar product
$S^{P}(\bullet,\bullet)$, and (3.5) with the upper bound
$\|\textbf{A}\|_{\infty}$ for the coefficient A in $a^{P}(\bullet,\bullet)$
lead to the estimate

$\displaystyle C_{s}^{-1}$ $\displaystyle
S^{P}\big{(}(1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Phi\big{)}\leq|(1-\Pi_{1})u_{h}|_{1,P}|(1-\Pi_{1})I_{h}\Phi|_{1,P}$
$\displaystyle\qquad\leq\|\textbf{A}\|_{\infty}|u_{h}|_{1,P}\Big{(}\|\nabla(I_{h}\Phi-\Phi)\|_{L^{2}(P)}+\|\nabla(1-\Pi_{1}I_{h})\Phi\|_{L^{2}(P)}\Big{)}$
$\displaystyle\qquad\leq\|\textbf{A}\|_{\infty}C_{\text{apx}}\Big{(}2+C_{\mathrm{PF}}+C_{\text{Itn}}\Big{)}h_{P}^{\sigma}|u_{h}|_{1,P}|\Phi|_{1+\sigma,P}$
(4.3)

with Theorem 2.8.b followed by (2.12) in the final step. This and Theorem 2.8
imply that

$\displaystyle|a^{P}(u_{h},I_{h}\Phi)-a_{h}^{P}(u_{h},I_{h}\Phi)|$
$\displaystyle\leq h_{P}^{\sigma}|u_{h}|_{1,P}\|\Phi\|_{1+\sigma,P}$
$\displaystyle\quad\times\Big{(}\|\textbf{A}\|_{\infty}C_{\text{apx}}(2+C_{\mathrm{PF}}+C_{\text{Itn}})(1+C_{s})+|\textbf{A}|_{1,\infty}C_{\text{Itn}}\Big{)}.$

The terms $b^{P}-b_{h}^{P}$ and $c^{P}-c_{h}^{P}$ are controlled by

$\displaystyle|b^{P}(u_{h},I_{h}\Phi)-b_{h}^{P}(u_{h},I_{h}\Phi)|+|c^{P}(u_{h},I_{h}\Phi)-c_{h}^{P}(u_{h},I_{h}\Phi)|$
$\displaystyle\leq
h_{P}^{\sigma}\|\Phi\|_{1+\sigma,P}\big{(}\|\textbf{b}\|_{\infty}(C_{\text{apx}}(2+C_{\mathrm{PF}}+C_{\text{Itn}})\|u_{h}\|_{L^{2}(P)}+C_{\mathrm{Itn}}C_{\mathrm{PF}}|u_{h}|_{1,P})$
$\displaystyle\hskip
71.13188pt+\|\gamma\|_{\infty}C_{\mathrm{PF}}(C_{\text{Itn}}\|u_{h}\|_{L^{2}(P)}+|u_{h}|_{1,P})\big{)}.$

The combination of the previous four displayed estimates with Lemma 2.6 leads
to an estimate for $P$. The sum over all polygonal domains $P\in\mathcal{T}$
reads

$\displaystyle B_{\mathrm{pw}}(u_{h},I_{h}\Phi)-B_{h}(u_{h},I_{h}\Phi)\leq
C_{d}h_{\text{max}}^{\sigma}|u_{h}|_{1,\mathrm{pw}}\|\Phi\|_{1+\sigma,\Omega}$
(4.4)

with a universal constant $C_{d}$. The bound for (4.2) results from Lemma
3.4.b for the first term, the boundedness of $B_{\mathrm{pw}}$ (with a
universal constant
$M_{b}:=\|\textbf{A}\|_{\infty}+C_{\mathrm{F}}\|\textbf{b}\|_{\infty}+C_{\mathrm{F}}^{2}\|\gamma\|_{\infty}$)
and (2.15) for the second term, (4.4) for the third term, and Theorem 2.8.a
for the last term on the right-hand side of (4.2). This shows

$\displaystyle\|u_{h}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq\Big{(}C^{*}_{\mathrm{NC}}+C_{\text{I}}M_{b}+C_{d}\Big{)}h_{\text{max}}^{\sigma}|u_{h}|_{1,\mathrm{pw}}\|\Phi\|_{1+\sigma,\Omega}+C_{\text{Itn}}\|f_{h}\|_{V_{h}^{*}}\|\Phi\|_{1,\Omega}.$

This and the regularity estimate (1.5) lead to
$C_{3}=C^{*}_{\mathrm{NC}}+C_{\text{I}}M_{b}+C_{d}$ in

$\displaystyle\|u_{h}\|_{L^{2}(\Omega)}\leq
C_{3}\,C^{*}_{\text{reg}}h_{\text{max}}^{\sigma}|u_{h}|_{1,\mathrm{pw}}+C_{\text{Itn}}\|f_{h}\|_{V_{h}^{*}}.$

The substitution of this in (4.1) proves

$\displaystyle\alpha|u_{h}|_{1,\mathrm{pw}}\leq\beta
C_{\text{F}}C_{3}C^{*}_{\text{reg}}h_{\text{max}}^{\sigma}|u_{h}|_{1,\mathrm{pw}}+(\beta
C_{\text{F}}C_{\text{Itn}}+1)\|f_{h}\|_{V_{h}^{*}}.$ (4.5)

For all $0<h_{\text{max}}\leq\delta:=(\frac{\alpha}{2\beta
C_{\text{F}}C_{3}C^{*}_{\text{reg}}})^{1/\sigma}$, the constant
$\overline{c}=(1-\frac{\beta}{\alpha}C_{\text{F}}C_{3}C^{*}_{\text{reg}}h_{\text{max}}^{\sigma})$
is positive and $C_{\text{stab}}:=\frac{\beta
C_{\text{F}}C_{\text{Itn}}+1}{\alpha-\beta
C_{\mathrm{F}}C_{3}C^{*}_{\text{reg}}h_{0}^{\sigma}}$ is well-defined. This
leads in (4.5) to

$\displaystyle|u_{h}|_{1,\mathrm{pw}}\leq
C_{\text{stab}}\|f_{h}\|_{V_{h}^{*}}.$ (4.6)

In the last part of the proof, suppose $f_{h}\equiv 0$ and let $u_{h}$ be any
solution to the resulting homogeneous linear discrete system. The stability
result (4.6) proves $u_{h}\equiv 0$. Hence, the linear system of equations
(3.8) has a unique solution and the coefficient matrix is regular. This proves
that there exists a unique solution $u_{h}$ to (3.8) for any right-hand side
$f_{h}\in V_{h}^{*}$. The combination of this with (4.6) concludes the proof.
∎

An immediate consequence of Theorem 4.1 is the following discrete inf-sup
estimate.

######  Theorem 4.2 (discrete inf-sup).

There exist $0<\delta\leq 1$ and $\overline{\beta}_{0}>0$ such that, for all
$\mathcal{T}\in\mathbb{T}(\delta)$,

$\displaystyle\overline{\beta}_{0}\leq\inf_{0\neq u_{h}\in V_{h}}\sup_{0\neq
v_{h}\in
V_{h}}\frac{B_{h}(u_{h},v_{h})}{|u_{h}|_{1,\mathrm{pw}}|v_{h}|_{1,\mathrm{pw}}}.$
(4.7)

###### Proof.

Define the operator ${\cal L}_{h}:V_{h}\to V_{h}^{*},$ $v_{h}\mapsto
B_{h}(v_{h},\bullet)$. The stability Theorem 4.1 can be interpreted as
follows: For any $f_{h}\in V_{h}^{*}$ there exists $u_{h}\in V_{h}$ such that
${\cal L}_{h}u_{h}=f_{h}$ and

$\displaystyle\overline{\beta}_{0}|u_{h}|_{1,\mathrm{pw}}\leq\|f_{h}\|_{V_{h}^{*}}=\sup_{0\neq
v_{h}\in V_{h}}\frac{(f_{h},v_{h})}{|v_{h}|_{1,\mathrm{pw}}}=\sup_{0\neq
v_{h}\in V_{h}}\frac{B_{h}(u_{h},v_{h})}{|v_{h}|_{1,\mathrm{pw}}}.$

The discrete problem $B_{h}(u_{h},\bullet)=(f_{h},\bullet)$ has a unique
solution in $V_{h}$. Therefore, $f_{h}$ and $u_{h}$ are in one to one
correspondence and the last displayed estimate holds for any $u_{h}\in V_{h}$.
The infimum over $u_{h}\in V_{h}$ therein proves (4.7) with
$\overline{\beta}_{0}=C_{\text{stab}}^{-1}$. ∎

###  4.2  A priori error estimates

This subsection establishes the error estimate in the energy norm
$|\cdot|_{1,\mathrm{pw}}$ and in the $L^{2}$ norm. The discrete inf-sup
condition allows for an error estimate in the $H^{1}$ norm and an Aubin-
Nitsche duality argument leads to an error estimate in the $L^{2}$ norm.

Recall $u\in H^{1}_{0}(\Omega)$ is a unique solution of (1.8) and $u_{h}\in
V_{h}$ is a unique solution of (3.8). Recall the definition of the bilinear
form $s_{h}(\cdot,\cdot)$ from Section 3.1 and define the induced seminorm
$|v_{h}|_{\mathrm{s}}:=s_{h}(v_{h},v_{h})^{1/2}$ for $v_{h}\in V_{h}$ as a
part of the norm $\|\cdot\|_{h}$ from Remark 6.

######  Theorem 4.3 (error estimate).

Set $\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u\in H(\text{div},\Omega)$.
There exist positive constants $C_{4},C_{5},$ and $\delta$ such that, for all
$\mathcal{T}\in\mathbb{T}(\delta)$, the discrete problem (3.8) has a unique
solution $u_{h}\in V_{h}$ and

$\displaystyle|u-u_{h}|_{1,\mathrm{pw}}+|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+h_{\mathrm{max}}^{-\sigma}(\|u-u_{h}\|_{L^{2}(\Omega)}+\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)})+|u_{h}|_{\mathrm{s}}+|I_{h}u-u_{h}|_{\mathrm{s}}$
$\displaystyle\quad\leq
C_{4}\Big{(}\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)}+\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})\Big{)}\leq
C_{5}h_{\mathrm{max}}^{\sigma}\|f\|_{L^{2}(\Omega)}.$ (4.8)

###### Proof.

Step 1 (initialization). Let $I_{h}u\in V_{h}$ be the interpolation of $u$
from Definition 2.7. The discrete inf-sup condition (4.7) for $I_{h}u-u_{h}\in
V_{h}$ leads to some $v_{h}\in V_{h}$ with $|v_{h}|_{1,\mathrm{pw}}\leq 1$
such that

$\displaystyle\overline{\beta}_{0}|I_{h}u-u_{h}|_{1,\mathrm{pw}}=B_{h}(I_{h}u-u_{h},v_{h}).$

Step 2 (error estimate for $|u-u_{h}|_{1,\mathrm{pw}}$). Rewrite the last
equation with the continuous and the discrete problem (1.8) and (3.8) as

$\displaystyle\overline{\beta}_{0}|I_{h}u-u_{h}|_{1,\mathrm{pw}}=B_{h}(I_{h}u,v_{h})-B(u,v)+(f,v)_{L^{2}(\Omega)}-(f_{h},v_{h})_{L^{2}(\Omega)}.$

This equality is rewritten with the definition of $B(u,v)$ in (1.7), the
definition of $B_{h}(I_{h}u,v_{h})$ in Section 3.1, and with $f_{h}=\Pi_{1}f$.
Recall $v:=Jv_{h}\in V$ from Lemma 3.3 and recall
$\nabla_{\mathrm{pw}}\Pi_{1}I_{h}u=\Pi_{0}\nabla u$ from (2.17). This results
in

LHS
$\displaystyle:=\overline{\beta}_{0}|I_{h}u-u_{h}|_{1,\mathrm{pw}}-s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})v_{h})$
$\displaystyle=(\textbf{A}\Pi_{0}\nabla
u+\textbf{b}\Pi_{1}I_{h}u,\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+(\gamma\Pi_{1}I_{h}u,\Pi_{1}v_{h})_{L^{2}(\Omega)}-(\bm{\sigma},\nabla
v)_{L^{2}(\Omega)}$ $\displaystyle\qquad+(f-\gamma
u,v)_{L^{2}(\Omega)}-(f,\Pi_{1}v_{h})_{L^{2}(\Omega)}.$

Abbreviate $w:=v-\Pi_{1}v_{h}$ and observe the orthogonalities
$\nabla_{\mathrm{pw}}w\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ in
$L^{2}(\Omega;\mathbb{R}^{2})$ and $w\perp\mathcal{P}_{1}(\mathcal{T})$ in
$L^{2}(\Omega)$ from Lemma 3.3.b-c and the definition of $\Pi_{1}$ with
$\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$. Lemma 3.3.d, the bound
$|(1-\Pi^{\nabla}_{1})v_{h}|_{1,\mathrm{pw}}\leq|v_{h}|_{1,\mathrm{pw}}\leq
1$, and the Poincaré-Friedrichs inequality for $v_{h}-\Pi^{\nabla}_{1}v_{h}$
from Lemma 2.1.a lead to

$\displaystyle|w|_{1,\mathrm{pw}}$
$\displaystyle\leq|v-v_{h}|_{1,\mathrm{pw}}+|v_{h}-\Pi_{1}v_{h}|_{1,\mathrm{pw}}\leq
C_{\mathrm{J}}+1,$ (4.9)
$\displaystyle\|h_{\mathcal{T}}^{-1}w\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|h_{\mathcal{T}}^{-1}(v-v_{h})\|_{L^{2}(\Omega)}+\|h_{\mathcal{T}}^{-1}(v_{h}-\Pi_{1}v_{h})\|_{L^{2}(\Omega)}\leq
C_{\mathrm{J}}+C_{\mathrm{PF}}.$ (4.10)

Elementary algebra and the above orthogonalities prove that

LHS $\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\Pi_{0}-1)\nabla
u+\textbf{b}(\Pi_{1}I_{h}u-u),\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}-((1-\Pi_{0})\bm{\sigma},\nabla_{\mathrm{pw}}w)_{L^{2}(\Omega)}$
$\displaystyle\qquad+(\gamma(\Pi_{1}I_{h}u-u),\Pi_{1}v_{h})_{L^{2}(\Omega)}+(h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma
u),h_{\mathcal{T}}^{-1}w)_{L^{2}(\Omega)}$
$\displaystyle\leq\Big{(}|\textbf{A}|_{1,\infty}+(1+C_{\mathrm{PF}})(\|\textbf{b}\|_{\infty}+C_{\mathrm{F}}\|\gamma\|_{\infty})\Big{)}h_{\text{max}}\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}$
$\displaystyle\quad+(C_{\mathrm{J}}+1)\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)}+(C_{\mathrm{J}}+C_{\text{PF}})\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})$ (4.11)

with the Lipschitz continuity of A, Lemma 2.8.b, the stabilities of $\Pi_{1}$
from (2.11), and (4.9)-(4.10) in the last step. The definition of stability
term (3.5) and Theorem 2.8.b lead to

$\displaystyle C_{s}^{-1}s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})v_{h})$
$\displaystyle\leq\|\textbf{A}\|_{\infty}|(1-\Pi_{1})I_{h}u|_{1,\mathrm{pw}}|(1-\Pi_{1})v_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq\|\textbf{A}\|_{\infty}(|I_{h}u-u|_{1,\mathrm{pw}}+|u-\Pi_{1}I_{h}u|_{1,\mathrm{pw}})|v_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq\|\textbf{A}\|_{\infty}(2+C_{\text{Itn}}+C_{\mathrm{PF}})\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}|v_{h}|_{1,\mathrm{pw}}.$ (4.12)

The triangle inequality, the bound (2.15) for the term
$|u-I_{h}u|_{1,\mathrm{pw}}$, and (4.11)-(4.12) for the term
$|I_{h}u-u_{h}|_{1,\mathrm{pw}}$ conclude the proof of (4.8) for the term
$|u-u_{h}|_{1,{\mathrm{pw}}}$.
Step $3$ (duality argument). To prove the bound for $u-u_{h}$ in the $L^{2}$
norm with a duality technique, let $g:=I_{h}u-u_{h}\in L^{2}(\Omega)$. The
solution $\Phi\in H^{1}_{0}(\Omega)\cap H^{1+\sigma}(\Omega)$ to the dual
problem (1.4) satisfies the elliptic regularity (1.5),

$\displaystyle\|\Phi\|_{1+\sigma,\Omega}\leq
C^{*}_{\text{reg}}\|I_{h}u-u_{h}\|_{L^{2}(\Omega)}.$ (4.13)

Step $4$ (error estimate for $\|u-u_{h}\|_{L^{2}(\Omega)}$). Let $I_{h}\Phi\in
V_{h}$ be the interpolation of $\Phi$ from Definition 2.7. Elementary algebra
reveals the identity

$\displaystyle\|g\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=((g,g)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(g,\Phi))+B_{\mathrm{pw}}(g,\Phi-
I_{h}\Phi)$
$\displaystyle\quad+(B_{\mathrm{pw}}(g,I_{h}\Phi)-B_{h}(g,I_{h}\Phi))+B_{h}(g,I_{h}\Phi).$
(4.14)

The bound (4.4) with $g$ as the first argument shows

$\displaystyle B_{\mathrm{pw}}(g,I_{h}\Phi)-B_{h}(g,I_{h}\Phi)\leq
C_{d}h_{\text{max}}^{\sigma}|g|_{1,\mathrm{pw}}\|\Phi\|_{1+\sigma,\Omega}.$

This controls the third term in (4.14), Lemma 3.4.b controls the first term,
the boundedness of $B_{\mathrm{pw}}$ and the interpolation error estimate
(2.15) control the second term on the right-hand side of (4.14). This results
in

$\displaystyle\|I_{h}u-u_{h}\|^{2}_{L^{2}(\Omega)}\leq(C^{*}_{\mathrm{NC}}+C_{\mathrm{I}}M_{b}+C_{d})h_{\mathrm{max}}^{\sigma}|g|_{1,\mathrm{pw}}\|\Phi\|_{1+\sigma,\Omega}+B_{h}(g,I_{h}\Phi).$
(4.15)

It remains to bound $B_{h}(g,I_{h}\Phi)$. The continuous and the discrete
problem (1.8) and (3.8) imply

$\displaystyle
B_{h}(g,I_{h}\Phi)=B_{h}(I_{h}u,I_{h}\Phi)-B(u,\Phi)+(f,\Phi)_{L^{2}(\Omega)}-(f_{h},I_{h}\Phi)_{L^{2}(\Omega)}.$

The definition of $B_{h}$ and $\Pi_{0}$ lead to

$\displaystyle
B_{h}(g,I_{h}\Phi)-s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})I_{h}\Phi)$
$\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\Pi_{0}-1)\nabla
u+\textbf{b}(\Pi_{1}I_{h}u-u),\nabla_{\mathrm{pw}}\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}+(\gamma(\Pi_{1}I_{h}u-u),\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}$
$\displaystyle\qquad-((1-\Pi_{0})\bm{\sigma},\nabla_{\mathrm{pw}}(1-\Pi_{1}I_{h})\Phi)_{L^{2}(\Omega)}+(f-\gamma
u,\Phi-\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}.$ (4.16)

The bound for the stability term as in (4.12) is

$\displaystyle s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})I_{h}\Phi)$
$\displaystyle\leq
C_{s}\|\textbf{A}\|_{\infty}|(1-\Pi_{1})I_{h}u|_{1,\mathrm{pw}}|(1-\Pi_{1})I_{h}\Phi|_{1,\mathrm{pw}}$
$\displaystyle\leq
C_{s}\|\textbf{A}\|_{\infty}(2+C_{\mathrm{Itn}}+C_{\mathrm{PF}})^{2}C_{\text{apx}}h_{\text{max}}^{\sigma}\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}|\Phi|_{1+\sigma,\Omega}.$ (4.17)

Step $5$ (oscillation). The last term in (4.16) is of optimal order
$O(h_{\text{max}}^{1+\sigma})$, but the following arguments allow to write it
as an oscillation. Recall the bubble-function $b_{\mathcal{T}}|_{P}:=b_{P}\in
H^{1}_{0}(P)$ from (3.17) extended by zero outside $P$. Given
$\Psi:=\Phi-\Pi_{1}I_{h}\Phi$, let $\Psi_{1}\in\mathcal{P}_{1}(\mathcal{T})$
be the Riesz representation of the linear functional
$\mathcal{P}_{1}(\mathcal{T})\to\mathbb{R}$ defined by
$w_{1}\mapsto(\Psi,w_{1})_{L^{2}(\Omega)}$ in the Hilbert space
$\mathcal{P}_{1}(\mathcal{T})$ endowed with the weighted scalar product
$(b_{\mathcal{T}}\bullet,\bullet)_{L^{2}(\Omega)}$. That means
$\Pi_{1}(b_{\mathcal{T}}\Psi_{1})=\Pi_{1}\Psi$. The identity $(f-\gamma
u,b_{\mathcal{T}}\Psi_{1})_{L^{2}(\Omega)}=(\bm{\sigma},\nabla(b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}$
follows from (1.8) with the test function $b_{\mathcal{T}}\Psi_{1}\in
H^{1}_{0}(\Omega)$. The $L^{2}$ orthogonalities $\Psi-
b_{\mathcal{T}}\Psi_{1}\perp\mathcal{P}_{1}(\mathcal{T})$ in $L^{2}(\Omega)$
and
$\nabla(b_{\mathcal{T}}\Psi_{1})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$
in $L^{2}(\Omega;\mathbb{R}^{2})$ allow the rewriting of the latter identity
as

$\displaystyle(f-\gamma
u,\Psi)_{L^{2}(\Omega)}=(h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma
u),h_{\mathcal{T}}^{-1}(\Psi-
b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}+((1-\Pi_{0})\bm{\sigma},\nabla(b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}$
$\displaystyle\qquad\leq\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})\|h_{\mathcal{T}}^{-1}(\Psi-
b_{\mathcal{T}}\Psi_{1})\|_{L^{2}(\Omega)}+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)}|b_{\mathcal{T}}\Psi|_{1,\mathrm{pw}}.$
(4.18)

It remains to control the terms $\|h_{\mathcal{T}}^{-1}(\Psi-
b_{\mathcal{T}}\Psi_{1})\|_{L^{2}(\Omega)}$ and
$|b_{\mathcal{T}}\Psi|_{1,\mathrm{pw}}$. Since the definition of $I_{h}$ and
the definition of $\Pi^{\nabla}_{1}$ with $\Pi_{1}=\Pi^{\nabla}_{1}$ in
$V_{h}$ imply $\int_{\partial P}\Psi\,ds=\int_{\partial
P}(\Phi-\Pi_{1}I_{h}\Phi)\,ds=0$, this allows the Poincaré-Friedrichs
inequality for $\Psi$ from Lemma 2.1.a on each $P\in\mathcal{T}$. This shows

$\displaystyle\|h_{\mathcal{T}}^{-1}\Psi\|_{L^{2}(\Omega)}\leq
C_{\mathrm{PF}}|\Psi|_{1,\mathrm{pw}}\leq
C_{\mathrm{PF}}C_{\text{apx}}h_{\text{max}}^{\sigma}|\Phi|_{1+\sigma,\Omega}$
(4.19)

with Theorem 2.8.b and (2.12) in the last inequality. Since $b_{P}\Psi_{1}\in
H^{1}_{0}(P)$ for $P\in\mathcal{T}$, the Poincaré-Friedrichs inequality from
Lemma 2.1.a leads to

$\displaystyle\|h_{P}^{-1}(b_{P}\Psi_{1})\|_{L^{2}(P)}\leq
C_{\mathrm{PF}}|b_{P}\Psi_{1}|_{1,P}.$ (4.20)

The first estimate in (3.20), the identity
$\Pi_{1}(b_{\mathcal{T}}\Psi_{1})=\Pi_{1}\Psi$, and the Cauchy-Schwarz
inequality imply

$\displaystyle
C_{b}^{-1}\|h_{P}^{-1}\Psi_{1}\|_{L^{2}(P)}^{2}\leq\|h_{P}^{-1}b_{P}^{1/2}\Psi_{1}\|_{L^{2}(P)}^{2}=(h_{P}^{-1}\Psi_{1},h_{P}^{-1}\Psi)_{L^{2}(P)}\leq\|h_{P}^{-1}\Psi_{1}\|_{L^{2}(P)}\|h_{P}^{-1}\Psi\|_{L^{2}(P)}.$

This proves $\|h_{P}^{-1}\Psi_{1}\|_{L^{2}(P)}\leq
C_{b}\|h_{P}^{-1}\Psi\|_{L^{2}(P)}$. The second estimate in (3.21) followed by
the first estimate in (3.20) leads to the first inequality and the arguments
as above lead to the second inequality in

$\displaystyle
C_{b}^{-3/2}|b_{P}\Psi_{1}|_{1,P}\leq\|h_{P}^{-1}b_{P}^{1/2}\Psi_{1}\|_{L^{2}(P)}\leq\|h_{P}^{-1}\Psi_{1}\|_{L^{2}(P)}^{1/2}\|h_{P}^{-1}\Psi\|_{L^{2}(P)}^{1/2}$
$\displaystyle\leq C_{b}^{1/2}\|h_{P}^{-1}\Psi\|_{L^{2}(P)}$

with $\|h_{P}^{-1}\Psi_{1}\|_{L^{2}(P)}^{1/2}\leq
C_{b}^{1/2}\|h_{P}^{-1}\Psi\|_{L^{2}(P)}^{1/2}$ from above in the last step.
The combination of the previous displayed estimate and (4.18)-(4.20) results
with $C_{6}:=C_{\mathrm{PF}}C_{\text{apx}}(1+C_{b}^{2}(1+C_{\mathrm{PF}}))$ in

$\displaystyle(f-\gamma u,\Psi)_{L^{2}(\Omega)}\leq
C_{6}(\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)})h_{\text{max}}^{\sigma}|\Phi|_{1+\sigma,\Omega}.$
(4.21)

Step $6$ (continued proof of estimate for $\|u-u_{h}\|_{L^{2}(\Omega)}$). The
estimate in Step 2 for $|g|_{1,\mathrm{pw}}$, (4.15)-(4.17), and (4.21) with
the regularity (4.13) show

$\displaystyle\|I_{h}u-u_{h}\|_{L^{2}(\Omega)}\lesssim
h_{\mathrm{max}}^{\sigma}\Big{(}\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})\Big{)}.$ (4.22)

Rewrite the difference $u-u_{h}=(u-I_{h}u)+(I_{h}u-u_{h})$, and apply the
triangle inequality with (2.15) for the first term

$\|u-I_{h}u\|_{L^{2}(\Omega)}\leq
C_{\mathrm{I}}h_{\text{max}}^{1+\sigma}|u|_{1+\sigma,\Omega}.$

This and (4.22) for the second term $I_{h}u-u_{h}$ conclude the proof of the
estimate for the term $h_{\text{max}}^{-\sigma}\|u-u_{h}\|_{L^{2}(\Omega)}$ in
(4.8) .
Step $7$ (stabilisation error $|u_{h}|_{\mathrm{s}}$ and
$|I_{h}u-u_{h}|_{\mathrm{s}}$). The triangle inequality and the upper bound of
the stability term (3.5) lead to

$\displaystyle|u_{h}|_{\mathrm{s}}\leq|I_{h}u-u_{h}|_{\mathrm{s}}+|I_{h}u|_{\mathrm{s}}\leq
C_{s}^{1/2}\|\textbf{A}\|_{\infty}^{1/2}(|I_{h}u-u_{h}|_{1,\mathrm{pw}}+|(1-\Pi_{1})I_{h}u|_{1,\mathrm{pw}})$

with
$|(1-\Pi_{1})(I_{h}u-u_{h})|_{1,\mathrm{pw}}\leq|I_{h}u-u_{h}|_{1,\mathrm{pw}}$
in the last inequality. The arguments as in (4.12) prove that
$|(1-\Pi_{1})I_{h}u|_{1,\mathrm{pw}}\leq(2+C_{\text{Itn}}+C_{\mathrm{PF}})\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}$. This and the arguments in Step 2 for the estimate of
$|I_{h}u-u_{h}|_{1,\mathrm{pw}}$ show the upper bound in (4.8) for the terms
$|u_{h}|_{\mathrm{s}}$ and $|I_{h}u-u_{h}|_{\mathrm{s}}$.
Step $8$ (error estimate for $u-\Pi_{1}u_{h}$). The VEM solution $u_{h}$ is
defined by the computed degrees of freedom given in (2.10), but the evaluation
of the function itself requires expansive additional calculations. The later
are avoided if $u_{h}$ is replaced by the Ritz projection $\Pi_{1}u_{h}$ in
the numerical experiments. The triangle inequality leads to

$\displaystyle|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}\leq|u-u_{h}|_{1,\mathrm{pw}}+|u_{h}-\Pi_{1}u_{h}|_{1,\mathrm{pw}}.$
(4.23)

A lower bound of the stability term (3.5) and the assumption (A2) imply

$\displaystyle|u_{h}-\Pi_{1}u_{h}|_{1,P}\leq
a_{0}^{-1/2}C_{s}^{1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}.$ (4.24)

This shows that the second term in (4.23) is bounded by
$|u_{h}|_{\mathrm{s}}$. Hence Step 2 and Step 7 prove the estimate for
$|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}$. Since $\int_{\partial
P}(u_{h}-\Pi_{1}u_{h})\,ds=0$ from the definition of $\Pi^{\nabla}_{1}$ and
$\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$, the combination of Poincaré-Friedrichs
inequality for $u_{h}-\Pi_{1}u_{h}$ from Lemma 2.1.a and (4.24) result in

$\displaystyle
C_{\mathrm{PF}}^{-1}a_{0}^{1/2}C_{s}^{-1/2}\|u_{h}-\Pi_{1}u_{h}\|_{L^{2}(P)}\leq
h_{P}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}.$ (4.25)

The analogous arguments for $\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}$, (4.25), and
the estimate for $|u_{h}|_{\mathrm{s}}$ prove the bound (4.8) for the term
$h_{\text{max}}^{-\sigma}\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}$. This concludes
the proof of Theorem 4.3. ∎

##  5 A posteriori error analysis

This section presents the reliability and efficiency of a residual-type a
posteriori error estimator.

###  5.1 Residual-based explicit a posteriori error control

Recall $u_{h}\in V_{h}$ is the solution to the problem (3.8), and the
definition of jump $[\cdot]_{E}$ along an edge $E\in\mathcal{E}$ from Section
2. For any polygonal domain $P\in\mathcal{T}$, set

$\displaystyle\eta_{P}^{2}:=h_{P}^{2}\|f-\gamma\Pi_{1}u_{h}\|_{L^{2}(P)}^{2}$ | (volume residual),
---|---
$\displaystyle\zeta_{P}^{2}:=S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})$ | (stabilization),
$\displaystyle\Lambda_{P}^{2}:=\|(1-\Pi_{0})(\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h})\|_{L^{2}(P)}^{2}$ | (inconsistency),
$\displaystyle\Xi_{P}^{2}:=\sum_{E\in\mathcal{E}(P)}|E|^{-1}\|[\Pi_{1}u_{h}]_{E}\|_{L^{2}(E)}^{2}$ | (nonconformity).

These local quantities $\bullet|_{P}$ form a family
($\bullet|_{P}:P\in\mathcal{T}$) over the index set $\mathcal{T}$ and their
Euclid vector norm $\bullet|_{\mathcal{T}}$ enters the upper error bound:
$\eta_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\eta_{P}^{2})^{1/2}$,
$\zeta_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\zeta_{P}^{2})^{1/2}$,
$\Lambda_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\Lambda_{P}^{2})^{1/2}$, and
$\Xi_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\Xi_{P}^{2})^{1/2}$. The following
theorem provides an upper bound to the error $u-u_{h}$ in the $H^{1}$ and the
$L^{2}$ norm. Recall the elliptic regularity (1.5) with the index
$0<\sigma\leq 1$, and recall the assumption $h_{\text{max}}\leq 1$ from
Subsection 2.1.

######  Theorem 5.1 (reliability).

There exist positive constants $C_{\text{rel}1}$ and $C_{\text{rel}2}$ (both
depending on $\rho$) such that

$\displaystyle
C_{\mathrm{rel}1}^{-2}|u-u_{h}|_{1,\mathrm{pw}}^{2}\leq\eta_{\mathcal{T}}^{2}+\zeta_{\mathcal{T}}^{2}+\Lambda_{\mathcal{T}}^{2}+\Xi_{\mathcal{T}}^{2}$
(5.1)

and

$\displaystyle\|u-u_{h}\|_{L^{2}(\Omega)}^{2}\leq
C_{\mathrm{rel}2}^{2}\sum_{P\in\mathcal{T}}\Big{(}h_{P}^{2\sigma}(\eta_{P}^{2}+\zeta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2})\Big{)}.$
(5.2)

The proof of this theorem in Subsection 5.3 relies on a conforming companion
operator elaborated in the next subsection. The upper bound in Theorem 5.1 is
efficient in the following local sense, where
$\omega_{E}:=\textrm{int}(\cup\mathcal{T}(E))$ denotes the patch of an edge
$E$ and consists of the one or the two neighbouring polygons in the set
$\mathcal{T}(E):=\\{P^{\prime}\in\mathcal{T}:E\subset\partial P^{\prime}\\}$
that share $E$. Recall $\bm{\sigma}=\textbf{A}\nabla u+\textbf{b}u$ from
Subsection 4.2 and the data-oscillation
$\mathrm{osc}_{1}(f,P):=\|h_{P}(1-\Pi_{1})f\|_{L^{2}(P)}$ from Subsection 2.1.

######  Theorem 5.2 (local efficiency up to oscillation).

The quantities $\eta_{P},\zeta_{P},\Lambda_{P},$ and $\Xi_{P}$ from Theorem
5.1 satisfy

$\displaystyle\zeta^{2}_{P}$
$\displaystyle\lesssim|u-u_{h}|^{2}_{1,P}+|u-\Pi_{1}u_{h}|^{2}_{1,P}$ (5.3)
$\displaystyle\eta_{P}^{2}$
$\displaystyle\lesssim\|u-u_{h}\|^{2}_{1,P}+|u-\Pi_{1}u_{h}|^{2}_{1,P}+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(P)}^{2}+\mathrm{osc}_{1}^{2}(f-\gamma
u,P),$ (5.4) $\displaystyle\Lambda_{P}^{2}$
$\displaystyle\lesssim\|u-u_{h}\|^{2}_{1,P}+|u-\Pi_{1}u_{h}|^{2}_{1,P}+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(P)}^{2},$
(5.5) $\displaystyle\Xi_{P}^{2}$
$\displaystyle\lesssim\sum_{E\in\mathcal{E}(P)}\sum_{P^{\prime}\in\omega_{E}}(\|u-u_{h}\|^{2}_{1,P^{\prime}}+|u-\Pi_{1}u_{h}|^{2}_{1,P^{\prime}}).$
(5.6)

The proof of Theorem 5.2 follows in Subsection 5.4. The reliability and
efficiency estimates in Theorem 5.1 and 5.2 lead to an equivalence up to the
approximation term

$\text{apx}:=\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T}).$

Recall the definition of $|u_{h}|_{\mathrm{s}}$ from Subsection 4.2. In this
paper, the norm $|\cdot|_{1,\mathrm{pw}}$ in the nonconforming space $V_{h}$
has been utilised for simplicity and one alternative is the norm
$\|\cdot\|_{h}$ from Remark 6 induced by $a_{h}$. Then it appears natural to
have the total error with the stabilisation term as

$\text{total
error}:=|u-u_{h}|_{1,\mathrm{pw}}+|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+h_{\text{max}}^{-\sigma}\|u-u_{h}\|_{L^{2}(\Omega)}+h_{\text{max}}^{-\sigma}\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}+|u_{h}|_{\mathrm{s}}.$

The point is that Theorem 4.3 assures that total error $+$ apx converges with
the expected optimal convergence rate.

######  Corollary 5.3 (equivalence).

The
$\mathrm{estimator}:=\eta_{\mathcal{T}}+\zeta_{\mathcal{T}}+\Lambda_{\mathcal{T}}+\Xi_{\mathcal{T}}\approx\mathrm{total\;error}+\mathrm{apx}$.

###### Proof.

∎Theorem 5.2 motivates apx and shows

$\mathrm{estimator}\lesssim\|u-u_{h}\|_{1,\mathrm{pw}}+\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma
u,\mathcal{T})+|u_{h}|_{\mathrm{s}}\leq\mathrm{total\;error}+\mathrm{apx}.$

This proves the first inequality $\lesssim$ in the assertion. Theorem 5.1, the
estimates in Subsection 5.3.3.1, and the definition of $|u_{h}|_{s}$ show
$\text{total error}\lesssim\text{estimator}$. The first of the terms in apx is

$\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\|_{L^{2}(\Omega)}\leq\|\bm{\sigma}-\Pi_{0}\bm{\sigma}_{h}\|_{L^{2}(\Omega)}\leq\|\bm{\sigma}-\bm{\sigma}_{h}\|_{L^{2}(\Omega)}+\|(1-\Pi_{0})\bm{\sigma}_{h}\|_{L^{2}(\Omega)}.$

The definition of $\bm{\sigma}$ and $\bm{\sigma}_{h}$ plus the triangle and
the Cauchy-Schwarz inequality show

$\displaystyle\|\bm{\sigma}-\bm{\sigma}_{h}\|_{L^{2}(\Omega)}\leq\|\textbf{A}\|_{\infty}|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+\|\textbf{b}\|_{\infty}\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}\lesssim\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}.$

The upper bound is $\lesssim$ estimator as mentioned above. Since the term
$\|(1-\Pi_{0})\bm{\sigma}_{h}\|_{L^{2}(\Omega)}=\Lambda_{\mathcal{T}}$ is a
part of the estimator,
$\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(\Omega)}\lesssim\mathrm{estimator}$. The
other term in apx is

$\displaystyle\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})$
$\displaystyle\leq\mathrm{osc}_{1}(f-\gamma\Pi_{1}u_{h},\mathcal{T})+\|h_{\mathcal{T}}\gamma(u-\Pi_{1}u_{h})\|_{L^{2}(\Omega)}$
$\displaystyle\leq\eta_{\mathcal{T}}+\|\gamma\|_{\infty}h_{\text{max}}\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}\lesssim\text{estimator}.\qed$

Section 5 establishes the a posteriori error analysis of the nonconforming
VEM. Related results are known for the conforming VEM and the nonconforming
FEM.

######  Remark 7 (comparison with nonconforming FEM).

Theorem 5.1 generalizes a result for the nonconforming FEM in [19, Thm. 3.4]
from triangulations into triangles to those in polygons (recall Example 2.2).
The only difference is the extra stabilization term that can be dropped in the
nonconforming FEM.

######  Remark 8 (comparison with conforming VEM).

The volume residual, the inconsistency term, and the stabilization also arise
in the a posteriori error estimator for the conforming VEM in [16, Thm. 13].
But it also includes an additional term with normal jumps compared to the
estimator (5.1). The extra nonconformity term in this paper is caused by the
nonconformity $V_{h}\not\subset V$ in general.

###  5.2 Enrichment and conforming companion operator

The link from the nonconforming approximation $u_{h}\in V_{h}$ to a global
Sobolev function in $H^{1}_{0}(\Omega)$ can be designed with the help of the
underlying refinement $\widehat{{\cal T}}$ of the triangulation $\mathcal{T}$
(from Section 2). The interpolation
$I_{\text{CR}}:V+V_{h}\to\textrm{CR}^{1}_{0}(\widehat{{\cal T}})$ in the
Crouzeix-Raviart finite element space $\textrm{CR}^{1}_{0}(\widehat{{\cal
T}})$ from Subsection 3.4 allows for a right-inverse $J^{\prime}$. A companion
operator $J^{\prime}\circ I_{\text{CR}}:V_{h}\to H^{1}_{0}(\Omega)$ acts as
displayed

$V_{h}$$\text{CR}^{1}_{0}(\widehat{{\cal
T}})$$H^{1}_{0}(\Omega)$$J^{\prime}$$I_{\text{CR}}$

Define an enrichment operator $E_{\mathrm{pw}}:\mathcal{P}_{1}(\widehat{{\cal
T}})\to S^{1}_{0}(\widehat{{\cal T}})$ by averaging nodal values: For any
vertex $z$ in the refined triangulation $\widehat{{\cal T}}$, let
$\widehat{{\cal T}}(z)=\\{T\in\widehat{{\cal T}}:z\in T\\}$ denote the set of
$|\widehat{{\cal T}}(z)|\geq 1$ many triangles that share the vertex $z$, and
define

$\displaystyle E_{\mathrm{pw}}v_{1}(z)=\frac{1}{|\widehat{{\cal
T}}(z)|}\sum_{T\in\widehat{{\cal T}}(z)}{v_{1}}|_{T}(z)$

for an interior vertex $z$ (and zero for a boundary vertex $z$ according to
the homogeneous boundary conditions). This defines $E_{\mathrm{pw}}v_{1}$ at
any vertex of a triangle $T$ in $\widehat{{\cal T}}$, and linear interpolation
then defines $E_{\mathrm{pw}}v_{1}$ in $T\in\widehat{{\cal T}}$, so that
$E_{\mathrm{pw}}v_{1}\in S^{1}_{0}(\widehat{{\cal T}})$. Huang et al. [31]
design an enrichment operator by an extension of [32] to polygonal domains,
while we deduce it from a sub-triangulation. The following lemma provides an
approximation property of the operator $E_{\mathrm{pw}}$.

######  Lemma 5.4.

There exists a positive constant $C_{En}$ that depends only on the shape
regularity of $\widehat{{\cal T}}$ such that any
$v_{1}\in\mathcal{P}_{1}(\mathcal{T})$ satisfies

$\displaystyle\|h_{\mathcal{T}}^{-1}(1-E_{\mathrm{pw}})v_{1}\|_{L^{2}(\Omega)}+|(1-E_{\mathrm{pw}})v_{1}|_{1,\mathrm{pw}}\leq
C_{\mathrm{En}}\left(\sum_{E\in{\mathcal{E}}}|E|^{-1}\|[v_{1}]_{E}\|_{L^{2}(E)}^{2}\right)^{1/2}.$
(5.7)

###### Proof.

There exists a positive constant $C_{En}$ independent of $h$ and $v_{1}$ [32,
p. 2378] such that

$\displaystyle\|h_{\widehat{{\cal
T}}}^{-1}(1-E_{\mathrm{pw}})v_{1}\|_{L^{2}(\Omega)}+\left(\sum_{T\in\widehat{{\cal
T}}}\|\nabla(1-E_{\mathrm{pw}})v_{1}\|_{L^{2}(T)}^{2}\right)^{1/2}\leq
C_{\mathrm{En}}\left(\sum_{E\in\widehat{\mathcal{E}}}|E|^{-1}\|[v_{1}]_{E}\|_{L^{2}(E)}^{2}\right)^{1/2}.$

Note that any edge $E\in\mathcal{E}$ is unrefined in the sub-triangulation
$\widehat{{\cal T}}$. Since $v_{1|P}\in H^{1}(P)$ is continuous in each
polygonal domain $P\in\mathcal{T}$ and $h_{T}\leq h_{P}$ for all
$T\in\widehat{{\cal T}}(P)$, the above inequality reduces to (5.7). This
concludes the proof. ∎

Recall the $L^{2}$ projection $\Pi_{1}$ onto the piecewise affine functions
$\mathcal{P}_{1}(\mathcal{T})$ from Section 2. An enrichment operator
$E_{\mathrm{pw}}\circ\Pi_{1}:V_{h}\to H^{1}_{0}(\Omega)$ acts as displayed

$V_{h}$$\mathcal{P}_{1}(\mathcal{T})\hookrightarrow\mathcal{P}_{1}(\widehat{{\cal
T}})$$H^{1}_{0}(\Omega)$$E_{\mathrm{pw}}$$\Pi_{1}$

###  5.3 Proof of Theorem 5.1

####  5.3.1 Reliable $H^{1}$ error control

Define $E_{1}u_{h}:=E_{\mathrm{pw}}\Pi_{1}u_{h}\in H^{1}_{0}(\Omega)$ so that
$u-E_{1}u_{h}\in H^{1}_{0}(\Omega)$. The inf-sup condition (1.9) leads to some
$v\in H^{1}_{0}(\Omega)$ with $\|v\|_{1,\Omega}\leq 1$ and

$\displaystyle\beta_{0}\|u-E_{1}u_{h}\|_{1,\Omega}=B(u-E_{1}u_{h},v)=((f,v)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},v))+B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{1}u_{h},v)$
(5.8)

with $B(u,v)=(f,v)$ from (1.8) and the piecewise version $B_{\mathrm{pw}}$ of
$B$ in the last step. The definition of $B_{h}$ from Subsection 3.1 and the
discrete problem (3.8) with $v_{h}=I_{h}v$ imply

$\displaystyle
B_{\mathrm{pw}}(\Pi_{1}u_{h},\Pi_{1}I_{h}v)+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)=B_{h}(u_{h},I_{h}v)=(f,\Pi_{1}I_{h}v)_{L^{2}(\Omega)}.$
(5.9)

Abbreviate $w:=v-\Pi_{1}I_{h}v$ and
$\bm{\sigma}_{h}:=\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$.
This and (5.9) simplify

$\displaystyle(f,v)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},v)=(f,w)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},w)+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$
$\displaystyle=(f-\gamma\Pi_{1}u_{h},w)_{L^{2}(\Omega)}-((1-\Pi_{0})\bm{\sigma}_{h},\nabla_{\mathrm{pw}}w)_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$
(5.10)

with $\int_{P}\nabla w\,dx=0$ for any $P\in\mathcal{T}$ from (2.17) in the
last step. Recall the notation $\eta_{P},\Lambda_{P}$, and $\zeta_{P}$ from
Subsection 5.1. The Cauchy-Schwarz inequality and Theorem 2.8.b followed by
$\|(1-\Pi_{0})\nabla v\|_{L^{2}(\Omega)}\leq|v|_{1,\Omega}\leq 1$ in the
second step show

$\displaystyle(f-\gamma\Pi_{1}u_{h},w)_{L^{2}(P)}$
$\displaystyle\leq\eta_{P}h_{P}^{-1}\|w\|_{L^{2}(P)}\leq(1+C_{\mathrm{PF}})\eta_{P},$
(5.11) $\displaystyle((1-\Pi_{0})\bm{\sigma}_{h},\nabla w)_{L^{2}(P)}$
$\displaystyle\leq\Lambda_{P}|w|_{1,P}\leq(1+C_{\mathrm{PF}})\Lambda_{P}.$
(5.12)

The upper bound $\|\textbf{A}\|_{\infty}$ of the coefficient A, (3.5), and the
Cauchy-Schwarz inequality for the stabilization term lead to the first
inequality in

$\displaystyle C_{s}^{-1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$
$\displaystyle\leq\|\textbf{A}\|_{\infty}^{1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}|(1-\Pi_{1})I_{h}v|_{1,P}$
$\displaystyle\leq\|\textbf{A}\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}})\zeta_{P}.$
(5.13)

The second inequality in (5.13) follows as in (4.3) and with
$\|(1-\Pi_{0})\nabla v\|_{L^{2}(P)}\leq 1$. Recall the boundedness constant
$M_{b}$ of $B_{\mathrm{pw}}$ from Subsection 4.1 and deduce from (5.7) and the
definition of $\Xi_{\mathcal{T}}$ from Subsection 5.1 that

$\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{1}u_{h},v)\leq
M_{b}|\Pi_{1}u_{h}-E_{1}u_{h}|_{1,\mathrm{pw}}\leq
M_{b}C_{\mathrm{En}}\Xi_{\mathcal{T}}.$ (5.14)

The substitution of (5.10)-(5.14) in (5.8) reveals that

$\displaystyle\|u-E_{1}u_{h}\|_{1,\Omega}$ $\displaystyle\leq
C_{7}(\eta_{\mathcal{T}}+\Lambda_{\mathcal{T}}+\zeta_{\mathcal{T}}+\Xi_{\mathcal{T}})$
(5.15)

with
$\beta_{0}C_{7}=1+C_{\mathrm{PF}}+C_{s}^{1/2}\|\textbf{A}\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}})+M_{b}C_{\mathrm{En}}.$

The combination of (4.24), (5.15) and (5.7) leads in the triangle inequality

$\displaystyle|u-u_{h}|_{1,\mathrm{pw}}\leq|u-E_{1}u_{h}|_{1,\Omega}+|E_{1}u_{h}-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+|\Pi_{1}u_{h}-u_{h}|_{1,\mathrm{pw}}$

to (5.1) with
$C_{\text{rel}1}/2=C_{7}+C_{\mathrm{En}}+a_{0}^{-1/2}C_{s}^{1/2}$. ∎

####  5.3.2 Reliable $L^{2}$ error control

Recall $I_{\text{CR}}$ from (3.14) and $J^{\prime}$ from the proof of Lemma
3.3, and define $E_{2}u_{h}:=J^{\prime}I_{\text{CR}}u_{h}\in
H^{1}_{0}(\Omega)$ from Subsection 5.2. Let $\Psi\in H^{1}_{0}(\Omega)\cap
H^{1+\sigma}(\Omega)$ solve the dual problem $B(v,\Psi)=(u-E_{2}u_{h},v)$ for
all $v\in V$ and recall (from (1.5)) the regularity estimate

$\displaystyle\|\Psi\|_{1+\sigma,\Omega}\leq
C^{*}_{\text{reg}}\|u-E_{2}u_{h}\|_{L^{2}(\Omega)}.$ (5.16)

The substitution of $v:=u-E_{2}u_{h}\in V$ in the dual problem shows

$\displaystyle\|u-E_{2}u_{h}\|^{2}_{L^{2}(\Omega)}=B(u-E_{2}u_{h},\Psi).$

The algebra in (5.8)-(5.10) above leads with $v=\Psi$ to the identity

$\displaystyle\|u-E_{2}u_{h}\|^{2}_{L^{2}(\Omega)}-s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Psi)=(f-\gamma\Pi_{1}u_{h},\Psi-\Pi_{1}I_{h}\Psi)_{L^{2}(\Omega)}$
$\displaystyle\quad\quad-((1-\Pi_{0})\bm{\sigma}_{h},\nabla_{\mathrm{pw}}(\Psi-\Pi_{1}I_{h}\Psi))_{L^{2}(\Omega)}+B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi).$
(5.17)

The definition of $I_{\text{CR}}$ and $J^{\prime}$ proves the first and second
equality in

$\int_{E}u_{h}\,ds=\int_{E}I_{\text{CR}}u_{h}\,ds=\int_{E}E_{2}u_{h}\,ds\quad\text{for
all}\;E\in\mathcal{E}.$

This and an integration by parts imply
$\int_{P}\nabla(u_{h}-E_{2}u_{h})\,dx=0$ for all $P\in\mathcal{T}$. Hence
Definition 2.2 of Ritz projection $\Pi^{\nabla}_{1}=\Pi_{1}$ in $V_{h}$ shows
$\int_{P}\nabla(\Pi_{1}u_{h}-E_{2}u_{h})\,ds=0$ for all $P\in\mathcal{T}$.
This $L^{2}$ orthogonality
$\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$
and the definition of $B_{\mathrm{pw}}$ in the last term of (5.17) result with
elementary algebra in

$\displaystyle
B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)=((\textbf{A}-\Pi_{0}\textbf{A})\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h}),\nabla\Psi)_{L^{2}(\Omega)}$
$\displaystyle\;+(\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h}),(\Pi_{0}\textbf{A})(1-\Pi_{0})\nabla\Psi)_{L^{2}(\Omega)}+(\Pi_{1}u_{h}-E_{2}u_{h},\textbf{b}\cdot\nabla\Psi+\gamma\Psi)_{L^{2}(\Omega)}.$
(5.18)

The triangle inequality and (c’) from the proof of Lemma 3.3 imply the first
inequality in

$\displaystyle|\Pi_{1}u_{h}-E_{2}u_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}\min_{v\in
V}|I_{\text{CR}}u_{h}-v|_{1,\mathrm{pw}}$
$\displaystyle\leq|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}|I_{\text{CR}}u_{h}-E_{1}u_{h}|_{1,\mathrm{pw}}$
$\displaystyle\leq|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}(|I_{\text{CR}}u_{h}-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+|\Pi_{1}u_{h}-E_{1}u_{h}|_{1,\mathrm{pw}})$
$\displaystyle\leq(1+C_{\mathrm{J^{\prime}}})|u_{h}-\Pi_{1}u_{h}|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}|\Pi_{1}u_{h}-E_{1}u_{h}|_{1,\mathrm{pw}}.$
(5.19)

The second estimate in (5.19) follows from $E_{1}u_{h}\in V$, the third is a
triangle inequality, and eventually
$|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}|_{1,\mathrm{pw}}\leq|u_{h}-\Pi_{1}u_{h}|_{1,\mathrm{pw}}$
results from the orthogonality
$\nabla_{\mathrm{pw}}(u_{h}-I_{\text{CR}})\perp\mathcal{P}_{0}(\widehat{{\cal
T}};\mathbb{R}^{2})$ and $\Pi_{1}u_{h}\in\mathcal{P}_{1}(\mathcal{T})$. The
Cauchy-Schwarz inequality, the Lipschitz continuity of A, and the
approximation estimate $\|(1-\Pi_{0})\nabla\Psi\|_{L^{2}(P)}\leq
C_{\text{apx}}h_{P}^{\sigma}|\Psi|_{1+\sigma,P}$ in (5.18) lead to the first
inequality in

$\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)$
$\displaystyle\leq\sum_{P\in\mathcal{T}}\Big{(}(h_{P}|\textbf{A}|_{1,\infty}+\|\textbf{A}\|_{\infty}C_{\text{apx}}h_{P}^{\sigma})|\Pi_{1}u_{h}-E_{2}u_{h}|_{1,P}$
$\displaystyle\qquad+\|\Pi_{1}u_{h}-E_{2}u_{h}\|_{L^{2}(P)}(\|\textbf{b}\|_{\infty}+\|\gamma\|_{\infty})\Big{)}\|\Psi\|_{1+\sigma,P}$
$\displaystyle\leq\sum_{P\in\mathcal{T}}\Big{(}h_{P}|\textbf{A}|_{1,\infty}+\|\textbf{A}\|_{\infty}C_{\text{apx}}h_{P}^{\sigma}+C_{\mathrm{PF}}(\|\textbf{b}\|_{\infty}+\|\gamma\|_{\infty})h_{P}\Big{)}|\Pi_{1}u_{h}-E_{2}u_{h}|_{1,P}\|\Psi\|_{1+\sigma,P}$
$\displaystyle\leq
C_{8}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}((1+C_{\mathrm{J^{\prime}}})|u_{h}-\Pi_{1}u_{h}|_{1,P}+C_{\mathrm{J^{\prime}}}|\Pi_{1}u_{h}-E_{1}u_{h}|_{1,P})\|\Psi\|_{1+\sigma,P}.$
(5.20)

The second inequality in (5.20) follows from the Poincaré-Friedrichs
inequality in Lemma 2.1.a for $\Pi_{1}u_{h}-E_{2}u_{h}$ with $\int_{\partial
P}(\Pi_{1}u_{h}-E_{2}u_{h})\,ds=0$ (from above); the constant
$C_{8}:=|\textbf{A}|_{1,\infty}+C_{\text{apx}}\|\textbf{A}\|_{\infty}+C_{\mathrm{PF}}(\|\textbf{b}\|_{\infty}+\|\gamma\|_{\infty})$
results from (5.19) and $h_{P}\leq h_{P}^{\sigma}$ (recall $h_{\text{max}}\leq
1$). Lemma 5.4 with $v_{1}=\Pi_{1}u_{h}$ and (4.24) in (5.20) show

$\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)$
$\displaystyle\leq
C_{8}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}\zeta_{P}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}\Xi_{P})\|\Psi\|_{1+\sigma,P}.$
(5.21)

Rewrite (5.11)-(5.13) with $w=\Psi-\Pi_{1}I_{h}\Psi$ and
$h_{P}^{-1}\|w\|_{L^{2}(P)}+|w|_{1,P}\leq(1+C_{\mathrm{PF}})\|(1-\Pi_{0})\nabla\Psi\|_{L^{2}(P)}\leq
C_{\text{apx}}(1+C_{\mathrm{PF}})h_{P}^{\sigma}|\Psi|_{1+\sigma,P}$ from
(2.12). This and (5.21) lead in (5.17) to

$\displaystyle\|u-E_{2}u_{h}\|_{L^{2}(\Omega)}^{2}\leq
C_{9}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}(\eta_{P}+\zeta_{P}+\Lambda_{P}+\Xi_{P})\|\Psi\|_{1+\sigma,P}$

for
$C_{9}:=C_{\text{apx}}(1+C_{\mathrm{PF}}+C_{s}^{1/2}\|\textbf{A}\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}}))+C_{8}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}).$
This and the regularity (5.16) result in

$\displaystyle\|u-E_{2}u_{h}\|_{L^{2}(\Omega)}\leq
C_{9}C^{*}_{\text{reg}}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}(\eta_{P}+\zeta_{P}+\Lambda_{P}+\Xi_{P}).$
(5.22)

The arguments in the proof of (5.20)-(5.21) also lead to

$\displaystyle\|E_{2}u_{h}-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}\leq
C_{\mathrm{PF}}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}})\sum_{P\in\mathcal{T}}h_{P}(\zeta_{P}+\Xi_{P}).$
(5.23)

The combination of (4.25), (5.22)-(5.23) and the triangle inequality

$\|u-u_{h}\|_{L^{2}(\Omega)}\leq\|u-E_{2}u_{h}\|_{L^{2}(\Omega)}+\|E_{2}u_{h}-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}+\|\Pi_{1}u_{h}-u_{h}\|_{L^{2}(\Omega)}$

lead to (5.2) with
$C_{rel2}/2=C_{9}C^{*}_{\text{reg}}+C_{\mathrm{PF}}\big{(}(2+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}\big{)}.$
This concludes the proof of the $L^{2}$ error estimate in Theorem 5.1. ∎

####  5.3.3 Comments

##### 5.3.3.1 Estimator for $u-\Pi_{1}u_{h}$

The triangle inequality with (5.1) and (4.24) provide an upper bound for
$H^{1}$ error

$\displaystyle\frac{1}{2}|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}^{2}\leq|u-u_{h}|_{1,\mathrm{pw}}^{2}+|(1-\Pi_{1})u_{h}|_{1,\mathrm{pw}}^{2}\leq
2C_{\text{rel}1}^{2}(\eta_{\mathcal{T}}^{2}+\zeta_{\mathcal{T}}^{2}+\Lambda_{\mathcal{T}}^{2}+\Xi_{\mathcal{T}}^{2}).$

The same arguments for an upper bound of the $L^{2}$ error in Theorem 5.1 show
that

$\displaystyle\frac{1}{2}\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq\|u-u_{h}\|_{L^{2}(\Omega)}^{2}+\|(1-\Pi_{1})u_{h}\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq
C_{\text{rel}2}^{2}\sum_{P\in\mathcal{T}}h_{P}^{2\sigma}(\eta_{P}^{2}+2\zeta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2}).$

The numerical experiments do not display $C_{\text{rel}1}$ and
$C_{\text{rel}2}$, and directly compare the error
$H1e:=|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}$ in the piecewise $H^{1}$ norm and the
error $L2e:=\|u-\Pi_{1}u_{h}\|_{L^{2}(\Omega)}$ in the $L^{2}$ norm with the
upper bound $H1\mu$ and $L2\mu$ (see, e.g., Figure 6.3).

##### 5.3.3.2 Motivation and discussion of apx

We first argue that those extra terms have to be expected and utilize the
abbreviations $\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u$ and $g:=f-\gamma
u$ for the exact solution $u\in H^{1}_{0}(\Omega)$ to (1.8), which reads

$\displaystyle(\bm{\sigma},\nabla
v)_{L^{2}(\Omega)}=(g,v)_{L^{2}(\Omega)}\quad\text{for all}\;v\in
H^{1}_{0}(\Omega).$ (5.24)

Recall the definition of $s_{h}(\cdot,\cdot)$ from Subsection 3.1. The
discrete problem (3.8) with the discrete solution $u_{h}\in V_{h}$ assumes the
form

$\displaystyle(\bm{\sigma}_{h},\nabla\Pi_{1}v_{h})_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h})=(g_{h},\Pi_{1}v_{h})_{L^{2}(\Omega)}\quad\text{for
all}\;v_{h}\in V_{h}$ (5.25)

for $\bm{\sigma}_{h}:=\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$,
and $g_{h}:=f-\gamma\Pi_{1}u_{h}$. Notice that $\bm{\sigma}_{h}$ and $g_{h}$
may be replaced in (5.25) by $\Pi_{0}\bm{\sigma}_{h}$ and $\Pi_{1}g_{h}$
because the test functions $\nabla\Pi_{1}v_{h}$ and $\Pi_{1}v_{h}$ belong to
$\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ and
$\mathcal{P}_{1}(\mathcal{T})$ respectively. In other words, the discrete
problems (3.8) and (5.25) do not see a difference of $\bm{\sigma}_{h}$ and
$g_{h}$ compared to $\Pi_{0}\bm{\sigma}_{h}$ and $\Pi_{1}g_{h}$ and so the
errors $\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}$ and $g_{h}-\Pi_{1}g_{h}$ may
arise in a posteriori error control. This motivates the a posteriori error
term
$\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}\|_{L^{2}(\Omega)}=\Lambda_{\mathcal{T}}$
as well as the approximation terms $\bm{\sigma}-\Pi_{0}\bm{\sigma}$ and
$g-\Pi_{1}g$ on the continuous level. The natural norm for the dual variable
$\bm{\sigma}$ is $L^{2}$ and that of $g$ is $H^{-1}$ and hence their norms
form the approximation term apx as defined in Subsection 5.1.

######  Example 5.1 ($\textbf{b}=0$).

The term $(1-\Pi_{0})\bm{\sigma}$ may not be visible in case of no advection
$\textbf{b}=0$ at least if A is piecewise constant. Suppose
$\textbf{A}\in\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2\times 2})$ and
estimate

$\|(1-\Pi_{0})(\textbf{A}\nabla
u)\|_{L^{2}(\Omega)}\leq\|\textbf{A}\|_{\infty}\|(1-\Pi_{0})\nabla
u\|_{L^{2}(\Omega)}\lesssim|u-\Pi_{1}u_{h}|_{1,\mathrm{pw}}.$

If A is not constant, there are oscillation terms that can be treated properly
in adaptive mesh-refining algorithms, e.g., in [27].

######  Example 5.2 ($\gamma$ piecewise constant).

While the data approximation term $\mathrm{osc}_{1}(f,\mathcal{T})$ [10] is
widely accepted as a part of the total error in the approximation of nonlinear
problems, the term $\mathrm{osc}_{1}(\gamma u,\mathcal{T})=\|\gamma
h_{\mathcal{T}}(u-\Pi_{1}u)\|_{L^{2}(\Omega)}\lesssim
h_{\text{max}}^{1+\sigma}\|f\|_{L^{2}(\Omega)}$ is of higher order and may
even be absorbed in the overall error analysis for a piecewise constant
coefficient $\gamma\in\mathcal{P}_{0}(\mathcal{T})$. In the general case
$\gamma\in L^{\infty}(\Omega)\backslash\mathcal{P}_{0}(\mathcal{T})$, however,
$\mathrm{osc}_{1}(u,\mathcal{T})$ leads in particular to terms with
$\|\gamma-\Pi_{0}\gamma\|_{L^{\infty}(\Omega)}$.

##### 5.3.3.3 Higher-order nonconforming VEM

The analysis applied in Theorem 5.1 can be extended to the nonconforming VEM
space of higher order $k\in\mathbb{N}$ (see [17, Sec. 4] for the definition of
discrete space). Since the projection operators $\nabla\Pi_{k}^{\nabla}$ and
$\Pi_{k-1}\nabla$ are not the same for general $k$, and the first operator
does not lead to optimal order of convergence for $k\geq 3$, the discrete
formulation uses $\Pi_{k-1}\nabla$ (cf. [6, Rem. 4.3] for more details). The
definition and approximation properties of the averaging operator
$E_{\mathrm{pw}}$ extend to the operator $E^{k}:\mathcal{P}_{k}(\widehat{{\cal
T}})\to H^{1}_{0}(\Omega)$ (see [32, p. 2378] for a proof). The identity (5.9)
does not hold in general, but algebraic calculations lead to

$\displaystyle\eta_{P}^{2}$
$\displaystyle:=h_{P}^{2}\|f-\gamma\Pi_{k}u_{h}\|_{L^{2}(P)}^{2},\qquad\Lambda_{P}^{2}:=\|(1-\Pi_{k-1})(\textbf{A}\Pi_{k-1}\nabla
u_{h}+\textbf{b}\Pi_{k}u_{h})\|_{L^{2}(P)}^{2}$ $\displaystyle\zeta_{P}^{2}$
$\displaystyle:=S^{P}((1-\Pi_{k})u_{h},(1-\Pi_{k})u_{h}),\qquad\Xi_{P}^{2}:=\sum_{E\in\mathcal{E}(P)}|E|^{-1}\|[\Pi_{k}u_{h}]_{E}\|^{2}_{L^{2}(E)}.$

The analysis developed for the upper bound of $L^{2}$ norm also extends to the
general case. The model problem is chosen in 2D for the simplicity of the
presentation. The results of this work can be extended to the three-
dimensional case with appropriate modifications. The present analysis holds
for any higher regularity index $\sigma>0$ and avoids any trace inequality for
higher derivatives. This is possible by a medius analysis in the form of
companion operators [26].

##### 5.3.3.4 Inhomogeneous boundary data

The error estimator for general Dirichlet condition $u|_{\partial\Omega}=g\in
H^{1/2}(\partial\Omega)$ can be obtained with some modifications of [33] in
Theorem 5.1. The only difference is in the modified jump contributions of the
boundary edges in the nonconformity term

$\displaystyle\Xi_{\mathcal{T}}^{2}=\sum_{E\in\mathcal{E}(\Omega)}|E|^{-1}\|[\Pi_{1}u_{h}]\|_{L^{2}(E)}^{2}+\sum_{E\in\mathcal{E}(\partial\Omega)}|E|^{-1}\|g-\Pi_{1}u_{h}\|_{L^{2}(E)}^{2}.$

###  5.4 Proof of Theorem 5.2

Recall the notation $\bm{\sigma}=\textbf{A}\nabla u+\textbf{b}u$ and
$\bm{\sigma}_{h}=\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$ from
Subsection 5.3.

###### Proof of (5.3).

The upper bound (3.5) for the stabilisation term and the triangle inequality
show

$\displaystyle\zeta_{P}^{2}\leq C_{s}|(1-\Pi_{1})u_{h}|^{2}_{1,P}\leq
2C_{s}(|u-u_{h}|^{2}_{1,P}+|u-\Pi_{1}u_{h}|^{2}_{1,P}).$

This concludes the proof of (5.3). ∎

###### Proof of (5.5).

The definition of $\Lambda_{P},\Pi_{0}$, and the triangle inequality lead to

$\displaystyle\Lambda_{P}=\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}\|_{L^{2}(P)}$
$\displaystyle\leq\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}\|_{L^{2}(P)}$
$\displaystyle\leq\|\textbf{A}\nabla(\Pi_{1}u_{h}-u)+\textbf{b}(\Pi_{1}u_{h}-u)\|_{L^{2}(P)}+\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(P)}.$
(5.26)

The upper bound $\|\textbf{A}\|_{\infty}$ and $\|\textbf{b}\|_{\infty}$ for
the coefficients and the triangle inequality lead to

$\displaystyle\Lambda_{P}-\|(1-\Pi_{0})\bm{\sigma}\|_{L^{2}(P)}\leq(\|\textbf{A}\|_{\infty}+\|\textbf{b}\|_{\infty})\|\Pi_{1}u_{h}-u\|_{1,P}$
$\displaystyle\qquad\leq(\|\textbf{A}\|_{\infty}+\|\textbf{b}\|_{\infty})(\|u_{h}-\Pi_{1}u_{h}\|_{1,P}+\|u-u_{h}\|_{1,P})\leq
C_{10}(\zeta_{P}+\|u-u_{h}\|_{1,P})$ (5.27)

with
$\|u_{h}-\Pi_{1}u_{h}\|_{1,P}\leq(1+h_{P}C_{\mathrm{PF}})a_{0}^{-1/2}C_{s}^{1/2}\zeta_{P}$
from (4.24)-(4.25) and with
$C_{10}:=(\|\textbf{A}\|_{\infty}+\|\textbf{b}\|_{\infty})((1+h_{P}C_{\mathrm{PF}})a_{0}^{-1/2}C_{s}^{1/2}+1)$.
This followed by (5.3) concludes the proof of (5.5). ∎

Recall the bubble-function $b_{\mathcal{T}}|_{P}=b_{P}$ supported on a
polygonal domain $P\in\mathcal{T}$ from (3.17) as the sum of interior bubble-
functions supported on each triangle $T\in\widehat{{\cal T}}(P)$.

###### Proof of (5.4).

Rewrite the term

$\displaystyle
f-\gamma\Pi_{1}u_{h}=\Pi_{1}(f-\gamma\Pi_{1}u_{h})+(1-\Pi_{1})(f-\gamma\Pi_{1}u_{h})=:R+\theta,$
(5.28)

and denote $R_{P}:=R|_{P}$ and $\theta_{P}:=\theta|_{P}$. The definition of
$B_{\mathrm{pw}}(u-\Pi_{1}u_{h},v)$ and the weak formulation $B(u,v)=(f,v)$
from (1.8) for any $v\in V$ imply

$\displaystyle B_{\mathrm{pw}}(u-\Pi_{1}u_{h},v)+(\bm{\sigma}_{h},\nabla
v)_{L^{2}(\Omega)}$
$\displaystyle=(f-\gamma\Pi_{1}u_{h},v)_{L^{2}(\Omega)}=(R+\theta,v)_{L^{2}(\Omega)}.$
(5.29)

Since $b_{P}R_{P}$ belongs to $H^{1}_{0}(\Omega)$ (extended by zero outside
$P$), $v:=b_{P}R_{P}\in V$ is admissible in (5.29). An integration by parts
proves that $(\Pi_{0}\bm{\sigma}_{h},\nabla(b_{P}R_{P}))_{L^{2}(P)}=0$.
Therefore, (5.29) shows

$\displaystyle(R_{P},b_{P}R_{P})_{L^{2}(P)}=B^{P}(u-\Pi_{1}u_{h},b_{P}R_{P})-(\theta_{P},b_{P}R_{P})_{L^{2}(P)}+((1-\Pi_{0})\bm{\sigma}_{h},\nabla(b_{P}R_{P}))_{L^{2}(P)}.$

The substitution of
$\chi=R_{P}=\Pi_{1}(f-\gamma\Pi_{1}u_{h})|_{P}\in\mathcal{P}_{1}(P)$ in (3.20)
and the previous identity with the boundedness of $B$ and the Cauchy-Schwarz
inequality lead to the first two estimates in

$\displaystyle
C_{b}^{-1}\|R_{P}\|_{L^{2}(P)}^{2}\leq(R_{P},b_{P}R_{P})_{L^{2}(P)}$
$\displaystyle\quad\leq\Big{(}M_{b}|u-\Pi_{1}u_{h}|_{1,P}+\|(1-\Pi_{0})\bm{\sigma}_{h}\|_{L^{2}(P)}\Big{)}|b_{P}R_{P}|_{1,P}+\|\theta_{P}\|_{L^{2}(P)}\|b_{P}R_{P}\|_{L^{2}(P)}$
$\displaystyle\quad\leq
C_{b}\Big{(}M_{b}|u-\Pi_{1}u_{h}|_{1,P}+\Lambda_{P}+h_{P}\|\theta_{P}\|_{L^{2}(P)}\Big{)}h_{P}^{-1}\|R_{P}\|_{L^{2}(P)}.$