# Nonconforming virtual elements for the biharmonic equation with Morley
degrees of freedom on polygonal meshes

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 lowest-order nonconforming virtual element extends the Morley triangular
element to polygons for the approximation of the weak solution $u\in
V:=H^{2}_{0}(\Omega)$ to the biharmonic equation. The abstract framework
allows (even a mixture of) two examples of the local discrete spaces
$V_{h}(P)$ and a smoother allows rough source terms $F\in
V^{*}=H^{-2}(\Omega)$. The a priori and a posteriori error analysis in this
paper circumvents any trace of second derivatives by some computable
conforming companion operator $J:V_{h}\to V$ from the nonconforming virtual
element space $V_{h}$. The operator $J$ is a right-inverse of the
interpolation operator and leads to optimal error estimates in piecewise
Sobolev norms without any additional regularity assumptions on $u\in V$. As a
smoother the companion operator modifies the discrete right-hand side and then
allows a quasi-best approximation. An explicit residual-based a posteriori
error estimator is reliable and efficient up to data oscillations. Numerical
examples display the predicted empirical convergence rates for uniform and
optimal convergence rates for adaptive mesh-refinement.

Keywords: biharmonic equation, virtual elements, nonconforming, polytopes,
enrichment,

a priori, a posteriori, adaptive mesh-refinement, companion operator, smoother

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

##  1 Introduction

The popular nonconforming Morley finite element method (FEM) for fourth-order
problems allows a generalization from triangular domains to polygons in the
class of nonconforming virtual element methods (ncVEM). Two ncVEM have been
introduced in [2, 21, 35] for $H^{3}$ regular solutions, while a medius
analysis in [30] allows minimal regularity. In comparison to the existing
literature on ncVEM for biharmonic problems, this paper presents an abstract
framework and identifies two hypotheses (H1)-(H2) for a unified stability and
a priori error analysis of at least two different ncVEM each with an
individual parameter $r=-1,0,1,2$ ($r=-1$ for original VE spaces and $r=0,1,2$
for enhanced VE spaces [1]) and even a mixture of those. This paper adds a new
analysis with a computable conforming companion that allows a quasi-best
approximation

$\displaystyle|u-Gu_{h}|_{2,\mathrm{pw}}\lesssim\min_{v_{h}\in
V_{h}}|u-Gv_{h}|_{2,\mathrm{pw}}$ (1.1)

with the local Galerkin projection $G$ onto piecewise quadratics and a general
source function with a smoother for the first time in ncVEM and completes the
a priori error convergence analysis in piecewise energy and weaker Sobolev
norms. The lower-order estimates are available in the literature for enhanced
VE spaces, e.g., Zhao et al. discuss piecewise $H^{1}$ error estimate in [35]
for an enhanced VE space ($r=0$) and this paper proves it also for original VE
spaces ($r=-1$). The design of companion operators started in [13] for second-
order and in [11, 20, 32] for fourth-order problems. It is related to
enrichments in multigrid methods [6] and to reliable a posteriori error
control [17]. Its role as a smoother in ncVEM generalizes [18, 19, 32] for the
Morley FEM. The first paper [16] on an a posteriori error analysis for ncVEM
is restricted to second-order problems and includes many references on an a
posteriori error analysis for the conforming VEM. This is the first paper on
an a posteriori virtual element error control for fourth-order problems with
reliable and efficient error estimators and a suggested adaptive mesh-refining
algorithm. The presented a posteriori error analysis also covers conforming
VEM [10].

Main results. This paper contributes to the understanding of the ncVEM for a
class of examples that includes the two known examples of discrete VE spaces
for fourth-order problems

  * •

a computable conforming companion operator,

  * •

a priori error estimates in piecewise $H^{1}$ and $H^{2}$ norms,

  * •

quasi-best approximation for a smoother for any source term $F\in
H^{-2}(\Omega)$,

  * •

reliable and (up to data oscillations) efficient a posteriori error control,

  * •

adaptive mesh-refinement algorithm with improved empirical convergence rates.

The results are displayed for 2D and the lowest-order case only corresponding
to the Morley degrees of freedom, but the arguments allow a higher dimension
and higher degrees.

Outline and organisation of the paper. Section 2 describes the admissible
partitions of the domain $\Omega$ into polygonal domains and defines the local
and global Morley degrees of freedom. Subsection 2.2 establishes a Poincaré-
Friedrichs inequality on polygons and Subsection 2.4 recall the local Galerkin
projection $G$ and establish associated error estimates. Section 3 explains an
abstract framework with hypotheses (H1)-(H2) and presents two affirmative
examples of virtual elements $V_{h}(P)$. The interpolation operator is defined
in Subsection 3.3 and its error estimates follow in Subsection 3.4. Section 4
designs a computable conforming companion operator, which is a right-inverse
of the interpolation operator, and provides the fundamental approximation
error estimates. Subsection 5.1 introduces the natural stabilization and
Subsection 5.2 the discrete problem with two choices for the right-hand side.
Subsection 5.3 provides the a priori error estimates in piecewise $H^{1}$ and
$H^{2}$ norms in the best-approximation form up to data oscillations; a
smoother in the right-hand side eliminates the oscillations. Section 6
developes an explicit residual-based reliable and (up to data oscillations)
efficient error analysis for ncVEM that also applies for the conforming VEM.
The stabilization term is efficient with respect to the sum of the error
$u-Gu_{h}$ and $u-u_{h}$ in their piecewise $H^{2}$ seminorms. Subsection 7.1
suggests an adaptive mesh-refinement algorithm. Numerical results support the
theoretical predictions in Subsections 7.2-7.3 and provide striking numerical
evidence of optimal empirical convergence rates for adaptive mesh-refining.
Supplement material accompanies this paper with details on the local virtual
element spaces and is referred to as Appendix A, B, and C throughout this
paper. This contains partly established or routine results that are somehow
standard but seemingly not available in the literature in this form.

Notation. Standard notation on Lebesgue and Sobolev spaces and norms applies
throughout this paper, e.g., $\|\cdot\|_{s,\cal{D}}$ (resp. seminorm
$|\cdot|_{s,\cal{D}}$) for $s\geq 0$ denotes norm on the Sobolev space
$H^{s}(\mathcal{D}):=H^{s}(\text{int}(\cal{D}))$ of order $s\in\mathbb{R}$
defined in the interior $\text{int}(\mathcal{D})$ of a domain $\mathcal{D}$,
while $(\cdot,\cdot)_{L^{2}({\cal D})}$ and $\|\cdot\|_{L^{2}({\cal D})}$
denote the $L^{2}$ scalar product and $L^{2}$ norm in ${\cal D}$. Let
$|\mathcal{D}|$ denote the area of a domain $\mathcal{D}$, 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_{\mathcal{D}}\bullet\,dx:=|\mathcal{D}|^{-1}\int_{\mathcal{D}}\bullet\,dx$
denote the integral mean on $\mathcal{D}$. Define the Sobolev space
$V=H^{2}_{0}(\Omega):=\\{v\in H^{2}(\Omega):v=v_{\textbf{n}}=0\\}$ for the
derivative $v_{\textbf{n}}=\nabla v\cdot\textbf{n}$ in the direction of
outward unit normal n along the boundary $\partial\mathcal{D}$. The vector
space $C^{r}(\mathcal{D})$ is the set of $C^{r}$-continuous functions defined
on a domain $\mathcal{D}$ for $r\in\mathbb{N}_{0}$. 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 $\mathcal{P}_{k}({\mathcal{M}})$ denote the
set of piecewise polynomials on an admissible partition
$\mathcal{M}\in\mathbb{M}$ (defined in Subsection 2.1). The piecewise seminorm
and norm in $H^{s}(\mathcal{M})$ for $s\in\mathbb{R}$ (see the definition of
$|\cdot|_{s,P}=|\cdot|_{H^{s}(P)}$ in, e.g., [8, Chapter 14]) read
$|\cdot|_{s,\text{pw}}:=\big{(}\sum_{P\in\mathcal{M}}|\cdot|_{s,P}^{2}\big{)}^{1/2}$
and
$\|\cdot\|_{s,\text{pw}}:=\big{(}\sum_{P\in\mathcal{M}}\|\cdot\|_{s,P}^{2}\big{)}^{1/2}$.
Let $\Pi_{k}$ denote the $L^{2}$ projection on $\mathcal{P}_{k}(\mathcal{M})$
for $k\in\mathbb{N}_{0}$. The oscillation of $f\in L^{2}(\Omega)$ reads

$\displaystyle\mathrm{osc}_{2}(f,{\mathcal{M}}):=\Big{(}\sum_{P\in{\mathcal{M}}}\mathrm{osc}_{2}^{2}(f,P)\Big{)}^{1/2}\quad\text{for}\quad\mathrm{osc}^{2}_{2}(f,P):=\|h_{P}^{2}(1-\Pi_{2})f\|_{L^{2}(P)}^{2}.$

Let $\mathbb{S}$ be the set of $2\times 2$ symmetric matrices in
$\mathbb{R}^{2\times 2}$ and let $\delta_{jk}$ denote the Kronecker delta
($\delta_{jk}=0$ if $j\neq k$ and $\delta_{jj}=1$). Let
$\alpha:=(\alpha_{1},\alpha_{2})$ denote a multi-index with
$\alpha_{j}\in\mathbb{N}_{0}$ for $j=1,2$ and
$|\alpha|:=\alpha_{1}+\alpha_{2}.$ The outward normal and tangential
derivatives of first and higher orders are written as subscripts
$\textbf{n},\bm{\tau},{\textbf{n}\textbf{n}},{\bm{\tau}\bm{\tau}},{\textbf{n}\bm{\tau}\bm{\tau}}$
etc. for the exterior unit normal vector n and the tangential vector
$\bm{\tau}$ along the boundary $\partial P$ of the (polygonal Lipschitz)
domain $P\in\mathcal{M}\in\mathbb{M}$ (from Subsection 2.1). An inequality
$A\lesssim B$ abbreviates $A\leq CB$ for a generic constant $C$, that
exclusively depends on the domain $\Omega$ and on the mesh-parameter $\rho$
(from (M2) below).

##  2 Virtual element method

###  2.1 Admissible partitions

Let $\mathbb{M}$ be a family of decompositions of $\overline{\Omega}$ into
polygonal domains satisfying the two mesh conditions (M1)-(M2) with a
universal positive constant $\rho$.

(M1) Admissibility. Any two distinct polygonal domains $P$ and $P^{\prime}$ in
$\mathcal{M}\in\mathbb{M}$ are disjoint or share a finite number of edges and
vertices.

(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 $h_{E}$ greater than equal
to $\rho h_{P}$.

Here and throughout this paper, $h_{\mathcal{M}}|_{P}:=h_{P}:=\text{diam}(P)$
denotes the piecewise constant mesh-size
$h_{\mathcal{M}}\in\mathcal{P}_{0}(\mathcal{M})$ and
$h_{\mathrm{max}}:=\max_{P\in\mathcal{M}}h_{P}$ denotes the maximum diameter
over all $P\in\mathcal{M}\in\mathbb{M}$. Let $\mathcal{V}(P)$ (resp.
$\mathcal{V}$) denote the set of vertices of $P$ (resp. of $\mathcal{M}$) and
let $\mathcal{E}(P)$ (resp. $\mathcal{E}$) denote the set of edges of $P$
(resp. of $\mathcal{M}$). Denote the interior and boundary edges of
$\mathcal{M}$ by $\mathcal{E}(\Omega)$ and $\mathcal{E}(\partial\Omega)$. Let
$|\mathcal{V}|$ (resp. $|\mathcal{E}|$) denote the number of vertices (resp.
edges) and $N:=|\mathcal{V}|+|\mathcal{E}|$.

The standard notation of the polygonal domain $P$ with $N_{P}$ edges and
$N_{P}$ vertices is depicted in Figure 2.1.a. Note that $3\leq N_{P}\leq
M(\rho)$ for a global number $M(\rho)$ that exclusively depends on $\rho$ [4].
We enumerate the vertices $\mathcal{V}(P):=\\{z_{1},\dots,z_{N_{P}}\\}$ and
edges $\mathcal{E}(P):=\\{E(1),\dots,E(N_{P})\\}$ consecutively, i.e.,
$E(j)=\text{conv}\\{z_{j},z_{j+1}\\}$ for $j=1,\dots,N_{P}$ with
$z_{N_{P}+1}:=z_{1}$ and enumerate $z_{1},\dots,z_{N_{P}}$ counterclockwise
along the boundary $\partial P$. (M2) implies that each polygonal domain
$P\in\mathcal{M}$ can be divided into triangles
$T(j):=\text{conv}\\{z_{0},z_{j},z_{j+1}\\}$ for all $j=1,\dots,N_{P}$ and for
the midpoint $z_{0}$ of the ball from (M2) in Figure 2.1.b. It is known [7]
that the resulting sub-triangulation
$\mathcal{T}|_{P}:=\mathcal{T}(P):=\cup_{j=1}^{N_{P}}T(j)$ of
$P\in\mathcal{M}$ is uniformly shape-regular; i.e, the minimum angle in each
triangle $T\in\mathcal{T}(P),P\in\mathcal{M}\in\mathbb{M}$, is bounded below
by some positive constant $w_{0}>0$ that exclusively depends on $\rho$. Let
$\widehat{\mathcal{V}}$ (resp. $\widehat{\mathcal{V}}(P)$) denote the set of
vertices and $\widehat{\mathcal{E}}$ (resp. $\widehat{\mathcal{E}}(P)$) denote
the set of edges in $\mathcal{T}$ (resp. $\mathcal{T}(P)$).

(a)

(b)

(c)

Figure 2.1: (a) Hexagon $P$ with vertices $z_{1},\dots,z_{6}$ and edges
$E(1),\dots,E(6)$ (b) its sub-triangulation $\mathcal{T}(P)$ and (c) $P$ with
corner points $\zeta_{1},\dots,\zeta_{4}$ and sides
$\gamma(1),\dots,\gamma(4)$.

With a counterclockwise orientation along the polygonal boundary $\partial P$,
assign the piecewise constant tangential unit vector $\bm{\tau}_{P}$ and the
outer normal unit vector $\textbf{n}_{P}$. Define the local degrees of freedom
(dofs) $\text{dof}_{1}(v),\dots,\text{dof}_{2N_{P}}(v)$, for $v\in H^{2}(P)$,
by

$\displaystyle\text{dof}_{j}(v)=\begin{cases}\text{$v(z_{j})\hskip 52.63777pt$
for $j=1,\dots,N_{P},$}\\\ \text{$\int_{E(k)}v_{\textbf{n}}\,ds\hskip
28.45274pt$ for $j=N_{P}+1,\dots,2N_{P}$ and $k=j-N_{P}$}\end{cases}$ (2.1)

with the vertices $z_{j}$ and the edges $E(k)\in\mathcal{E}(P)$ of the
polygonal domain $P\in\mathcal{M}$ in Figure 2.1.a. Given a polygonal domain
$P\in\mathcal{M}$, the dofs in (2.1) are collected in the linear map
$\text{Dof}:=(\text{dof}_{1},\dots,\text{dof}_{2N_{P}}):H^{2}(P)\to\mathbb{R}^{2N_{P}}$.

Fix the orientation of a unit normal vector $\textbf{n}_{E}$ to each edge
$E\in\mathcal{E}$. The sign of the jump $[\bullet]_{E}$ across an edge
$E\in\mathcal{E}$ follows from the subsequent convention of the orientation of
a unit normal vector $\textbf{n}_{E}$ along an edge $E$: Label the two
neighbouring polygons $P_{\pm}$ sharing the interior edge $E=\partial
P_{+}\cap\partial P_{-}$ such that $\textbf{n}_{P_{+}}|_{E}=\textbf{n}_{E}$
and $\textbf{n}_{P_{-}}|_{E}=-\textbf{n}_{E}$. This defines the sign in the
jump $[\bullet]_{E}:=\bullet|_{P_{+}}-\bullet|_{P_{-}}$ across
$E\in\mathcal{E}(\Omega)$. For a boundary edge
$E\in\mathcal{E}(\partial\Omega)$, set
$\textbf{n}_{\Omega}|_{E}=\textbf{n}_{E}$ and $[\bullet]_{E}:=\bullet|_{E}$.
For $v_{\mathrm{nc}}\in V_{\mathrm{nc}}:=\\{v\in
H^{2}(\mathcal{M}):v\;\text{is continuous at interior vertices and zero at
boundary}$ $\text{vertices of $\mathcal{M}$,
and}\int_{E}[v_{\textbf{n}}]_{E}\,ds=0\quad\text{for
all}\;E\in\mathcal{E}\\}$, the global Morley dofs read

$\displaystyle\text{dof}_{j}(v_{\mathrm{nc}})=\begin{cases}\text{$v_{\mathrm{nc}}(z_{j})\hskip
52.63777pt$ for $j=1,\dots,|\mathcal{V}|,$}\\\
\text{$\int_{E(k)}(v_{\mathrm{nc}})_{\textbf{n}}\,ds\hskip 21.33955pt$ for
$j=|\mathcal{V}|+1,\dots,N$ and $k=j-|\mathcal{V}|.$}\end{cases}$ (2.2)

The global degrees of freedom $\text{dof}_{j}(v_{\mathrm{nc}})$ from (2.2)
coincide with the local degrees of freedom $\text{Dof}(v_{\mathrm{nc}}|_{P})$
from (2.1) for each polygonal domain $P\in\mathcal{M}$ up to a (known) change
of signs of $\textbf{n}_{E}$ and $\textbf{n}_{P}|_{E}$ for an interior edge
$E\in\mathcal{E}(\Omega)$.

######  Lemma 2.1.

$|\cdot|_{2,\mathrm{pw}}$ defines a norm on $V_{\mathrm{nc}}$ equivalent to
$\|\cdot\|_{2,\mathrm{pw}}$.

###### Proof.

This is shown in [2, Lemma 3.1] and in [35, Lemma 5.1] based on the Poincaré-
Friedrichs inequality for piecewise $H^{2}$ functions in [9]. ∎

###  2.2 Poincaré-Friedrichs inequality

This subsection provides a Poincaré-Friedrichs inequality for a polygonal
domain $P\in\mathcal{M}\in\mathbb{M}$ with explicit constants that exclusively
depend on $\rho$ from Subsection 2.1.

######  Theorem 2.2 (Poincaré-Friedrichs inequality).

There exists a positive constant $C_{\mathrm{PF}}$ (that exclusively depends
on $\rho$) such that
$(a)$ $\|f\|_{L^{2}(P)}\leq C_{\mathrm{PF}}h_{P}|f|_{1,P}$ holds for any $f\in
H^{1}(P)$ with
$0\in\mathrm{conv}\Big{\\{}\int_{E(1)}f\,ds,\dots,\int_{E(N_{P})}f\,ds\Big{\\}},$
$(b)$ $\sum_{m=0}^{1}h_{P}^{m-2}|f|_{m,P}\leq C_{\mathrm{PF}}|f|_{2,P}$ holds
for any $f\in H^{2}(P)$ with
$0\in\mathrm{conv}\\{f(z_{1}),\dots,f(z_{N_{P}})\\}$ and
$0\in\mathrm{conv}\left\\{\int_{E(1)}\frac{\partial f}{\partial
x_{j}}\,ds,\dots,\int_{E(N_{P})}\frac{\partial f}{\partial
x_{j}}\,ds\right\\}$ for $j=1$ and $j=2$.

###### Proof.

The proof of $(a)$ is included in [16, Lemma 2.1]. The proof of $(b)$
considers the sub-triangulation $\mathcal{T}(P)$ of the polygonal domain $P$
from Figure 2.1.b. Define the linear interpolation $I_{1}f\in
S_{1}(\mathcal{T}(P)):=\mathcal{P}_{1}(\mathcal{T}(P))\cap C^{0}(P)$ with
$(f-I_{1}f)(z_{k})=0$ for all $k=0,\dots,N_{P}$. The triangle inequality shows

$\displaystyle\|f\|_{L^{2}(P)}\leq\|f-I_{1}f\|_{L^{2}(P)}+\|I_{1}f\|_{L^{2}(P)}.$
(2.3)

The Bramble Hilbert lemma [7, 22] leads to a positive constant $C_{\text{BH}}$
(that exclusively depends on the shape of the triangles $T(1),\dots,T(N_{P})$
and so merely on $\rho$) in the error estimate

$\displaystyle\|f-I_{1}f\|_{L^{2}(P)}\leq C_{\text{BH}}h_{P}^{2}|f|_{2,P}.$
(2.4)

(Explicit formulas for $C_{\text{BH}}$ in terms of the maximal angle in a
triangle can be found in [14].) Let $\varphi_{0},\dots,\varphi_{N_{P}}$ be the
nodal basis functions of $S_{1}(\mathcal{T}(P))$ with
$\varphi_{k}(z_{\ell})=\delta_{k\ell}$ for $k,\ell=0,\dots,N_{P}$; whence
$I_{1}f=\sum_{k=0}^{N_{P}}f(z_{k})\varphi_{k}$. The local mass matrix with
entries $\int_{T}\lambda_{j}\lambda_{k}\,dx=(1+\delta_{jk})|T|/12$ for
$j,k=1,2,3$ and the barycentric coordinates
$\lambda_{1},\lambda_{2},\lambda_{3}$ in a triangle $T\in\mathcal{T}(P)$ has
the eigenvalues $|T|/12$ (twice) and $|T|/3$. Rayleigh quotients with the
local mass matrix reveal

$\displaystyle\|I_{1}f\|^{2}_{L^{2}(P)}$
$\displaystyle=\sum_{T\in\mathcal{T}(P)}\|\sum_{z\in\mathcal{V}(T)}f(z)\varphi_{z}\|^{2}_{L^{2}(T)}\leq\sum_{T\in\mathcal{T}(P)}\frac{|T|}{3}\sum_{z\in\mathcal{V}(T)}f(z)^{2}$
$\displaystyle\leq\frac{1}{3}\sum_{j=0}^{N_{P}}f(z_{j})^{2}\sum_{\begin{subarray}{c}T\in\mathcal{T}(P)\\\
z_{j}\in\mathcal{V}(T)\end{subarray}}|T|\leq\frac{|P|}{3}\sum_{k=0}^{N_{P}}f(z_{k})^{2}.$
(2.5)

Since $0\in\text{conv}\\{f(z_{1}),\dots,f(z_{N_{P}})\\}$, there exists convex
coefficients $0\leq\mu_{1},\dots,\mu_{N_{P}}\leq 1$ with
$\sum_{\ell=1}^{N_{P}}\mu_{\ell}=1$ and
$\sum_{\ell=1}^{N_{P}}\mu_{\ell}f(z_{\ell})=0$. This implies
$\min_{j=1}^{N_{P}}f(z_{j})\leq 0\leq\max_{j=1}^{N_{P}}f(z_{j})$ and so
$x=\\{f(z_{1}),\dots,f(z_{N_{P}})\\}$ in [15, Lemma 4.2] guarantees

$\displaystyle\sum_{k=0}^{N_{P}}f(z_{k})^{2}\leq
f(z_{0})^{2}+M\sum_{j=1}^{N_{P}}(f(z_{j}))-f(z_{j+1}))^{2}$ (2.6)

for the constant $M:=(2(1-\cos(\pi/N_{P})))^{-1}$ (that exclusively depends on
$M(\rho)\geq N_{P}$ and so on $\rho$). The underlying inequality
$\sum_{j=1}^{N_{P}}x_{j}^{2}\leq M\sum_{j=1}^{N_{P}}(x_{j+1}-x_{j})^{2}$ with
$x_{N_{P}+1}=x_{1}$ follows for $x_{\ell}:=\min\\{x_{1},\dots,x_{N_{P}}\\}\leq
0\leq\max\\{x_{1},\dots,x_{N_{P}}\\}=:x_{m}$ for some indices
$\ell,m\in\\{1,\dots,N_{P}\\}$ immediately from
$\max\\{|x_{1}|,\dots,|x_{N_{P}}|\\}\leq|x_{\ell}|+|x_{m}|=|x_{\ell}-x_{m}|\leq\sum_{j=1}^{N_{P}}|x_{j+1}-x_{j}|\leq
N_{P}^{1/2}(\sum_{j=1}^{N_{P}}|x_{j+1}-x_{j}|^{2})^{1/2}$ with triangle and
Cauchy-Schwarz inequalities. The optimal constants in [15] require little
matrix analysis. The above coefficients $\mu_{\ell}$ and a Cauchy-Schwarz
inequality show that

$\displaystyle
f(z_{0})^{2}=\left(\sum_{\ell=1}^{N_{P}}\mu_{\ell}(f(z_{0})-f(z_{\ell}))\right)^{2}\leq
N_{P}\sum_{\ell=1}^{N_{P}}(f(z_{0})-f(z_{\ell}))^{2}.$ (2.7)

The combination (2.6)-(2.7) results in

$\displaystyle\sum_{k=0}^{N_{P}}f(z_{k})^{2}\leq
N_{P}\sum_{\ell=1}^{N_{P}}(f(z_{0}))-f(z_{\ell}))^{2}+M\sum_{j=1}^{N_{P}}(f(z_{j}))-f(z_{j+1}))^{2}.$
(2.8)

For any edge $E=\text{conv}\\{a,b\\}\in\mathcal{E}(\mathcal{T}(P))$ with
vertices $a,b\in\mathcal{V}(E)\subset\mathcal{V}(\mathcal{T}(P))$ and an
aligned triangle $T(E)=\text{conv}\\{z_{0},E\\}\supset E$, the tangential
derivative shows $|f(a)-f(b)|=\bigg{|}\int_{E}f_{\bm{\tau}}\,ds\bigg{|}$. The
trace identity $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}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(E)}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(E)}(x-z_{0})\cdot\nabla f(x)\,dx$ [15, Lemma
2.6] implies that

$\displaystyle 2h_{E}^{-1}|T(E)|\;|f(a)-f(b)|\leq 2\bigg{|}\int_{T(E)}\nabla
f(x)\cdot\bm{\tau}_{E}\,dx\bigg{|}+\bigg{|}\int_{T(E)}(x-z_{0})\cdot
D^{2}f(x)\bm{\tau}_{E}\;dx\bigg{|}.$

A Cauchy-Schwarz inequality provides

$\displaystyle|f(a)-f(b)|\leq
h_{E}|T(E)|^{-1/2}(|f|_{1,T(E)}+h_{T(E)}|f|_{2,T(E)}).$ (2.9)

Recall $\omega_{0}$ from Subsection 2.1 and note that
$\sin(\omega_{0})h_{T(E)}^{2}\leq 4|T(E)|$. This and $h_{E}\leq h_{T(E)}\leq
h_{P}$ for all $E\in\mathcal{E}(P)$ lead in (2.9) to

$\displaystyle|f(a)-f(b)|^{2}\leq\frac{8}{\sin(\omega_{0})}(|f|^{2}_{1,T(E)}+h_{P}^{2}|f|^{2}_{2,T(E)}).$

This applies to the triangles $T(E)=T(j)\in\mathcal{T}(P)$ with the edges
$\text{conv}\\{a,b\\}=\text{conv}\\{z_{j},z_{j+1}\\}\;$
and $\text{conv}\\{z_{0},z_{j}\\}$ for $j=1,\dots,N_{P}$ from Figure 2.1.b.
Hence (2.8) shows that

$\displaystyle(N_{P}+M)^{-1}\frac{\sin{\omega_{0}}}{8}\sum_{k=0}^{N_{P}}f(z_{k})^{2}\leq\sum_{E\in\mathcal{E}(P)}(|f|^{2}_{1,T(E)}+h_{P}^{2}|f|^{2}_{2,T(E)})=|f|^{2}_{1,P}+h_{P}^{2}|f|^{2}_{2,P}.$

This and (2.5) result in $\|I_{1}f\|_{L^{2}(P)}\leq
C(\rho)(h_{P}|f|_{1,P}+h_{P}^{2}|f|_{2,P})$ with $|P|\leq\pi h_{P}^{2}$ from
[di2019hybrid, Lemma 1.12], $N_{P}\leq M(\rho)$ from Subsection 2.1, and
$C(\rho)^{2}:=\frac{8\pi(M(\rho)+M)}{3\sin(\omega_{0})}$. The part $(a)$ of
the lemma applies to $\partial f/\partial x_{j}$ for $j=1$ and for $j=2$, and
so controls the term $|f|_{1,P}\leq C_{\mathrm{PF}}h_{P}|f|_{2,P}$.
Consequently, $\|I_{1}f\|_{L^{2}(P)}\leq
C(\rho)(1+C_{\mathrm{PF}})h_{P}^{2}|f|_{2,P}.$ This and (2.3)-(2.4) show
$\|f\|_{L^{2}(P)}\leq(C_{\text{BH}}+C(\rho)(1+C_{\mathrm{PF}}))h_{P}^{2}|f|_{2,P}$
and conclude the proof of $(b)$ with a re-labelled constant $C_{\mathrm{PF}}$.
∎

Recall $\text{Dof}:H^{2}(P)\to\mathbb{R}^{2N_{P}}$ for a polygonal domain
$P\in\mathcal{M}\in\mathbb{M}$ from (2.1).

######  Lemma 2.3.

Any $v\in H^{2}(P)$ with $\mathrm{Dof}(v)=0$ satisfies
$h_{P}^{-2}\|v\|_{L^{2}(P)}+h_{P}^{-1}|v|_{1,P}\leq C_{\mathrm{PF}}|v|_{2,P}.$

###### Proof.

If $\textbf{n}:=(\textbf{n}_{x},\textbf{n}_{y})$ is an outward unit normal to
an edge $E$, then the unit tangential vector along $E$ is
$\bm{\tau}:=(\bm{\tau}_{x},\bm{\tau}_{y})=(-\textbf{n}_{y},\textbf{n}_{x})$.
This leads to the split $\nabla
v=v_{\textbf{n}}\textbf{n}+v_{\bm{\tau}}\bm{\tau}$. An integration along
$E(k)$ of the tangential derivative implies for all $k=1,\dots,N_{P}$ that

$\displaystyle\int_{E(k)}\nabla
v\,ds=\Big{(}\int_{E(k)}v_{\textbf{n}}\,ds\Big{)}\textbf{n}_{E(k)}+(v(z_{k+1})-v(z_{k}))\bm{\tau}_{E(k)}.$
(2.10)

This and the re-summation
$\sum_{k=1}^{N_{P}}(v(z_{k+1})-v(z_{k}))\bm{\tau}_{E(k)}=\sum_{k=1}^{N_{P}}(\bm{\tau}_{E(k-1)}-\bm{\tau}_{E(k)})v(z_{k})$
lead to

$\displaystyle\int_{E(k)}\nabla
v\,ds=\text{dof}_{N_{P}+k}(v)\textbf{n}_{E(k)}+(\bm{\tau}_{E(k-1)}-\bm{\tau}_{E(k)})\text{dof}_{k}(v).$
(2.11)

If $\text{Dof}(v)=0$, then $\int_{E(k)}\nabla v\,ds=0$ for all
$k=1,\dots,N_{P}$ from (2.11). This and $v(z_{j})=\text{dof}_{j}(v)=0$ for all
$j=1,\dots,N_{P}$ allow the Poincaré-Friedrichs inequality in Theorem 2.2.b. ∎

###  2.3 Biharmonic model problem

Define the scalar product $a(\cdot,\cdot):V\times V\to\mathbb{R}$ for $u,v\in
V=H^{2}_{0}(\Omega)$ by

$\displaystyle
a(u,v):=\int_{\Omega}D^{2}u:D^{2}v\,dx\quad\text{with}\;D^{2}u:=\begin{pmatrix}u_{11}&u_{12}\\\
u_{12}&u_{22}\end{pmatrix}$ (2.12)

with $D^{2}u:D^{2}v:=u_{11}v_{11}+2u_{12}v_{12}+u_{22}v_{22}$. The subscripts
$\alpha,\beta=1,2$ abbreviate the second-order partial derivatives
$u_{\alpha\beta}:=\frac{\partial^{2}u}{\partial x_{\alpha}\partial
x_{\beta}}$. The bilinear form $a_{\mathrm{pw}}$ and the differential operator
$D^{2}_{\mathrm{pw}}$ denote the corresponding piecewise versions (with
respect to a partition $\mathcal{M}$ or $\mathcal{T}$ suppressed in the
notation). The local contribution $a^{P}(\cdot,\cdot)$ is the semi-scalar
product

$a^{P}(u,v):=\int_{P}D^{2}u:D^{2}v\,dx\quad\text{for}\;u,v\in H^{2}(P).$

The scalar product $a(\cdot,\cdot)$ induces the energy norm
$|v|_{2,\Omega}:=a(v,v)^{1/2}$ equivalent to the Sobolev norm
$\|\cdot\|_{2,\Omega}$ owing to the Friedrichs inequality [8, Sec. 10.6] and
$(V,a(\cdot,\cdot))$ is a Hilbert space. Given any $F\in
V^{*}=H^{-2}(\Omega)$, the Riesz representation is the weak solution $u\in V$
to

$\displaystyle a(u,v)=F(v)\quad\text{for all}\;v\in V.$ (2.13)

Elliptic regularity. For the fixed polygonal bounded Lipschitz domain
$\Omega$, there exist positive constants $\sigma_{\text{reg}}>1/2$ and
$C_{\text{reg}}$ [3, 5, 19] such that $F\in H^{-s}(\Omega)$ and $2-\sigma\leq
s\leq 2$ for $\sigma:=\min\\{\sigma_{\text{reg}},1\\}$ imply $u\in V\cap
H^{4-s}$ and

$\displaystyle\|u\|_{4-s,\Omega}\leq C_{\text{reg}}\|F\|_{-s,\Omega}.$ (2.14)

###  2.4 Galerkin projection

The $H^{2}$ elliptic projection operator $G:H^{2}(P)\to\mathcal{P}_{2}(P)$ is
defined, for any $v\in H^{2}(P)$, by $Gv\in\mathcal{P}_{2}(P)$ and

$\displaystyle a^{P}(Gv,\chi)=a^{P}(v,\chi)\quad\text{for
all}\;\chi\in\mathcal{P}_{2}(P)$ (2.15)

with the additional conditions (i.e., three equations to fix the affine
contribution)

$\displaystyle\frac{1}{N_{P}}\sum_{j=1}^{N_{P}}Gv(z_{j})=\frac{1}{N_{P}}\sum_{j=1}^{N_{P}}v(z_{j})\quad\text{and}\quad\int_{\partial
P}\nabla Gv\,ds=\int_{\partial P}\nabla v\,ds.$ (2.16)

Equation (2.15) determines $Gv\in\mathcal{P}_{2}(P)$ up to affine functions
and the additional three equations in (2.16) define $Gv\in\mathcal{P}_{2}(P)$
uniquely for $v\in H^{2}(P)$. The linear operator
$G:H^{2}(P)\to\mathcal{P}_{2}(P)$ is a projection onto $\mathcal{P}_{2}(P)$.
An integration by parts and (2.16) imply, for all $v\in H^{2}(P)$, that

$\displaystyle\Pi_{0}D^{2}v=\frac{1}{|P|}\int_{\partial P}\nabla
v\,ds=\frac{1}{|P|}\int_{\partial P}\nabla Gv\,ds=\Pi_{0}D^{2}Gv=D^{2}Gv.$
(2.17)

######  Lemma 2.4 (approximation error of $G$).

Any $v\in H^{2}(P)$ with $Gv\in\mathcal{P}_{2}(P)$ from (2.15)-(2.16)
satisfies
$C_{\mathrm{PF}}^{-1}\sum_{m=0}^{1}h_{P}^{m-2}|v-Gv|_{m,P}\leq|v-Gv|_{2,P}\leq|v|_{2,P}$
and there exists a positive constant $C_{\mathrm{apx}}$ (that exclusively
depends on $\rho$) such that $|v-Gv|_{2,P}\leq
C_{\mathrm{apx}}h_{P}^{s}|v|_{2+s,P}$ for $v\in H^{2+s}(P)$ and $0<s\leq 1$.

###### Proof.

The condition (2.16) implies that $\sum_{j=1}^{N_{P}}(v-Gv)(z_{j})=0$ and
$\int_{\partial P}\nabla(v-Gv)\,ds=0$. Hence the Poincaré-Friedrichs
inequality in Theorem 2.2.b proves that
$C_{\mathrm{PF}}^{-1}\sum_{m=0}^{1}h_{P}^{m-2}|v-Gv|_{m,P}\leq|v-Gv|_{2,P}$.
The Pythagoras identity $|v-Gv|^{2}_{2,P}+|Gv|^{2}_{2,P}=|v|^{2}_{2,P}$ from
(2.15) provides $|v-Gv|_{2,P}\leq|v|_{2,P}$. The definition of $G$ in (2.15)
shows that $|v-Gv|_{2,P}\leq|v-\chi|_{2,P}$ for any
$\chi\in\mathcal{P}_{2}(P)$. The Bramble-Hilbert lemma [25, Thm. 6.1]
concludes the proof. ∎

Recall $\text{Dof}:H^{2}(P)\to\mathbb{R}^{2N_{P}}$ for a polygonal domain
$P\in\mathcal{M}\in\mathbb{M}$ from (2.1).

######  Lemma 2.5 (boundedness of Dof).

There exists a positive constant $C_{d}$ (that exclusively depends on $\rho$)
such that any $v\in H^{2}(P)$ satisfies $|\mathrm{Dof}(v-Gv)|_{\ell^{2}}\leq
C_{d}h_{P}|v-Gv|_{2,P}.$

###### Proof.

The scaled Sobolev inequality from [7, Sec. 2.1.3] leads for $w:=v-Gv$ to

$\displaystyle|w(z_{0})|\leq\|w\|_{L^{\infty}(P)}\leq
C_{S}\sum_{m=0}^{2}h_{P}^{m-1}|w|_{m,P}$ (2.18)

with a positive constant $C_{S}$ (that exclusively depends on $\rho$). A
Cauchy-Schwarz inequality and the trace inequality $\|w\|^{2}_{L^{2}(E)}\leq
C_{T}(h_{E}^{-1}\|w\|^{2}_{L^{2}(P)}+h_{E}\|\nabla w\|^{2}_{L^{2}(P)})$ (e.g.,
from [7, p. 554]) for any edge $E\in\mathcal{E}(P)$ result in

$\displaystyle\Big{|}\int_{E}w_{\textbf{n}}\,ds\Big{|}\leq
h_{E}^{1/2}\|w_{\textbf{n}}\|_{L^{2}(\partial P)}\leq
C_{T}(|w|_{1,P}+h_{E}|w|_{2,P}).$ (2.19)

The combination of (2.18)-(2.19) and Lemma 2.4 imply that

$\displaystyle|w(z_{0})|\leq
C_{S}(1+C_{\mathrm{PF}})h_{P}|w|_{2,P}\quad\text{and}\quad\Big{|}\int_{E}w_{\textbf{n}}\,ds\Big{|}\leq
C_{T}(1+C_{\mathrm{PF}})h_{P}|w|_{2,P}.$ (2.20)

This concludes the proof of the lemma with
$C_{d}:=(C_{S}+C_{T})(1+C_{\mathrm{PF}})$. ∎

The following lemma estimates $|Gv|_{m,P}$ for $m=0,1,2$ and the Galerkin
projection $G$ in terms of dofs of $v$ for any $v\in H^{2}(P)$.

######  Lemma 2.6 ($G$ as a function of Dof).

The projection operator $Gv$ is computable in terms of the degrees of freedom
$\mathrm{Dof}(v)\in\mathbb{R}^{2N_{P}}$ for any $v\in H^{2}(P)$ and
$\sum_{m=0}^{2}h_{P}^{m-1}|Gv|_{m,P}\leq C_{g}|\mathrm{Dof}(v)|_{\ell^{2}}$
holds with a positive constant $C_{g}$ (that exclusively depends on $\rho$).

###### Proof.

Antonietti et al. discuss the proof [2, Lemma 3.3] of the computability of the
projection operator $G$ in terms of the dofs from (2.1). Appendix A provides
details of this first part and the proof of the estimates of $|Gv|_{m,P}$ for
$m=0,1,2$. ∎

##  3 Abstract framework and fundamental estimates

###  3.1 Hypotheses

Given any polygonal domain $P\in\mathcal{M}\in\mathbb{M}$, recall the geometry
and the local degrees of freedom from Subsection 2.1 and merely suppose the
hypotheses (H1)-(H2) throughout this paper.

  1. (H1)

The vector space $V_{h}(P)$ is of dimension $2N_{P}$, satisfies
$\displaystyle\mathcal{P}_{2}(P)\subseteq V_{h}(P)\subset H^{2}(P)$, and the
triplet $(P,V_{h}(P),(\text{dof}_{1},\dots,\text{dof}_{2N_{P}}))$ is a finite
element in the sense of Ciarlet.

The unique existence of a nodal basis $\psi_{1},\dots,\psi_{2N_{P}}$ of
$V_{h}(P)$ with $\text{dof}_{k}(\psi_{j})=\delta_{jk}$ for all
$j,k=1,\dots,2N_{P}$ is a consequence for any finite element in the sense of
Ciarlet [8, Chapter 3].

  1. (H2)

The aforementioned nodal basis functions $\psi_{1},\dots,\psi_{2N_{P}}$
satisfy $h_{P}\Big{(}\sum_{j=1}^{2N_{P}}|\psi_{j}|^{2}_{2,P}\Big{)}^{1/2}\leq
C_{\text{stab}}$ for a positive constant $C_{\text{stab}}$ (that exclusively
depends on $\rho$).

Notice that (H1)-(H2) also imply the uniform stability of the discrete problem
for the natural stabilization term $s_{h}$ in (5.1) below. The road map of the
proofs in the two examples below is outlined in the seminal contribution [21]
for a related virtual element space $V_{h}(P)$. Appendix B and C independently
provide details for the two examples below.

###  3.2 Examples of the discrete space $V_{h}(P)$

This subsection presents two examples [2, 35] of the local discrete space
$V_{h}(P)$ with (H1)-(H2). Recall that $\mathcal{P}_{r}(P)$ is the vector
space of polynomials of degree $\leq r$ regarded as functions in $P$, and fix
the parameter $r=-1,0,1,2$ with the convention $\mathcal{P}_{-1}(P)=\\{0\\}$.

####  3.2.1 First example of $V_{h}(P)$ from [2, 21]

The discrete space in [2, Sec. 3] and in [21, Sec. 2.3] solves the biharmonic
problem with boundary conditions for $\mathcal{P}_{0}(\mathcal{E}(P)):=\\{q\in
L^{\infty}(\partial P):\forall E\in\mathcal{E}(P)\quad
q|_{E}\in\mathcal{P}_{0}(E)\\}$,

$\displaystyle\widehat{V}_{h}(P)$
$\displaystyle:=\begin{rcases}\begin{dcases}v\in
H^{2}(P):\exists\;f\in\mathcal{P}_{r}(P)\;\exists\;g\in\mathcal{P}_{0}(\mathcal{E}(P))\;\exists\;a_{1},\dots,a_{N_{P}}\in\mathbb{R}\\\
\quad\forall\;w\in H^{2}(P)\quad
a^{P}(v,w)=(f,w)_{L^{2}(P)}+(g,w_{\textbf{n}})_{L^{2}(\partial
P)}+\sum_{j=1}^{N_{P}}a_{j}w(z_{j})\end{dcases}\end{rcases},$ (3.1)
$\displaystyle V_{h}(P)$
$\displaystyle:=\\{v\in\widehat{V}_{h}(P):v-Gv\perp\mathcal{P}_{r}(P)\quad\text{in}\;L^{2}(P)\\}.$
(3.2)

######  Proposition 3.1.

The discrete space $V_{h}(P)$ from (3.1)-(3.2) satisfies (H1)-(H2).

###### Proof.

The arguments in [21, Lemma 3.4-3.5] and in [21, Appendix A] can be adopted
for the proof of (H1) and of (H2) for $r=-1$. Appendix B presents a simpler
proof that also covers $r=0,1,2$. ∎

####  3.2.2 Second example of $V_{h}(P)$ generalizes [35]

Recall the set $\mathcal{E}(P)=\\{E(1),\dots,E(N_{P})\\}$ of edges and the set
$\mathcal{V}(P)=\\{z_{1},\dots,z_{N_{P}}\\}$ of vertices along the polygon
$\partial P=E(1)\cup\dots\cup E(N_{P})$ as in Subsection 2.1. The following
generalization of $V_{h}(P)$ in [35, Sec. 4] requires further notation with
corners as depicted in Figure 2.1.c. The boundary of the polygon $\partial
P:=\cup_{j=1}^{J}\text{conv}\\{\zeta_{j},\zeta_{j+1}\\}$ is also a polygon of
the corner points
$\zeta_{1},\dots,\zeta_{J}\subset\\{z_{1},\dots,z_{N_{P}}\\}$ with indices
$\zeta_{j}=z_{k(j)}$, $\zeta_{j+1}=z_{k(j)+m(j)}$, $k(j+1)=k(j)+m(j)$, and
$k(J+1):=k(1)$. By definition, the interior angle at a corner $\zeta_{j}$ is
different from $0,\pi,2\pi$, while it is equal to $\pi$ at all other vertices
$z_{j}\in\mathcal{V}(P)\setminus\\{\zeta_{1},\dots,\zeta_{J}\\}$. Given the
one-dimensional side
$\overline{\gamma(j)}:=\text{conv}\\{\zeta_{j},\zeta_{j+1}\\}=E(k(j))\cup\dots\cup
E(k(j)+m(j))$, consider the $(m(j)+2)$-dimensional quadratic $C^{1}$ spline
space

$S(j):=\mathcal{P}_{2}(\mathcal{E}(\gamma(j)))\cap
C^{1}(\gamma(j))\quad\text{for}\;j=1,\dots,J.$

The vertices $z_{k(j)},\dots,z_{k(j)+m(j)}$ on $\gamma(j)$ lead to a partition
of $\gamma(j)$, written as $\mathcal{E}(\gamma(j))$, and the subset of
functions in $S(j)$ that vanish at all those vertices form a one-dimensional
subspace span$\\{\psi_{j}\\}$ of $S(j)$. This is elementary to verify and
Appendix C exploits pictures and norms of $\psi_{j}$. It turns out that two
conditions on the sign $\psi_{j}|_{E(k(j))}\geq 0$ and on the scaling
$\|\psi_{j}\|_{L^{\infty}(\gamma(j))}=1$ determine $\psi_{j}$ uniquely. So
$\psi_{j}\in S(j)$ is fixed by the geometry of $P$ in Figure 2.1.c. The second
class of VEs is generalized through a linear functional
$\Lambda_{j}:S(j)\to\mathbb{R}$ with the normalization
$\Lambda_{j}(\psi_{j})=1$ and the boundedness $\|\Lambda_{j}\|\leq
C_{\Lambda}$ of the operator norm $\|\Lambda_{j}\|:=\sup\\{\Lambda_{j}(f):f\in
S(j),\|f\|_{L^{\infty}(\gamma(j))}=1\\}$ of $\Lambda_{j}$, provided $S(j)$ is
endowed with the maximum norm. We suppose that the upper bound $C_{\Lambda}$
exclusively depends on $\rho$. Abbreviate $W:=H^{1}_{0}(P)\cap H^{2}(P)$ and
define

$\displaystyle\widehat{W}_{h}(P)$
$\displaystyle:=\begin{cases}\begin{rcases}w\in H^{2}(P)&:w|_{\partial P}\in
C^{0}(\partial P),\quad\forall{j=1,\dots,J}\quad w|_{\gamma(j)}\in
S(j),\;\text{and}\\\ &\exists f\in\mathcal{P}_{r}(P)\quad\exists
g\in\mathcal{P}_{0}(\mathcal{E}(P))\quad\forall\phi\in W\\\
&a^{P}(w,\phi)=(f,\phi)_{L^{2}(P)}+(g,\phi_{\textbf{n}})_{L^{2}(\partial
P)}\end{rcases},\end{cases}$ (3.3) $\displaystyle W_{h}(P)$
$\displaystyle:=\begin{cases}\begin{rcases}w\in\widehat{W}_{h}(P):&\forall
j=1,\dots,J\quad\Lambda_{j}((w-Gw)|_{\gamma(j)})=0\;\text{and}\\\
&w-Gw\perp\mathcal{P}_{r}(P)\quad\text{in}\;L^{2}(P)\end{rcases}.\end{cases}$
(3.4)

######  Proposition 3.2.

The discrete space $W_{h}(P)$ from (3.3)-(3.4) satisfies (H1)-(H2).

###### Proof.

The proof of (H1) for $\Lambda_{j}$ from Example 3.4 below and for $r=-1$ is
given in [35, Lemma 4.1]. A proof of (H2) with clear dependence of the
constant $C_{\text{stab}}$ on the mesh regularity parameter $\rho$ seems
missing in the literature. Appendix C contains further details for the general
case and the proof of (H2) with explicit constant. ∎

######  Example 3.3 (Example of $\Lambda_{j}$).

Given the first edge $E(k(j))$ in $\gamma(j)$, define
$\Lambda_{j}(v):=\frac{3}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(k(j))}v\,ds$ for $v\in L^{1}(\gamma(j))$. Then
$\Lambda_{j}(\psi_{j})=1$ and $\|\Lambda_{j}\|\leq C_{\Lambda}=3/2$.

######  Example 3.4 (Comparison with [35]).

Zhao et al. consider merely convex polygons in [35] and we interpret that this
implies that the sides are the edges ($\zeta_{j}=z_{j}$ for all $j=1,\dots,J$
and $J=N_{P}$; all interior angles in $P$ are different from $\pi$). Then
their choice $\Lambda_{j}(\bullet)=\mathchoice{{\vbox{\hbox{$\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)}\bullet\,ds$ coincides with Example 3.3 and
recovers $V_{h}(P)$ in [35] for certain geometries.

We continue the more general discussion and point out that, for each
$P\in\mathcal{M}$, $V_{h}(P)$ can even be selected from either (3.1)-(3.2) or
(3.3)-(3.4) and a mixture is allowed in the abstract framework at hand. In
particular, for different polygons $P\in\mathcal{M}$, the space $V_{h}(P)$ and
the parameter $r$ could be different.

###  3.3 Interpolation

Given $P\in\mathcal{M}\in\mathbb{M}$, recall from (H1)-(H2) the nodal basis
$(\psi_{1},\dots,\psi_{2N_{P}})$ of $V_{h}(P)$ with
dof${}_{j}(\psi_{k})=\delta_{jk}$ for $j,k=1,\dots,2N_{P}$.

######  Definition 3.5 (local interpolation operator).

Define $I_{h}^{P}:H^{2}(P)\to V_{h}(P)$ by

$\displaystyle
I_{h}^{P}v=\sum_{j=1}^{2N_{P}}\text{dof}_{j}(v)\psi_{j}\quad\text{for
all}\;v\in H^{2}(P).$ (3.5)

Recall the notation $V_{\mathrm{nc}}$ from Subsection 2.1 for
$\mathcal{M}\in\mathbb{M}$.

######  Definition 3.6 (global discrete space).

The (nonconforming) global virtual element space is the collection of all
local spaces $V_{h}(P)$ for $P\in\mathcal{M}$ with well-defined global dofs
from (2.2), namely

$\displaystyle V_{h}:=\begin{rcases}\begin{dcases}v_{h}\in
V_{\mathrm{nc}}:&\forall\;P\in\mathcal{M}\quad v_{h}|_{P}\in
V_{h}(P)\end{dcases}\end{rcases}.$

######  Definition 3.7 (global interpolation operator).

Define the global interpolation operator $I_{h}:V_{\mathrm{nc}}\to V_{h}$ by
$(I_{h}v_{\mathrm{nc}})|_{P}:=I_{h}^{P}(v_{\mathrm{nc}}|_{P})$ for all
$P\in\mathcal{M}$. (The global interpolation $I_{h}$ is well-defined because
the dofs from (2.1) are uniquely defined for any $v_{\mathrm{nc}}\in
V_{\mathrm{nc}}$.)

###  3.4 Interpolation error estimates

The main results about $I_{h}$ are summarized as follows.

######  Theorem 3.8 (interpolation).

There exists a positive constant $C_{\mathrm{I}}$ (that exclusively depends on
$\rho$) such that any $v\in H^{2}(P)$ and its interpolation $I_{h}^{P}v\in
V_{h}(P)$ from Definition 3.5 satisfy

  1. $\left(a\right)$

$\displaystyle
D^{2}_{\mathrm{pw}}(v-I_{h}^{P}v)\perp\mathcal{P}_{0}(P;\mathbb{S})\quad\text{in}\;L^{2}(P;\mathbb{S}),$

  2. $\left(b\right)$

$GI_{h}^{P}v=Gv$ and
$\displaystyle\sum_{m=0}^{2}h_{P}^{m-2}|v-GI_{h}^{P}v|_{m,P}\leq(1+C_{\mathrm{PF}})|v-Gv|_{2,P},$

  3. $\left(c\right)$

$\displaystyle|I_{h}^{P}v|_{2,P}\leq C_{\mathrm{Ib}}|v|_{2,P},$

  4. $\left(d\right)$

$\displaystyle\sum_{m=0}^{2}h_{P}^{m-2}|v-I_{h}^{P}v|_{m,P}\leq
C_{\mathrm{I}}|v-Gv|_{2,P}\leq
C_{\mathrm{I}}C_{\mathrm{apx}}h_{P}^{s}|v|_{2+s,P}$

provided $v\in H^{2+s}(P)$ for $0<s\leq 1$ in the last estimate in $(d)$ with
$C_{\mathrm{apx}}$ from Lemma 2.4.

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

Since $v$ and $I_{h}^{P}v$ coincide at the vertices from Definition 3.5 of
$I_{h}^{P}$, their tangential derivatives satisfy

$\displaystyle\int_{E}(I_{h}^{P}v)_{\bm{\tau}}\,ds=I_{h}^{P}v(z_{2})-I_{h}^{P}v(z_{1})=v(z_{2})-v(z_{1})=\int_{E}v_{\bm{\tau}}\,ds$
(3.6)

for the vertices $z_{1},z_{2}$ of an edge $E$ directed from $z_{1}$ to
$z_{2}=z_{1}+h_{E}\bm{\tau}_{E}$. An integration by parts proves the first
step and the split $\nabla v=v_{\textbf{n}}\textbf{n}+v_{\bm{\tau}}\bm{\tau}$
proves the second step in

$\displaystyle\int_{P}D^{2}v\,dx$
$\displaystyle=\sum_{E\in\mathcal{E}(P)}\left(\int_{E}\nabla
v\,ds\right)\otimes\textbf{n}_{P}|_{E}=\sum_{E\in\mathcal{E}(P)}\left(\int_{E}(v_{\textbf{n}}\textbf{n}+v_{\bm{\tau}}\bm{\tau})\,ds\right)\otimes\textbf{n}_{P}|_{E}.$
(3.7)

The combination of (3.6)-(3.7) and
$\int_{E}(I_{h}^{P}v)_{\textbf{n}}\,ds=\int_{E}(v)_{\textbf{n}}\,ds$ from
Definition 3.5 shows

$\displaystyle\int_{P}D^{2}v\,dx=\sum_{E\in\mathcal{E}(P)}\left(\int_{E}((I_{h}^{P}v)_{\textbf{n}}\textbf{n}+(I_{h}^{P}v)_{\bm{\tau}}\bm{\tau})\,ds\right)\otimes\textbf{n}_{P}|_{E}=\int_{P}D^{2}I_{h}^{P}v\,dx$
(3.8)

with the split for $\nabla
I_{h}^{P}v=(I_{h}^{P}v)_{\textbf{n}}\textbf{n}+(I_{h}^{P}v)_{\bm{\tau}}\bm{\tau}$
and an integration by parts in the last step. ∎

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

Since the dofs of $I_{h}^{P}v$ and $v$ coincide by Definition 3.5 and $G$ is
uniquely determined by Lemma 2.6, $GI_{h}^{P}v=Gv$ for $v\in H^{2}(P)$. Hence
Lemma 2.4 concludes the proof. ∎

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

Since $I_{h}^{P}v-Gv\in V_{h}(P)$ for any $v\in H^{2}(P)$, the nodal basis
functions $\psi_{j}\in V_{h}(P)$ for $j=1,\dots,2N_{P}$ and the dofs from
(2.1) with $\text{Dof}(I_{h}^{P}v)=\text{Dof}(v)$ lead to

$\displaystyle
I_{h}^{P}v-Gv=\sum_{j=1}^{2N_{P}}\text{dof}_{j}(I_{h}^{P}v-Gv)\psi_{j}=\sum_{j=1}^{2N_{P}}\text{dof}_{j}(v-Gv)\psi_{j}.$

The Cauchy-Schwarz inequality $|I_{h}^{P}v-Gv|_{2,P}\leq
h_{P}^{-1}|\text{Dof}(v-Gv)|_{\ell^{2}}h_{P}\Big{(}\sum_{j=1}^{2N_{P}}|\psi_{j}|_{2,P}^{2}\Big{)}^{1/2},$
Lemma 2.4-2.5, and (H2) result in

$\displaystyle|I_{h}^{P}v-Gv|_{2,P}\leq C_{d}C_{\text{stab}}|v|_{2,P}.$ (3.9)

Since $|Gv|_{2,P}\leq|v|_{2,P}$ (from (2.15)), (3.9) and the triangle
inequality $|I_{h}^{P}v|_{2,P}\leq|I_{h}^{P}v-Gv|_{2,P}+|Gv|_{2,P}$ conclude
the proof with $C_{\text{Ib}}:=1+C_{d}C_{\text{stab}}$. ∎

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

Substitute $w:=v-I_{h}^{P}v\in H^{2}(P)$ for $v$ in $(b)$ and observe
$I_{h}^{P}w=0$, so $GI_{h}^{P}w=0,$ to derive

$\displaystyle(1+C_{\mathrm{PF}})^{-1}\sum_{m=0}^{2}h_{P}^{m-2}|w|_{m,P}\leq|w-Gw|_{2,P}\leq|v-Gv|_{2,P}+|I_{h}^{P}v-GI_{h}^{P}v|_{2,P}$
(3.10)

with a triangle inequality in the last step. Let $g:=v-GI_{h}^{P}v\in
H^{2}(P)$ with $I_{h}^{P}g=I_{h}^{P}v-GI_{h}^{P}v$ from
$I_{h}^{P}(GI_{h}^{P}v)=GI_{h}^{P}v$. Then $(b)$-$(c)$ result in
$|I_{h}^{P}g|_{2,P}\leq C_{\text{Ib}}|g|_{2,P}\leq C_{\text{Ib}}|v-Gv|_{2,P}$.
This and (3.10) conclude the proof of $(d)$ with
$C_{\text{I}}:=(1+C_{\mathrm{PF}})(1+C_{\text{Ib}})$. ∎

##  4 Conforming companion

Recall that $\mathcal{T}$ is a shape-regular sub-triangulation of the
polygonal mesh $\mathcal{M}\in\mathbb{M}$ of the domain $\Omega$ from
Subsection 2.1 with the set of edges
$\widehat{\mathcal{E}}\supset\mathcal{E}$.

###  4.1 Morley interpolation

The Morley finite element space $\mathrm{M}(\mathcal{T})$ is defined as

$\displaystyle\mathrm{M}(\mathcal{T}):=\begin{rcases}\begin{dcases}v_{\mathrm{M}}\in\mathcal{P}_{2}(\mathcal{T}):&v_{\mathrm{M}}\;\text{is
continuous at interior vertices and zero at boundary}\\\ &\text{vertices of
$\mathcal{T}$, and $(v_{\mathrm{M}})_{\textbf{n}}$ is continuous at midpoints
of interior}\\\ &\text{edges and zero at midpoints of boundary edges of
$\widehat{\mathcal{E}}$}\end{dcases}\end{rcases}.$

The Morley interpolation operator
$I_{\mathrm{M}}:V_{h}\to\mathrm{M}(\mathcal{T})$ is (uniquely) defined by

$\displaystyle
I_{\mathrm{M}}v_{h}(z)=v_{h}(z)\quad\text{at}\;z\in\widehat{\mathcal{V}}\;\text{and}\;\int_{E}(I_{\mathrm{M}}v_{h})_{\textbf{n}}\,ds=\int_{E}(v_{h})_{\textbf{n}}\,ds\quad\text{on}\;E\in\widehat{\mathcal{E}}.$
(4.1)

A well-known consequence of (4.1) [19, Lemma 3.1] (follows as the proof of
Theorem 3.8.a) is

$\displaystyle
D^{2}_{\mathrm{pw}}(v_{h}-I_{\mathrm{M}}v_{h})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{S})\quad\text{in}\;L^{2}(\Omega;\mathbb{S}).$
(4.2)

The interpolation error estimate from [11, Theorem 3] (for functions in
$H^{2}(T)$) implies

$\displaystyle\frac{1}{2}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{h}-I_{\mathrm{M}}v_{h})|_{m,\mathrm{pw}}\leq|v_{h}-Gv_{h}|_{2,\mathrm{pw}}.$
(4.3)

######  Lemma 4.1 (conforming companion for Morley [27]).

There exists a linear map $J^{\prime}:\mathrm{M}(\mathcal{T})\to
H^{2}_{0}(\Omega)$ and a constant $C_{J^{\prime}}$ (that exclusively depends
on $\rho$) such that any $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ satisfies
$(a)$ $J^{\prime}v_{\mathrm{M}}(z)=v_{\mathrm{M}}(z)$ at all vertices
$z\in\widehat{\mathcal{V}}$, $(b)$ $\mathchoice{{\vbox{\hbox{$\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_{\mathrm{M}})_{\textbf{n}}\,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_{\mathrm{M}})_{\textbf{n}}\,ds$ on all edges
$E\in{\widehat{\mathcal{E}}}$,
$(c)$ $\displaystyle
D^{2}_{\mathrm{pw}}(v_{\mathrm{M}}-J^{\prime}v_{\mathrm{M}})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{S})$
in $L^{2}(\Omega;\mathbb{S})$,
$(d)$
$\displaystyle\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\mathrm{M}}-J^{\prime}v_{\mathrm{M}})|_{m,\mathrm{pw}}\leq
C_{J^{\prime}}\min_{v\in V}|v_{\mathrm{M}}-v|_{2,\mathrm{pw}}$.∎

###  4.2 Computable Morley interpolation

The virtual element function $v_{h}\in V_{h}$ is given implicitly such that
the computation of $I_{\mathrm{M}}v_{h}$ is possible in principle, but too
costly. The aim in this section is the analysis of a function
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ that is computable in terms of dofs
of $v_{h}\in V_{h}$. Given any $v_{h}\in V_{h}$ set

$\displaystyle
v_{\mathrm{M}}(z):=\begin{cases}v_{h}(z)\;\text{at}\;z\in\mathcal{V},\\\
Gv_{h}(z)\;\text{at}\;z\in\widehat{\mathcal{V}}\setminus\mathcal{V}\end{cases}\;\text{and}\quad\int_{E}(v_{\mathrm{M}})_{\textbf{n}}\,ds:=\begin{cases}\int_{E}(v_{h})_{\textbf{n}}\,ds\;\text{on}\;E\in\mathcal{E},\\\
\int_{E}(Gv_{h})_{\textbf{n}}\,ds\;\text{on}\;E\in\widehat{\mathcal{E}}\setminus\mathcal{E}.\end{cases}$
(4.4)

Notice that $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ is well-defined and
computable in terms of the dofs of $v_{h}\in V_{h}$ (for
$z\in\mathcal{V},E\in\mathcal{E}$) and from
$Gv_{h}\in\mathcal{P}_{2}(\mathcal{T})$ with Lemma 2.6 (for
$z\in\widehat{\mathcal{V}}\setminus\mathcal{V},E\in\widehat{\mathcal{E}}\setminus\mathcal{E}$).
The following estimates control the difference between $I_{\mathrm{M}}v_{h}$
and $v_{\mathrm{M}}$ in $\mathrm{M}(\mathcal{T})$ for $v_{h}\in V_{h}$.

######  Lemma 4.2 (key).

There exists a positive constant $C_{\mathrm{M}}$ (that exclusively depends on
$\rho$) with

$\displaystyle\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(I_{\mathrm{M}}v_{h}-v_{\mathrm{M}})|_{m,\mathrm{pw}}\leq
C_{\mathrm{M}}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}\quad\text{for any}\;v_{h}\in
V_{h}\;\text{and}\;v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})\;\text{with}\;\eqref{newm}.$

###### Proof.

Recall the sub-triangulation $\mathcal{T}(P)$ of $P$ with mid-point $z_{0}$
and the set of edges $\widehat{\mathcal{E}}(P)$ (resp. $\mathcal{E}(P)$) in
$\mathcal{T}(P)$ (resp. $P$) from Subsection 2.1 for a polygonal domain
$P\in\mathcal{M}$. Let $\psi_{z}$ and $\psi_{E}$ be the nodal basis functions
of $\mathrm{M}(\mathcal{T})$ for $z\in\widehat{\mathcal{V}}$ and
$E\in\widehat{\mathcal{E}}$, and write
$I_{\mathrm{M}}v_{h}-v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ as

$\displaystyle(I_{\mathrm{M}}v_{h}-v_{\mathrm{M}})|_{P}=(v_{h}-Gv_{h})(z_{0})\psi_{z_{0}}+\sum_{E\in\widehat{\mathcal{E}}(P)\setminus\mathcal{E}(P)}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}(v_{h}-Gv_{h})_{\textbf{n}}\,ds\right)\psi_{E}.$
(4.5)

The basis functions $\psi_{z}$ and $\psi_{E}$ are written explicitly, e.g., in
[12, Sec. 6] and scale like

$\displaystyle|\psi_{z_{0}}|_{m,T}\approx
h_{T}^{1-m}\quad\text{and}\quad|\psi_{E}|_{m,T}\approx
h_{T}^{2-m}\quad\text{for}\;m=0,1,2$ (4.6)

in a triangle $T\in\mathcal{T}(P)$ with $E\subset\partial T$. Note that the
integral means of normal derivatives of $\psi_{E}$ over edges enter (4.5) and
not the dofs for the ncVEM from (2.1). Lemma 2.5 with integral means along
edges $E\in\widehat{\mathcal{E}}(P)\setminus\mathcal{E}(P)$ in (2.20) and
$h_{P}\leq\rho^{-1}h_{E}$ from (M2) imply that

$\displaystyle
h_{P}^{-1}|(v_{h}-Gv_{h})(z_{0})|+\Big{|}\mathchoice{{\vbox{\hbox{$\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}-Gv_{h})_{\textbf{n}}\,ds\Big{|}\lesssim|v_{h}-Gv_{h}|_{2,P}.$
(4.7)

The combination (4.5)-(4.7) concludes the proof. ∎

###  4.3 Computable companion for VEM

The new tool in this paper is the conforming companion operator $J$
established in Theorem 4.3. The estimate in part (c) of which is sharp in the
sense that the reverse inequality also holds (with a multiplicative constant
$2$ from a triangle inequality).

######  Theorem 4.3 (companion operator for VEM).

There exist a linear map $J:V_{h}\to H^{2}_{0}(\Omega)$, which is a right-
inverse to the interpolation operator $I_{h}$, and a universal constant
$C_{J}$ (that exclusively depends on $\rho$) such that any $v_{h}\in V_{h}$
satisfies $(a)$-$(c)$.
$(a)$
$D^{2}_{\mathrm{pw}}(v_{h}-Jv_{h})\perp\mathcal{P}_{0}(\mathcal{M};\mathbb{S})\quad\text{in}\;L^{2}(\Omega;\mathbb{S}),\qquad$
$(b)$
$Gv_{h}-Jv_{h}\perp\mathcal{P}_{2}(\mathcal{M})\quad\text{in}\;L^{2}(\Omega)$,
$(c)$
$\displaystyle\sum_{m=0}^{2}(|h_{\mathcal{M}}^{m-2}(Gv_{h}-Jv_{h})|_{m,\mathrm{pw}}+|h_{\mathcal{M}}^{m-2}(v_{h}-Jv_{h})|_{m,\mathrm{pw}})\leq
C_{J}\big{(}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+\min_{v\in
V}|v_{h}-v|_{2,\mathrm{pw}}\big{)}.$

###### Proof.

Construction of $J$. For a given $v_{h}\in V_{h}$, a composite function
$J^{\prime}v_{\mathrm{M}}$ belongs to $V=H^{2}_{0}(\Omega)$ and we need a
modification to achieve $(b)$. For each $T\in\mathcal{T}(P)$ from Subsection
2.1, the cubic bubble-function $b_{T}:=27\lambda_{1}\lambda_{2}\lambda_{3}\in
H_{0}^{1}(T)$ for the barycentric co-ordinates
$\lambda_{1},\lambda_{2},\lambda_{3}\in\mathcal{P}_{1}(T)$ of $T$, leads to
$b_{T}^{2}\in H^{2}_{0}(T)$ with $0\leq b_{T}^{2}\leq 1$ 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_{T}b_{T}^{2}\,dx=81/280$. Hence
$\mathchoice{{\vbox{\hbox{$\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$ follows for

$\displaystyle b_{P}:=\frac{280}{81}\sum_{T\in\mathcal{T}(P)}b_{T}^{2}\in
H_{0}^{2}(P)\subset H^{2}_{0}(\Omega)\quad\text{for}\;P\in\mathcal{M}.$ (4.8)

Let $v_{P}\in\mathcal{P}_{2}(P)$ be the Riesz representation of the linear
functional $\mathcal{P}_{2}(P)\to\mathbb{R}$,
$w_{2}\mapsto(Gv_{h}-J^{\prime}v_{\mathrm{M}},w_{2})_{L^{2}(P)}$ in the
Hilbert space $\mathcal{P}_{2}(P)$ endowed with the weighted scalar product
$(b_{P}\bullet,\bullet)_{L^{2}(P)}$. In other words, given $v_{h}\in V_{h}$,
$v_{\mathcal{M}}\in\mathcal{P}_{2}(\mathcal{M})$ with
$v_{\mathcal{M}}|_{P}:=v_{P}$ satisfies

$\displaystyle(b_{P}v_{\mathcal{M}},w_{2})_{L^{2}(\Omega)}=(Gv_{h}-J^{\prime}v_{\mathrm{M}},w_{2})_{L^{2}(\Omega)}\quad\text{for
all}\;w_{2}\in\mathcal{P}_{2}(\mathcal{M});$ (4.9)

whence $\Pi_{2}(Gv_{h}-J^{\prime}v_{\mathrm{M}})=\Pi_{2}(b_{P}v_{P})$. Given
$v_{h}\in V_{h}$ with $v_{\mathcal{M}}\in\mathcal{P}_{2}(\mathcal{M})$ define

$\displaystyle
Jv_{h}:=J^{\prime}v_{\mathrm{M}}+\sum_{P\in\mathcal{M}}b_{P}v_{\mathcal{M}}\in
V.$ (4.10)

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

The definitions of $J$ and $b_{P}\in H^{2}_{0}(P)$ imply for any vertex
$z\in\mathcal{V}$ and for any edge $E\in\mathcal{E}$ that
$Jv_{h}(z)=J^{\prime}v_{\mathrm{M}}(z)$ 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_{E}(Jv_{h})_{\textbf{n}}\,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_{\mathrm{M}})_{\textbf{n}}\,ds.$
This, Lemma 4.1.a-b as $\mathcal{V}\subset\widehat{\mathcal{V}}$ and
$\mathcal{E}\subset\widehat{\mathcal{E}}$, and (4.4) lead to

$\displaystyle Jv_{h}(z)$
$\displaystyle=J^{\prime}v_{\mathrm{M}}(z)=v_{\mathrm{M}}(z)=v_{h}(z),$ (4.11)
$\displaystyle\mathchoice{{\vbox{\hbox{$\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})_{\textbf{n}}\,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_{E}(J^{\prime}v_{\mathrm{M}})_{\textbf{n}}\,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_{\mathrm{M}})_{\textbf{n}}\,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})_{\textbf{n}}\,ds.$ (4.12)

The proof of $(a)$ follows from (4.11)-(4.12) as in the proof of Theorem
3.8.a. ∎

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

The definition of $J$ in (4.10) directly shows
$\Pi_{2}(Gv_{h})=\Pi_{2}(Jv_{h})$. ∎

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

Abbreviate $\overline{v}:=N_{P}^{-1}\sum_{j=1}^{N_{P}}v(z_{j})$ for any $v\in
H^{2}(P)$. The definition of $G$ in (2.16) and (4.11) imply
$\overline{Gv_{h}-Jv_{h}}=\overline{Gv_{h}-v_{h}}+\overline{v_{h}-Jv_{h}}=0$.
The split $\nabla
Jv_{h}:=(Jv_{h})_{\textbf{n}}\textbf{n}+(Jv_{h})_{\bm{\tau}}\bm{\tau}$ and an
integration of the tangential component show for $k=1,\dots,N_{P}$ that

$\displaystyle\int_{E(k)}\nabla Jv_{h}\,ds$
$\displaystyle=\int_{E(k)}(Jv_{h})_{\textbf{n}}\textbf{n}_{E(k)}\,ds+(Jv_{h}(z_{k+1})-Jv_{h}(z_{k})))\bm{\tau}_{E(k)}$
$\displaystyle=\int_{E(k)}(v_{h})_{\textbf{n}}\textbf{n}_{E(k)}\,ds+(v_{h}(z_{k+1})-v_{h}(z_{k})))\bm{\tau}_{E(k)}=\int_{E(k)}\nabla
v_{h}\,ds$ (4.13)

with (4.11)-(4.12) in the second step and $\nabla
v_{h}:=(v_{h})_{\textbf{n}}\textbf{n}+(v_{h})_{\bm{\tau}}\bm{\tau}$ in the
last step. This and (2.16) lead to $\int_{\partial
P}\nabla(Gv_{h}-Jv_{h})\,ds=\int_{\partial
P}\nabla(Gv_{h}-v_{h})\,ds+\int_{\partial P}\nabla(v_{h}-Jv_{h})\,ds=0$. Hence
the Poincaré-Friedrich inequality in Theorem 2.2.b implies

$\displaystyle\sum_{m=0}^{2}|h_{\mathcal{M}}^{m-2}(Gv_{h}-Jv_{h})|_{m,\mathrm{pw}}\leq(1+C_{\mathrm{PF}})|Gv_{h}-Jv_{h}|_{2,\mathrm{pw}}.$
(4.14)

It remains to control $|Gv_{h}-Jv_{h}|_{2,\mathrm{pw}}$. There exists a
positive constant $C_{b}$ in the inverse estimates

$\displaystyle C_{b}^{-1}\|\chi\|^{2}_{L^{2}(P)}\leq$
$\displaystyle(b_{P},\chi^{2})_{L^{2}(P)}\leq C_{b}\|\chi\|^{2}_{L^{2}(P)},$
(4.15) $\displaystyle C_{b}^{-1}\|\chi\|_{L^{2}(P)}\leq$
$\displaystyle\sum_{m=0}^{2}h_{P}^{m}|b_{P}\chi|_{m,P}\leq
C_{b}\|\chi\|_{L^{2}(P)}\quad\text{for any}\;\chi\in\mathcal{P}_{2}(P).$
(4.16)

The inequalities (4.15)-(4.16) are standard inverse estimates for a (shape-
regular) triangle (replace $P$ by $T\in\mathcal{T}(P)$ therein) and (4.8)
reveals (4.15)-(4.16) as a sum of those. Hence $C_{b}$ depends exclusively on
$\rho$. The definition of $J$ in (4.10) shows
$|J^{\prime}v_{\mathrm{M}}-Jv_{h}|_{2,P}=|b_{P}v_{P}|_{2,P}$ and the inverse
inequality (4.16) proves $|b_{P}v_{P}|_{2,P}\leq
C_{b}h_{P}^{-2}\|v_{P}\|_{L^{2}(P)}.$ The first inequality in (4.15) and the
definition of $v_{P}\in\mathcal{P}_{2}(P)$ in (4.9) lead to

$\displaystyle
C_{b}^{-1}\|v_{P}\|^{2}_{L^{2}(P)}\leq(b_{P}v_{P},v_{P})_{L^{2}(P)}=(Gv_{h}-J^{\prime}v_{\mathrm{M}},v_{P})_{L^{2}(P)}\leq\|Gv_{h}-J^{\prime}v_{\mathrm{M}}\|_{L^{2}(P)}\|v_{P}\|_{L^{2}(P)}$

with a Cauchy-Schwarz inequality in the last step. Hence
$\|v_{P}\|_{L^{2}(P)}\leq C_{b}\|Gv_{h}-J^{\prime}v_{\mathrm{M}}\|_{L^{2}(P)}$
and the combination with the above estimates verifies

$|J^{\prime}v_{\mathrm{M}}-Jv_{h}|_{2,P}=|b_{P}v_{P}|_{2,P}\leq
C_{b}^{2}h_{P}^{-2}\|(Gv_{h}-J^{\prime}v_{\mathrm{M}})\|_{L^{2}(P)}.$

Triangle inequalities and Lemma 4.1.d lead, for any $v\in V$, to

$\displaystyle C_{b}^{-2}|J^{\prime}v_{\mathrm{M}}-Jv_{h}|_{2,\mathrm{pw}}$
$\displaystyle\leq\|h_{\mathcal{M}}^{-2}(Gv_{h}-I_{\mathrm{M}}v_{h})\|_{L^{2}(\Omega)}+\|h_{\mathcal{M}}^{-2}(I_{\mathrm{M}}v_{h}-v_{\mathrm{M}})\|_{L^{2}(\Omega)}$
$\displaystyle\quad+C_{J^{\prime}}(|v_{\mathrm{M}}-I_{\mathrm{M}}v_{h}|_{2,\mathrm{pw}}+|I_{\mathrm{M}}v_{h}-v_{h}|_{2,\mathrm{pw}}+|v_{h}-v|_{2,\mathrm{pw}}).$
(4.17)

The estimation of the upper bound in (4.17) involves a few arguments like the
triangle inequality
$\|h_{\mathcal{M}}^{-2}(Gv_{h}-I_{\mathrm{M}}v_{h})\|_{L^{2}(\Omega)}\leq\|h_{\mathcal{M}}^{-2}(Gv_{h}-v_{h})\|_{L^{2}(\Omega)}+\|h_{\mathcal{M}}^{-2}(v_{h}-I_{\mathrm{M}}v_{h})\|_{L^{2}(\Omega)}$,
$\|h_{\mathcal{M}}^{-2}(Gv_{h}-v_{h})\|_{L^{2}(\Omega)}\leq
C_{\mathrm{PF}}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$ from Lemma 2.4,
$\|h_{\mathcal{M}}^{-2}(v_{h}-I_{\mathrm{M}}v_{h})\|_{L^{2}(\Omega)}+|v_{h}-I_{\mathrm{M}}v_{h}|_{2,\mathrm{pw}}\leq
2|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$ from (4.3), and
$|v_{\mathrm{M}}-I_{\mathrm{M}}v_{h}|\leq
C_{\mathrm{M}}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$ from Lemma 4.2. This and the
fact that $v\in V$ is arbitrary eventually result in

$\displaystyle|J^{\prime}v_{\mathrm{M}}-Jv_{h}|_{2,\mathrm{pw}}\leq
C_{b}^{2}(2+C_{\mathrm{PF}}+C_{\mathrm{M}}+C_{J^{\prime}}(2+C_{\mathrm{M}}))\Big{(}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+\min_{v\in
V}|v_{h}-v|_{2,\mathrm{pw}}\Big{)}$ (4.18)

as well as
$|v_{\mathrm{M}}-v_{h}|_{2,\mathrm{pw}}\leq|v_{\mathrm{M}}-I_{\mathrm{M}}v_{h}|_{2,\mathrm{pw}}+|I_{\mathrm{M}}v_{h}-v_{h}|_{2,\mathrm{pw}}\leq(2+C_{\mathrm{M}})|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$.
The latter estimate, Lemma 4.1.d, and a triangle inequality lead, for any
$v\in V$, to

$\displaystyle
C_{J^{\prime}}^{-1}|v_{\mathrm{M}}-J^{\prime}v_{\mathrm{M}}|_{2,\mathrm{pw}}\leq|v_{\mathrm{M}}-v|_{2,\mathrm{pw}}\leq(2+C_{\mathrm{M}})|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+|v_{h}-v|_{2,\mathrm{pw}}.$
(4.19)

Recall $|I_{\mathrm{M}}v_{h}-v_{\mathrm{M}}|_{2,\mathrm{pw}}\leq
C_{\mathrm{M}}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$ from Lemma 4.2 and deduce
$|Gv_{h}-I_{\mathrm{M}}v_{h}|_{2,\mathrm{pw}}\leq|v_{h}-Gv_{h}|_{2,\mathrm{pw}}$
from (4.2). The aforementioned estimates and (4.18)-(4.19) allow the
estimation of all terms in the upper bound of the triangle inequality

$\displaystyle|Gv_{h}-Jv_{h}|_{2,\mathrm{pw}}$
$\displaystyle\leq|Gv_{h}-I_{\mathrm{M}}v_{h}|_{2,\mathrm{pw}}+|I_{\mathrm{M}}v_{h}-v_{\mathrm{M}}|_{2,\mathrm{pw}}+|v_{\mathrm{M}}-J^{\prime}v_{\mathrm{M}}|_{2,\mathrm{pw}}+|J^{\prime}v_{\mathrm{M}}-Jv_{h}|_{2,\Omega}$
$\displaystyle\leq C_{JP}\big{(}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+\min_{v\in
V}|v_{h}-v|_{2,\mathrm{pw}}\big{)}$

with
$C_{JP}:=1+C_{\mathrm{M}}+C_{J^{\prime}}(2+C_{\mathrm{M}})+C_{b}^{2}(2+C_{\mathrm{PF}}+C_{\mathrm{M}}+C_{J^{\prime}}(2+C_{\mathrm{M}}))$.
This and (4.14) show

$\displaystyle\sum_{m=0}^{2}|h_{\mathcal{M}}^{m-2}(Gv_{h}-Jv_{h})|_{m,\mathrm{pw}}\leq(1+C_{\mathrm{PF}})C_{JP}\big{(}|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+\min_{v\in
V}|v_{h}-v|_{2,\mathrm{pw}}\big{)}.$ (4.20)

Lemma 2.4, (4.20), and triangle inequalities

$\sum_{m=0}^{2}|h_{\mathcal{M}}^{m-2}(v_{h}-Jv_{h})|_{m,\mathrm{pw}}\leq\sum_{m=0}^{2}|h_{\mathcal{M}}^{m-2}(v_{h}-Gv_{h})|_{m,\mathrm{pw}}+\sum_{m=0}^{2}|h_{\mathcal{M}}^{m-2}(Gv_{h}-Jv_{h})|_{m,\mathrm{pw}}$

conclude the proof of $(c)$ with $C_{J}:=(1+C_{\mathrm{PF}})(1+C_{JP})$. ∎

Proof of $I_{h}J=\mathrm{id}$ in $V_{h}$. Recall
$N=|\mathcal{V}|+|\mathcal{E}|$ from Subsection 2.1. Definition 3.6 and
(H1)-(H2) imply the existence of a (global) nodal basis
$(\psi_{1},\dots,\psi_{N})$ of $V_{h}$ with dof${}_{j}(\psi_{k})=\delta_{jk}$
for $j,k=1,\dots,N$. Definition 3.7 of $I_{h}$ and
$\text{dof}_{j}(Jv_{h})=\text{dof}_{j}(v_{h})$ from (4.11)-(4.12) show that

$I_{h}(Jv_{h})=\sum_{j=1}^{N}\text{dof}_{j}(Jv_{h})\psi_{j}=\sum_{j=1}^{N}\text{dof}_{j}(v_{h})\psi_{j}=v_{h}.\qed$

######  Remark 1 (another companion).

The properties $(a)$-$(c)$ in Theorem 4.3 can also be satisfied by other
smoothers, e.g., by $J^{\prime}I_{\mathrm{M}}$. The latter is not immediately
computable; whence the new definition of $J$ is adopted in this paper to be
used as a smoother $Q$ in Section 5.

######  Remark 2 (generalizations).

Any linear operator $J:V_{h}\to V$ with $(a)$-$(c)$ allow the a priori and a
posteriori error estimates in Section 5-6. The analog design of such a
conforming companion is possible in 3D with the Morley companion operator
$J^{\prime}$ from [20] in 3D.

##  5 Discrete problem and a priori error analysis

###  5.1 Stabilization

The discrete VE functions $v_{h}\in V_{h}$ will not be computed explicitly,
but $Gv_{h}$ will. The resulting discrete counterpart $a_{h}(\cdot,\cdot)$ in
$V_{h}$ of the scalar product $a(\cdot,\cdot)$ in (2.12) requires a
stabilization. Recall the semi-scalar product $a^{P}$ from Subsection 2.3, the
number $N_{P}$ of vertices for $P\in\mathcal{M}$,
$\text{dof}_{1},\dots,\text{dof}_{2N_{P}}$ from (2.1), and define the semi-
scalar product $S^{P}:V_{h}(P)\times V_{h}(P)\to\mathbb{R}$,

$\displaystyle
S^{P}(v_{h},w_{h})=h_{P}^{-2}\sum_{j=1}^{2N_{P}}\text{dof}_{j}(v_{h})\text{dof}_{j}(w_{h})\quad\text{for}\;v_{h},w_{h}\in
V_{h}(P).$ (5.1)

######  Lemma 5.1.

There exists a positive constant $C_{s}$ (that exclusively depends on $\rho$)
such that

$\displaystyle C_{s}^{-1}a^{P}(w_{h},w_{h})\leq S^{P}(w_{h},w_{h})\leq
C_{s}a^{P}(w_{h},w_{h})\quad\text{for all}\;w_{h}\in(1-G)V_{h}(P).$ (5.2)

###### Proof.

For any $w_{h}=\sum_{j=1}^{2N_{P}}\text{dof}_{j}(w_{h})\psi_{j}\in V_{h}(P)$
with the nodal basis functions $\psi_{1},\dots,\psi_{2N_{P}}$ of $V_{h}(P)$
from (H1)-(H2), a Cauchy-Schwarz inequality and (H2) show that

$\displaystyle|w_{h}|_{2,P}\leq|\text{Dof}(w_{h})|_{\ell^{2}}\Big{(}\sum_{j=1}^{2N_{P}}|\psi_{j}|_{2,P}^{2}\Big{)}^{1/2}\leq
C_{\text{stab}}h_{P}^{-1}|\text{Dof}(w_{h})|_{\ell^{2}}.$

The sum over all squared estimates for $P\in\mathcal{M}$ proves the first
inequality in (5.2) for $C_{s}=C_{\text{stab}}^{2}$. Note that $Gw_{h}=0$ for
each $w_{h}\in(1-G)V_{h}(P)$ and Lemma 2.4-2.5 result in

$h_{P}^{-1}|\text{Dof}(w_{h})|_{\ell^{2}}=h_{P}^{-1}|\text{Dof}(w_{h}-Gw_{h})|_{\ell^{2}}\leq
C_{d}|w_{h}-Gw_{h}|_{2,P}\leq C_{d}|w_{h}|_{2,P}.$

The sum over all squared estimates for $P\in\mathcal{M}$ proves the second
inequality in (5.2). This concludes the proof with
$C_{s}:=\max\\{C_{\text{stab}}^{2},C_{d}^{2}\\}$. ∎

######  Remark 3 (generalization).

The a priori and a posteriori error analysis in Section 5-6 hold for any semi-
scalar product $S^{P}:V_{h}(P)\times V_{h}(P)\to\mathbb{R}$ with (5.2).

###  5.2 Discrete problem

Recall $a_{\mathrm{pw}}$ from Subsection 2.3 and $(S^{P}:P\in\mathcal{M})$
from (5.1). Define the discrete semi-scalar products, for $v_{h},w_{h}\in
V_{h}$, by

$\displaystyle a_{h}(v_{h},w_{h})$
$\displaystyle:=\sum_{P\in\mathcal{M}}a_{h}^{P}(v_{h},w_{h}):=a_{\mathrm{pw}}(Gv_{h},Gw_{h})+s_{h}(v_{h},w_{h}),$
(5.3) $\displaystyle s_{h}(v_{h},w_{h})$
$\displaystyle:=\sum_{P\in\mathcal{M}}S^{P}((1-G)v_{h},(1-G)w_{h}).$ (5.4)

Recall that $|\cdot|_{2,\mathrm{pw}}$ defines a norm in $V_{h}$ (cf. Lemma
2.1), so $a_{h}(\cdot,\cdot)$ is a scalar product in $V_{h}$.

######  Lemma 5.2 (boundedness and ellipticity of $a_{h}$).

Any $v_{h},w_{h}\in V_{h}$ satisfy

$\displaystyle
a_{h}(v_{h},w_{h})\leq(1+C_{s})|v_{h}|_{2,\mathrm{pw}}|w_{h}|_{2,\mathrm{pw}}\quad\text{and}\quad
C_{s}^{-1}|v_{h}|^{2}_{2,\mathrm{pw}}\leq a_{h}(v_{h},v_{h}).$

###### Proof.

The boundedness follows from the definition of $a_{h}$ and (5.2). The
ellipticity follows from (5.2),(2.15), and the Pythagoras identity
$|v_{h}|^{2}_{2,P}=|(1-G)v_{h}|_{2,P}^{2}+|Gv_{h}|_{2,P}^{2}$. ∎

The Riesz representation theorem guarantees the unique existence of a solution
$u_{h}\in V_{h}$ to

$\displaystyle a_{h}(u_{h},v_{h})=F_{h}(v_{h})\quad\text{for all}\;v_{h}\in
V_{h}$ (5.5)

for the right-hand side $F_{h}(v_{h}):=F(Qv_{h})$ with $Q=G$ (standard VEM) or
$Q=J$ (smoother).

###  5.3 A priori error estimates

This subsection establishes an error estimate with respect to the norms
$|\cdot|_{2,\mathrm{pw}}$ and $|\cdot|_{1,\mathrm{pw}}$. Recall elliptic
regularity and $0<\sigma\leq 1$ from (2.14) for the weak solution $u\in
H^{2}_{0}(\Omega)\cap H^{2+\sigma}(\Omega)$ to (B.15) provided $f\in
L^{2}(\Omega)$ with $F(\cdot):=(f,\cdot)_{L^{2}(\Omega)}$ in $V$. Let
$u_{h}\in V_{h}$ solve (5.5) and recall the maximal mesh-size
$h_{\text{max}}$. The second part of the assertion implies (1.1).

######  Theorem 5.3 (error estimates).

There exist positive constants $C_{1}$ and $C_{2}$ (that exclusively depend on
$\rho$) such that $f\in L^{2}(\Omega)$ and $Q=G$ or $Q=J$ imply

$\displaystyle
h^{-\sigma}_{\text{max}}(|u-u_{h}|_{1,\mathrm{pw}}+|u-Gu_{h}|_{1,\mathrm{pw}})+|u-u_{h}|_{2,\mathrm{pw}}+|u-Gu_{h}|_{2,\mathrm{pw}}$
$\displaystyle\qquad\leq
C_{1}(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M}))\leq
C_{2}h^{\sigma}_{\text{max}}\|f\|_{L^{2}(\Omega)}.$

For $F\in V^{*}=H^{-2}(\Omega)$ and solely for $Q=J$, it holds (even without
extra regularity of $u\in V$)

$\displaystyle
h^{-\sigma}_{\text{max}}(|u-u_{h}|_{1,\mathrm{pw}}+|u-Gu_{h}|_{1,\mathrm{pw}})+|u-u_{h}|_{2,\mathrm{pw}}+|u-Gu_{h}|_{2,\mathrm{pw}}\leq
C_{1}|u-Gu|_{2,\mathrm{pw}}.$

###### Proof.

Key identity. Let $e_{h}:=I_{h}u-u_{h}\in V_{h}$ and $v_{h}\in V_{h}$ with the
interpolation $I_{h}$ from Definition 3.7 and $GI_{h}u=Gu$ from Theorem 3.8.b.
The discrete problem (5.5) and (5.3) imply

$\displaystyle
a_{h}(e_{h},v_{h})=a_{\mathrm{pw}}(Gu,Gv_{h})+s_{h}(I_{h}u,v_{h})-F(Qv_{h})=a_{\mathrm{pw}}(Gu,Jv_{h})+s_{h}(I_{h}u,v_{h})-F(Qv_{h})$

with
$a_{\mathrm{pw}}(Gu,Gv_{h})=a_{\mathrm{pw}}(Gu,v_{h})=a_{\mathrm{pw}}(Gu,Jv_{h})$
from (2.15) and Theorem 4.3.a in the last step. The continuous problem (B.15)
with a test function $v=Jv_{h}$ reveals

$\displaystyle
a_{h}(e_{h},v_{h})=a_{\mathrm{pw}}(Gu-u,Jv_{h})+s_{h}(I_{h}u,v_{h})+F(Jv_{h})-F(Qv_{h}).$
(5.6)

Estimate of $a_{\mathrm{pw}}(Gu-u,Jv_{h})$. A triangle inequality, Theorem
4.3.c, and $|v_{h}-Gv_{h}|_{2,\mathrm{pw}}\leq|v_{h}|_{2,\mathrm{pw}}$ (cf.
Lemma 2.4) show

$|Jv_{h}|_{2,\Omega}\leq|v_{h}-Jv_{h}|_{2,\mathrm{pw}}+|v_{h}|_{2,\mathrm{pw}}\leq
C_{J}(|v_{h}-Gv_{h}|_{2,\mathrm{pw}}+|v_{h}|_{2,\mathrm{pw}})+|v_{h}|_{2,\mathrm{pw}}\leq(1+2C_{J})|v_{h}|_{2,\mathrm{pw}}.$

This and a Cauchy-Schwarz inequality provide

$\displaystyle
a_{\mathrm{pw}}(Gu-u,Jv_{h})\leq(1+2C_{J})|u-Gu|_{2,\mathrm{pw}}|v_{h}|_{2,\mathrm{pw}}.$
(5.7)

Estimates of $s_{h}(I_{h}u,v_{h})$. A Cauchy-Schwarz inequality for the semi-
scalar product $s_{h}(\cdot,\cdot)$ and (5.2) provide
$C_{s}^{-1}s_{h}(I_{h}u,v_{h})\leq|(1-G)I_{h}u|_{2,\mathrm{pw}}|(1-G)v_{h}|_{2,\mathrm{pw}}.$
A triangle equality, Lemma 2.4, and Theorem 3.8.b-d in the end result in

$\displaystyle
C_{s}^{-1}s_{h}(I_{h}u,v_{h})\leq(|u-I_{h}u|_{2,\mathrm{pw}}+|u-Gu|_{2,\mathrm{pw}})|v_{h}|_{2,\mathrm{pw}}\leq(1+C_{\text{I}})|u-Gu|_{2,\mathrm{pw}}|v_{h}|_{2,\mathrm{pw}}.$
(5.8)

Estimate of $F(Jv_{h})-F(Qv_{h})$. The term $F(Jv_{h})-F(Qv_{h})$ vanishes for
$Q=J$, so let $Q=G$. The orthogonality
$Jv_{h}-Gv_{h}\perp\mathcal{P}_{2}(\mathcal{M})$ in $L^{2}(\Omega)$ from
Theorem 4.3.b result in

$\displaystyle
F(Jv_{h})-F(Qv_{h})=(h_{\mathcal{M}}^{2}(f-\Pi_{2}f),h_{\mathcal{M}}^{-2}(Jv_{h}-Gv_{h}))_{L^{2}(\Omega)}\leq
2C_{J}\mathrm{osc}_{2}(f,\mathcal{M})|v_{h}|_{2,\mathrm{pw}}.$ (5.9)

The last step follows from a Cauchy-Schwarz inequality, Theorem 4.3.c, and
Lemma 2.4.
Estimate of $|u-u_{h}|_{2,\mathrm{pw}}$. The key identity (5.6) with
$v_{h}=e_{h}$, the coercivity of $a_{h}$ from Lemma 5.2, and (5.7)-(5.8) lead
to $C_{3}:=1+2C_{J}+C_{s}(1+C_{\text{I}}))$ in

$\displaystyle|e_{h}|_{2,\mathrm{pw}}\leq
C_{3}(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M})).$ (5.10)

A triangle inequality for $u-u_{h}=(u-I_{h}u)+e_{h}$, Theorem 3.8.d, and
(5.10) show that

$\displaystyle|u-u_{h}|_{2,\mathrm{pw}}\leq(C_{\text{I}}+C_{3})(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M})).$
(5.11)

Recall that $\mathrm{osc}_{2}(f,\mathcal{M})$ can be omitted in (5.9)-(5.11)
(and below in (5.18)-(5.19)) if $Q=J$.
Duality solution and its regularity. Let $z\in H^{2}_{0}(\Omega)\cap
H^{2+\sigma}(\Omega)$ be the weak solution to $\Delta^{2}z=-\Delta Je_{h}\in
L^{2}(\Omega)$ and recall $\|z\|_{2+\sigma,\Omega}\leq C_{\mathrm{reg}}|\Delta
Je_{h}|_{-1,\Omega}\leq C_{\mathrm{reg}}|Je_{h}|_{1,\Omega}$ from (2.14). The
weak formulation of $\Delta^{2}z=-\Delta Je_{h}\in L^{2}(\Omega)$ leads to
$|Je_{h}|^{2}_{1,\Omega}=a(Je_{h},z).$
Reduction to the key term $a_{h}(e_{h},I_{h}z)$. Elementary algebra reveals
(with $GI_{h}z=Gz$)

$\displaystyle|Je_{h}|^{2}_{1,\Omega}=a_{\mathrm{pw}}(Je_{h}-e_{h},z)+a_{\mathrm{pw}}(e_{h},z-Gz)+a_{\mathrm{pw}}(e_{h},Gz).$
(5.12)

Theorem 4.3.a implies
$a_{\mathrm{pw}}(Je_{h}-e_{h},z)=a_{\mathrm{pw}}(Je_{h}-e_{h},z-Gz)$. Hence a
Cauchy-Schwarz inequality and Theorem 4.3.c imply that

$\displaystyle a_{\mathrm{pw}}(Je_{h}-e_{h},z)$
$\displaystyle\leq(|e_{h}-Ge_{h}|_{2,\mathrm{pw}}|+|e_{h}|_{2,\mathrm{pw}})|z-Gz|_{2,\mathrm{pw}}$
$\displaystyle\leq
2C_{\mathrm{apx}}C_{J}h^{\sigma}_{\text{max}}|e_{h}|_{2,\mathrm{pw}}|z|_{2+\sigma,\Omega}$
(5.13)

with Lemma 2.4 in the end. A Cauchy-Schwarz inequality and Lemma 2.4 show

$\displaystyle
a_{\mathrm{pw}}(e_{h},z-Gz)\leq|e_{h}|_{2,\mathrm{pw}}|z-Gz|_{2,\mathrm{pw}}\leq
C_{\text{apx}}h^{\sigma}_{\text{max}}|e_{h}|_{2,\mathrm{pw}}|z|_{2+\sigma,\Omega}.$
(5.14)

Key identity revisited. The definition (5.3) and the identity (5.6) lead, for
$v_{h}=I_{h}z$, to

$\displaystyle
a_{\mathrm{pw}}(e_{h},Gz)=a_{h}(e_{h},I_{h}z)-s_{h}(e_{h},I_{h}z)=a_{\mathrm{pw}}(Gu-u,JI_{h}z)+F(JI_{h}z)-F(QI_{h}z).$

Estimate of $a_{\mathrm{pw}}(Gu-u,JI_{h}z)$. The definition of G in (2.15)
shows $a_{\mathrm{pw}}(Gu-u,JI_{h}z)=a_{\mathrm{pw}}(Gu-u,JI_{h}z-GI_{h}z)$.
Theorem 4.3.c (with $v_{h}=I_{h}z$ and $v=z$) and Theorem 3.8.b imply
$|JI_{h}z-GI_{h}z|_{2,\mathrm{pw}}\leq
C_{J}(|I_{h}z-Gz|_{2,\mathrm{pw}}+|I_{h}z-z|_{2,\mathrm{pw}})$. A triangle
inequality reveals

$\displaystyle
C_{J}^{-1}|JI_{h}z-GI_{h}z|_{2,\mathrm{pw}}\leq(2|z-I_{h}z|+|z-Gz|_{2,\mathrm{pw}})\leq(2C_{\text{I}}+C_{\text{apx}})h^{\sigma}_{\text{max}}|z|_{2+\sigma,\Omega}$
(5.15)

with Theorem 3.8.d and Lemma 2.4 in the last step. This and a Cauchy-Schwarz
inequality prove

$\displaystyle a_{\mathrm{pw}}(Gu-u,JI_{h}z)\leq
C_{J}(2C_{\text{I}}+C_{\text{apx}})h^{\sigma}_{\text{max}}|u-Gu|_{2,\mathrm{pw}}|z|_{2+\sigma,\Omega}.$
(5.16)

Estimate of $F(JI_{h}z)-F(QI_{h}z)$. The term $F(JI_{h}z)-F(QI_{h}z)$ vanishes
for $Q=J$, so let $Q=G$. The estimate (5.9) for $v_{h}=I_{h}z$ provides
$F(JI_{h}z)-F(QI_{h}z)\leq\mathrm{osc}_{2}(f,\mathcal{M})\|h_{\mathcal{M}}^{-2}(JI_{h}z-GI_{h}z)\|_{L^{2}(\Omega)}.$
Theorem 4.3.c (with $v=z$), Theorem 3.8.b, and (5.15) show

$\displaystyle F(JI_{h}z)-F(QI_{h}z)$ $\displaystyle\leq
C_{J}\mathrm{osc}_{2}(f,\mathcal{M})(|I_{h}z-Gz|_{2,\mathrm{pw}}+|I_{h}z-z|_{2,\mathrm{pw}})$
$\displaystyle\leq
C_{J}(2C_{\text{I}}+C_{\text{apx}})h_{\text{max}}^{\sigma}\mathrm{osc}_{2}(f,\mathcal{M})|z|_{2+\sigma,\Omega}.$
(5.17)

Estimate of $|u-u_{h}|_{1,\mathrm{pw}}$. The combination of (5.12)-(5.14) and
(5.16)-(5.17) imply

$|Je_{h}|^{2}_{1,\Omega}\leq
C_{4}h_{\text{max}}^{\sigma}(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M}))|z|_{2+\sigma,\Omega}$

with $C_{4}:=C_{\text{apx}}(1+2C_{J})+C_{J}(2C_{\text{I}}+C_{\text{apx}}))$.
This and the aforementioned regularity show

$\displaystyle|Je_{h}|_{1,\Omega}\leq
C_{\text{reg}}C_{4}h_{\text{max}}^{\sigma}(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M})).$
(5.18)

A triangle inequality for $u-u_{h}=(u-I_{h}u)+(e_{h}-Je_{h})+Je_{h}$, Theorem
3.8.d for the first term, Theorem 4.3.c and (5.10) for the second term, and
(5.18) for the last term result in

$\displaystyle|u-u_{h}|_{1,\mathrm{pw}}$ $\displaystyle\leq
C_{\text{I}}h_{\text{max}}|u-Gu|_{2,\mathrm{pw}}+2C_{J}h_{\text{max}}|e_{h}|_{2,\mathrm{pw}}+|Je_{h}|_{1,\Omega}$
$\displaystyle\leq(C_{\text{I}}+2C_{J}C_{3}+C_{4}C_{\text{reg}})h^{\sigma}_{\text{max}}(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M})).$
(5.19)

Estimate of $|u-Gu_{h}|_{2,\mathrm{pw}}$. Recall that
$|\cdot|_{s}:=s_{h}(\cdot,\cdot)^{1/2}$ for $s_{h}(\cdot,\cdot)$ from
Subsection 4.1 defines a seminorm in $V_{h}$ that is equivalent to
$|(1-G)\cdot|_{2,\mathrm{pw}}$ owing to (5.2). It follows that

$\displaystyle|u_{h}-Gu_{h}|_{2,P}\leq
C_{s}^{1/2}S^{P}((1-G)u_{h},(1-G)u_{h})^{1/2}.$ (5.20)

This, triangle inequalities, and
$|(1-G)(u_{h}-I_{h}u)|_{2,\mathrm{pw}}\leq|u_{h}-I_{h}u|_{2,\mathrm{pw}}$ from
Lemma 2.4 with Theorem 3.8.b show

$\displaystyle|u_{h}|_{s}\leq|u_{h}-I_{h}u|_{s}+|I_{h}u|_{s}$
$\displaystyle\leq
C_{s}^{1/2}(|u_{h}-I_{h}u|_{2,\mathrm{pw}}+|(1-G)I_{h}u|_{2,\mathrm{pw}})$
$\displaystyle\leq
C_{s}^{1/2}(|u_{h}-I_{h}u|_{2,\mathrm{pw}}+|u-I_{h}u|_{2,\mathrm{pw}}+|u-Gu|_{2,\mathrm{pw}}).$
(5.21)

The combination of (5.10)-(5.11) and (5.20)-(5.21) with the triangle
inequality
$|u-Gu_{h}|_{2,\mathrm{pw}}\leq|u-u_{h}|_{2,\mathrm{pw}}+|u_{h}-Gu_{h}|_{2,\mathrm{pw}}$
results in

$|u-Gu_{h}|_{2,\mathrm{pw}}\leq(C_{s}+(1+C_{s})(C_{\text{I}}+C_{3}))(|u-Gu|_{2,\mathrm{pw}}+\mathrm{osc}_{2}(f,\mathcal{M})).$

Estimate of $|u-Gu_{h}|_{1,\mathrm{pw}}$. A triangle inequality and Lemma 2.4
provide

$\displaystyle|u-Gu_{h}|_{1,\mathrm{pw}}\leq|u-u_{h}|_{1,\mathrm{pw}}+C_{\mathrm{PF}}h_{\text{max}}|u_{h}-Gu_{h}|_{2,\mathrm{pw}}.$

This and the combination of (5.19)-(5.21) conclude the proof. ∎

##  6 A posteriori error analysis

This section establishes a reliable and (up to data oscillations) efficient
explicit residual-based a posteriori error estimator for a source term $f\in
L^{2}(\Omega)$ for both cases $Q=G$ and $Q=J$. Given any polygon
$P\in\mathcal{M}$ and the discrete solution $u_{h}\in V_{h}$ to (5.5), define
the computable terms

$\eta_{P}^{2}:=h_{P}^{4}\|f\|_{L^{2}(P)}^{2}$ | (volume residual),
---|---
$\zeta_{P}^{2}:=S^{P}((1-G)u_{h},(1-G)u_{h})$ | (stabilization),
$\Xi_{P}^{2}:=\sum_{E\in\mathcal{E}(P)}\Big{(}h_{E}^{-3}\|[Gu_{h}]_{E}\|_{L^{2}(E)}^{2}+h_{E}^{-1}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}^{2}\Big{)}$ | (nonconformity),
$\mu_{P}^{2}:=\eta_{P}^{2}+\zeta_{P}^{2}+\Xi_{P}^{2}$ | (error estimator).

Those local quantities $\bullet|_{P}$ form a family
($\bullet|_{P}:P\in\mathcal{M}$) over the index set $\mathcal{M}$ and their
Euclid vector norms $\bullet|_{\mathcal{M}}$ enter the upper error bounds

$\displaystyle\eta_{\mathcal{M}}:=(\sum_{P\in\mathcal{M}}\eta_{P}^{2})^{1/2},\quad\zeta_{\mathcal{M}}:=(\sum_{P\in\mathcal{M}}\zeta_{P}^{2})^{1/2},\quad\Xi_{\mathcal{M}}:=(\sum_{P\in\mathcal{M}}\Xi_{P}^{2})^{1/2},\quad\mu_{\mathcal{M}}:=(\sum_{P\in\mathcal{M}}\mu_{P}^{2})^{1/2}.$

######  Theorem 6.1 (reliability).

There exist positive constants $C_{\mathrm{r1}}$ and $C_{\mathrm{r2}}$ (that
exclusively depend on $\rho$), such that, for $m=1,2$,

$\displaystyle
C_{\mathrm{rm}}^{-2}(|u-u_{h}|_{m,\mathrm{pw}}^{2}+|u-Gu_{h}|_{m,\mathrm{pw}}^{2})$
$\displaystyle\leq\sum_{P\in\mathcal{M}}h_{P}^{2\sigma(2-m)}\mu_{P}^{2}.$
(6.1)

The proof of Theorem 6.1 uses an enrichment operator
$E_{h}:\mathcal{P}_{2}(\mathcal{T})\to H^{2}_{0}(\Omega)$ from [28].

######  Lemma 6.2.

There exists a positive constant $C_{a}$ (that exclusively depends on $\rho$)
such that any $v_{2}\in\mathcal{P}_{2}(\mathcal{T})$ satisfies

$\displaystyle|v_{2}-E_{h}v_{2}|^{2}_{2,\mathrm{pw}}\leq
C_{a}^{2}\sum_{E\in\mathcal{E}}\Big{(}h_{E}^{-3}\|[v_{2}]_{E}\|_{L^{2}(E)}^{2}+h_{E}^{-1}\|[(v_{2})_{\textbf{n}}]_{E}\|_{L^{2}(E)}^{2}\Big{)}.$
(6.2)

###### Proof.

There exists a positive constant $C_{a}$ (that exclusively depends on $\rho$)
such that any $v_{2}\in\mathcal{P}_{2}(\mathcal{T})$ and its enrichment
$E_{h}v_{2}\in H^{1}_{0}(\Omega)$ from [28, Lemma 3.1] satisfy

$\displaystyle|v_{2}-E_{h}v_{2}|^{2}_{2,\mathcal{T}}\leq
C_{a}^{2}\sum_{E\in\widehat{\mathcal{E}}}\Big{(}h_{E}^{-3}\|[v_{2}]_{E}\|_{L^{2}(E)}^{2}+h_{E}^{-1}\|[(v_{2})_{\textbf{n}}]_{E}\|_{L^{2}(E)}^{2}\Big{)}.$

The constant $C_{a}$ depends on the shape-regularity of the sub-triangulation
$\mathcal{T}$ from Subsection 2.1 and so depends on $\rho$. Since any edge
$E\in\mathcal{E}$ is unrefined in the sub-triangulation $\mathcal{T}$, the
above inequality reduces to (6.2) for any $v_{2|P}\in H^{2}(P)$ and
$P\in\mathcal{M}$. This concludes the proof. ∎

The composition $E_{h}\circ G:V_{h}\to V$ connects a given function $v_{h}\in
V_{h}$ to a conforming function,

$V_{h}$$\mathcal{P}_{2}(\mathcal{M})\hookrightarrow\mathcal{P}_{2}(\mathcal{T})$$V$$E_{h}$$G$

###### Proof of Theorem 6.1 for $m=2$.

Let $e:=u-E_{h}Gu_{h}\in V=H^{2}_{0}(\Omega)$. The scalar product
$a(\cdot,\cdot)$ and the continuous problem (B.15) lead to
$|u-E_{h}Gu_{h}|^{2}_{2,\Omega}=a(u-E_{h}Gu_{h},e)=F(e)-a(E_{h}Gu_{h},e).$
Recall
$a_{\mathrm{pw}}(Gu_{h},GI_{h}e)=a_{\mathrm{pw}}(Gu_{h},I_{h}e)=a_{\mathrm{pw}}(Gu_{h},e)$
from (2.15) and Theorem 3.8.a. This and the discrete problem (5.5) result in

$\displaystyle|u-E_{h}Gu_{h}|^{2}_{2,\Omega}=F(e)-F(QI_{h}e)+a_{\mathrm{pw}}(Gu_{h}-E_{h}Gu_{h},e)+s_{h}(u_{h},I_{h}e).$
(6.3)

Estimate of $F(e)-F(QI_{h}e)$. Case 1 ($Q=G$). Theorem 3.8.b and Lemma 2.4
show

$\|e-GI_{h}e\|_{L^{2}(P)}\leq C_{\mathrm{PF}}h_{P}^{2}|e-Ge|_{2,P}\leq
C_{\mathrm{PF}}h_{P}^{2}|e|_{2,P}.$

Case 2 ($Q=J$). A triangle inequality and Theorem 4.3.c lead to

$\|e-JI_{h}e\|_{L^{2}(P)}\leq\|e-I_{h}e\|_{L^{2}(P)}+C_{J}h_{P}^{2}(|I_{h}e-GI_{h}e|_{2,P}+|I_{h}e|_{2,P}).$

Theorem 3.8.c and Lemma 2.4 show $|I_{h}e-GI_{h}e|_{2,P}+|I_{h}e|_{2,P}\leq
2C_{\text{Ib}}|e|_{2,P}$. The previous estimates and Theorem 3.8.d result in
$\|e-JI_{h}e\|_{L^{2}(P)}\leq(C_{\text{I}}+2C_{J}C_{\text{Ib}})h_{P}^{2}|e|_{2,P}.$
A Cauchy-Schwarz inequality in the right-hand side of
$F(e)-F(QI_{h}e)=(f,e-QI_{h}e)_{L^{2}(\Omega)}$ and the above estimates in
case $Q=G$ or $Q=J$ provide an estimate for each $P\in\mathcal{M}$. Their sum
reads

$\displaystyle
F(e)-F(QI_{h}e)\leq(C_{\mathrm{PF}}+C_{\text{I}}+2C_{J}C_{\text{Ib}})\eta_{\mathcal{M}}|e|_{2,\Omega}.$
(6.4)

Estimate of $a_{\mathrm{pw}}(Gu_{h}-E_{h}Gu_{h},e)$. A Cauchy-Schwarz
inequality and Lemma 6.2 lead to

$\displaystyle a_{\mathrm{pw}}(Gu_{h}-E_{h}Gu_{h},e)\leq
C_{a}\Xi_{\mathcal{M}}|e|_{2,\Omega}.$ (6.5)

Estimate of $s_{h}(u_{h},I_{h}e)$. A Cauchy-Schwarz inequality and (5.2)
result in

$\displaystyle s_{h}(u_{h},I_{h}e)\leq
C_{s}^{1/2}s_{h}^{1/2}(u_{h},u_{h})|(1-G)I_{h}e|_{2,\mathrm{pw}}\leq
C_{s}^{1/2}C_{\text{Ib}}s_{h}^{1/2}(u_{h},u_{h})|e|_{2,\Omega}$ (6.6)

with $|(1-G)I_{h}e|_{2,\mathrm{pw}}\leq|I_{h}e|_{2,\mathrm{pw}}\leq
C_{\text{Ib}}|e|_{2,\Omega}$ from Lemma 2.4 and Theorem 3.8.c in the last
step.
Estimate of $|u-Gu_{h}|_{2,\mathrm{pw}}$. The combination of (6.3)-(6.6) shows
$C_{5}:=C_{\text{Ip}}+C_{\text{I}}+2C_{J}C_{\text{Ib}}+C_{a}+C_{s}^{1/2}C_{\text{Ib}}$
in $|u-E_{h}Gu_{h}|_{2,\Omega}\leq C_{5}\mu_{\mathcal{M}}.$ This, Lemma 6.2,
and a triangle inequality for the error
$u-Gu_{h}=(u-E_{h}Gu_{h})+(E_{h}Gu_{h}-Gu_{h})$ reveal

$\displaystyle|u-Gu_{h}|_{2,\mathrm{pw}}\leq(C_{5}+C_{a})\mu_{\mathcal{M}}.$
(6.7)

Estimate of $|u-u_{h}|_{2,\mathrm{pw}}$. Recall (5.20) in the form
$C_{s}^{1/2}|u_{h}-Gu_{h}|_{2,\mathrm{pw}}\leq\zeta_{\mathcal{M}}.$ This,
(6.7), and a triangle inequality lead to

$|u-u_{h}|_{2,\mathrm{pw}}\leq|u-Gu_{h}|_{2,\mathrm{pw}}+|u_{h}-Gu_{h}|_{2,\mathrm{pw}}\leq(C_{5}+C_{a}+C_{s}^{1/2})\mu_{\mathcal{M}}.$

This and (6.7) verify (6.1) for $m=2$ with
$C_{\mathrm{r2}}:=\sqrt{3}(2(C_{5}+C_{a})+C_{s}^{1/2})$. ∎

###### Proof of Theorem 6.1 for $m=1$.

Let $\Psi\in V=H^{2}_{0}(\Omega)$ solve the dual problem
$a(v,\Psi)=(\Delta(u-J^{\prime}I_{\mathrm{M}}u_{h}),v)_{L^{2}(\Omega)}$ for
all $v\in V$. Elliptic regularity (2.14) provides $\Psi\in
H^{2+\sigma}(\Omega)$ and the estimate

$\displaystyle\|\Psi\|_{2+\sigma,\Omega}\leq
C_{\text{reg}}|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}.$ (6.8)

The test function $v=u-J^{\prime}I_{\mathrm{M}}u_{h}\in V$ in the dual problem
shows

$\displaystyle|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}^{2}=a(u-J^{\prime}I_{\mathrm{M}}u_{h},\Psi)=F(\Psi)-F(QI_{h}\Psi)+a_{h}(u_{h},I_{h}\Psi)-a(J^{\prime}I_{\mathrm{M}}u_{h},\Psi)$
(6.9)

with the continuous problem (B.15) and the discrete problem (5.5) in the last
step. Theorem 3.8.b provides $GI_{h}\Psi=G\Psi$ and (2.15) shows
$a_{\mathrm{pw}}(Gu_{h},GI_{h}\Psi)=a_{\mathrm{pw}}(Gu_{h},\Psi)$. Notice that
$a_{\mathrm{pw}}(Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h},G\Psi)=a_{\mathrm{pw}}(Gu_{h}-u_{h},G\Psi)+a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},G\Psi)+a_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-J^{\prime}I_{\mathrm{M}}u_{h},G\Psi)=0$
follows from (2.15), (4.2), and Lemma 4.1.c. This and (6.9) result in

$\displaystyle|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}^{2}$
$\displaystyle=F(\Psi)-F(QI_{h}\Psi)+a_{\mathrm{pw}}(Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h},\Psi-G\Psi)+s_{h}(u_{h},I_{h}\Psi).$
(6.10)

Estimate of $F(\Psi)-F(QI_{h}\Psi)$. Case1 ($Q=G$). Theorem 3.8.b and Lemma
2.4 show that

$\|\Psi-GI_{h}\Psi\|_{L^{2}(P)}\leq
C_{\mathrm{PF}}C_{\text{apx}}h_{P}^{2+\sigma}|\Psi|_{2+\sigma,P}.$

Case 2 ($Q=J$). A triangle inequality and Theorem 4.3.c (with $v=\Psi$) reveal
that

$\displaystyle\|\Psi-JI_{h}\Psi\|_{L^{2}(P)}$ $\displaystyle\leq\|\Psi-
I_{h}\Psi\|_{L^{2}(P)}+C_{J}h_{P}^{2}(|I_{h}\Psi-
GI_{h}\Psi|_{2,P}+|I_{h}\Psi-\Psi|_{2,P})$
$\displaystyle\leq(C_{\text{I}}+C_{J}(2C_{\text{I}}+C_{\text{apx}}))h_{P}^{2+\sigma}|\Psi|_{2+\sigma,P}$

with Theorem 3.8.d and (5.15) in the last step. The two estimates for $\|\Psi-
QI_{h}\Psi\|_{L^{2}(P)}$ (for $Q=G$ and $Q=J$) and the Cauchy-Schwarz
inequality $\int_{P}f(\Psi-QI_{h}\Psi)\,dx\leq\|f\|_{L^{2}(P)}\|\Psi-
QI_{h}\Psi\|_{L^{2}(P)}$ prove an estimate for each $P\in\mathcal{M}$. The sum
of all those provides

$\displaystyle
F(\Psi)-F(QI_{h}\Psi)\leq(C_{\mathrm{PF}}C_{\text{apx}}+C_{\text{I}}+C_{J}(2C_{\text{I}}+C_{\text{apx}}))|\Psi|_{2+\sigma,\Omega}\sum_{P\in\mathcal{M}}h_{P}^{\sigma}\eta_{P}.$
(6.11)

Estimate of
$a_{\mathrm{pw}}(Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h},\Psi-G\Psi)$. A triangle
inequality and Lemma 4.1.d (with $v=E_{h}Gu_{h}$) imply
$|Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h}|_{2,\mathrm{pw}}\leq|Gu_{h}-I_{\mathrm{M}}u_{h}|_{2,\mathrm{pw}}+C_{J^{\prime}}|I_{\mathrm{M}}u_{h}-E_{h}Gu_{h}|_{2,\mathrm{pw}}$.
This and
$|Gu_{h}-I_{\mathrm{M}}u_{h}|_{2,\mathrm{pw}}\leq|u_{h}-Gu_{h}|_{2,\mathrm{pw}}$
from (4.2) lead to

$\displaystyle|Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h}|_{2,\mathrm{pw}}$
$\displaystyle\leq(1+C_{J^{\prime}})|u_{h}-Gu_{h}|_{2,\mathrm{pw}}+C_{J^{\prime}}|Gu_{h}-E_{h}Gu_{h}|_{2,\mathrm{pw}}$
$\displaystyle\leq(1+C_{J^{\prime}})C_{s}^{1/2}\zeta_{\mathcal{M}}+C_{J^{\prime}}C_{a}\Xi_{\mathcal{M}}$
(6.12)

with (5.20) and Lemma 6.2 in the last step. A Cauchy-Schwarz inequality,
(6.12), and Lemma 2.4 prove with
$C_{6}:=C_{\text{apx}}((1+C_{J^{\prime}})C_{s}^{1/2}+C_{J^{\prime}}C_{a})$
that

$\displaystyle
a_{\mathrm{pw}}(Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h},\Psi-G\Psi)\leq
C_{6}|\Psi|_{2+\sigma,\Omega}\sum_{P\in\mathcal{M}}h_{P}^{\sigma}(\zeta_{P}+\Xi_{P}).$
(6.13)

Estimate of $s_{h}(u_{h},I_{h}\Psi)$. Argue as in (6.6) for the stability term
and proceed with
$|(1-G)I_{h}\Psi|_{2,\mathrm{pw}}\leq|I_{h}\Psi-G\Psi|_{2,\mathrm{pw}}$ from
Theorem 3.8.b to deduce that

$\displaystyle s_{h}(u_{h},I_{h}\Psi)\leq
C_{s}^{1/2}\zeta_{\mathcal{M}}|I_{h}\Psi-G\Psi|_{2,\mathrm{pw}}\leq
C_{s}^{1/2}(C_{\text{I}}+C_{\text{apx}})|\Psi|_{2+\sigma,\Omega}\sum_{P\in\mathcal{M}}h_{P}^{\sigma}\zeta_{P}$
(6.14)

with a triangle inequality, Theorem 3.8.d, and Lemma 2.4 in the last step.
Estimate of $|u-Gu_{h}|_{1,\mathrm{pw}}$. The combination of (6.10)-(6.11) and
(6.13)-(6.14) provides

$\displaystyle|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}^{2}\leq
C_{7}|\Psi|_{2+\sigma,\Omega}\sum_{P\in\mathcal{M}}h_{P}^{\sigma}\mu_{P}$
(6.15)

with
$C_{7}:=C_{\mathrm{PF}}C_{\text{apx}}+C_{\text{I}}+C_{J}(2C_{\text{I}}+C_{\text{apx}})+C_{6}+C_{s}^{1/2}(C_{\text{I}}+C_{\text{apx}})$.
Argue as in (4.13) to prove $\int_{\partial
P}\nabla(J^{\prime}I_{\mathrm{M}}u_{h}-u_{h})\,ds=0$ so that the Poincaré-
Friedrichs inequality from Theorem 2.2.a applies. This and the triangle
inequality
$|u-Gu_{h}|_{1,\mathrm{pw}}\leq|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}+|J^{\prime}I_{\mathrm{M}}u_{h}-Gu_{h}|_{1,\mathrm{pw}}$
result in

$|u-Gu_{h}|_{1,\mathrm{pw}}\leq|u-J^{\prime}I_{\mathrm{M}}u_{h}|_{1,\Omega}+C_{\mathrm{PF}}|h_{\mathcal{M}}(J^{\prime}I_{\mathrm{M}}u_{h}-Gu_{h})|_{2,\mathrm{pw}}.$

The regularity estimate (6.8), (6.15), and (6.12) lead in the previous
displayed estimate to

$\displaystyle|u-Gu_{h}|_{1,\mathrm{pw}}\leq
C_{\text{reg}}(C_{7}+C_{\mathrm{PF}}((1+C_{J^{\prime}})C_{s}^{1/2}+C_{J^{\prime}}C_{a}))\sum_{P\in\mathcal{M}}h_{P}^{\sigma}\mu_{P}.$

Estimate of $|u-u_{h}|_{1,\mathrm{pw}}$. The triangle inequality
$|u-u_{h}|_{1,\mathrm{pw}}\leq|u-Gu_{h}|_{1,\mathrm{pw}}+|Gu_{h}-u_{h}|_{1,\mathrm{pw}}$,
Lemma 2.4, and (5.20) conclude the proof of (6.1) for $m=1$ with
$C_{\mathrm{r1}}:=\sqrt{3}(2C_{\text{reg}}(C_{7}+C_{\mathrm{PF}}((1+C_{J^{\prime}})C_{s}^{1/2}+C_{J^{\prime}}C_{a})+C_{s}^{1/2})$.
∎

Let $z\in{\mathcal{V}}$ be a vertex in $\mathcal{M}$ with the neighbouring
polygons $\mathcal{M}(z):=\\{P^{\prime}\in\mathcal{M}:z\in P^{\prime}\\}$ and
define the vertex patch $\omega_{z}:=\text{int}(\cup\mathcal{M}(z))$ and the
larger neighbourhood $\Omega(P):=\cup_{z\in{\mathcal{V}}(P)}\omega_{z}$. The
edge patch $\mathcal{M}(E):=\\{P^{\prime}\in\mathcal{M}:E\subset\partial
P^{\prime}\\}$ consists of one or two neighbouring polygons that share an edge
$E\in\mathcal{E}$ and this defines
$\omega(E):=\text{int}(\cup\mathcal{M}(E))$.

######  Theorem 6.3 (local efficiency up to data oscillations).

For any $P\in\mathcal{M}$ it holds

$\displaystyle\zeta_{P}^{2}$
$\displaystyle\lesssim|u-u_{h}|^{2}_{2,P}+|u-Gu_{h}|^{2}_{2,P},$ (6.16)
$\displaystyle\eta_{P}^{2}$
$\displaystyle\lesssim|u-Gu_{h}|^{2}_{2,P}+\mathrm{osc}_{2}^{2}(f,P),$ (6.17)
$\displaystyle\Xi_{P}^{2}$
$\displaystyle\lesssim\sum_{E\in\mathcal{E}(P)}\sum_{P^{\prime}\in\Omega(\omega(E))}(|u-u_{h}|^{2}_{2,P^{\prime}}+|u-Gu_{h}|^{2}_{2,P^{\prime}}).$
(6.18)

###### Proof of (6.16).

The upper bound in (5.2) and a triangle inequality lead to

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

###### Proof of (6.17).

Abbreviate $\theta_{P}:=(f-\Pi_{2}f)|_{P}$, recall the bubble-function $b_{P}$
from (4.8), and substitute $v=b_{P}\Pi_{2}f\in V$ in (B.15) to obtain

$\displaystyle(\Pi_{2}f,v)_{L^{2}(P)}=a^{P}(u,v)-(\theta_{P},v)_{L^{2}(P)}=a^{P}(u-Gu_{h},v)-(\theta_{P},v)_{L^{2}(P)}$

with $a^{P}(Gu_{h},v)=a^{P}(Gu_{h},Gv)=0$ from (2.15) and $Gv=0$ (from Lemma
2.6 for $\text{Dof}(v)=0$) in the last step. This and Cauchy-Schwarz
inequalities show

$\displaystyle(\Pi_{2}f,v)_{L^{2}(P)}\leq|u-Gu_{h}|_{2,P}|v|_{2,P}+\|\theta_{P}\|_{L^{2}(P)}\|v\|_{L^{2}(P)}.$
(6.19)

The first inequality in (4.15) shows
$C_{b}^{-1}\|\Pi_{2}f\|_{L^{2}(P)}^{2}\leq(\Pi_{2}f,v)_{L^{2}(P)}$ and the
second inequality in (4.16) verifies $\sum_{m=0}^{2}h_{P}^{m}|v|_{2,P}\leq
C_{b}\|\Pi_{2}f\|_{L^{2}(P)}$. Those estimates prove in (6.19) that

$\displaystyle
C_{b}^{-2}h_{P}^{2}\|\Pi_{2}f\|_{L^{2}(P)}\leq|u-Gu_{h}|_{2,P}+\mathrm{osc}_{2}(f,P).$

This and the triangle inequality
$\|\eta_{P}^{2}\|_{L^{2}(P)}\leq\|h_{P}^{2}(f-\Pi_{2}f)\|_{L^{2}(P)}+\|h_{P}^{2}\Pi_{2}f\|_{L^{2}(P)}$
conclude the proof of (6.17). ∎

###### Proof of (6.18).

Since $[Gu_{h}]_{E}=[Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h}]_{E}$, the trace
inequality leads to

$\displaystyle C_{T}^{-1}\|[Gu_{h}]_{E}\|_{L^{2}(E)}\leq
h_{E}^{-1/2}\|Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h}\|_{L^{2}(\omega(E))}+h_{E}^{1/2}\|\nabla_{\mathrm{pw}}(Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h})\|_{L^{2}(\omega(E))}.$

Rewrite
$Gu_{h}-J^{\prime}I_{\mathrm{M}}u_{h}=(Gu_{h}-u_{h})+(u_{h}-I_{\mathrm{M}}u_{h})+(I_{\mathrm{M}}u_{h}-J^{\prime}I_{\mathrm{M}}u_{h})$.
Abbreviate
$|\cdot|_{2,\mathcal{T}(P^{\prime})}:=\sum_{T\in\mathcal{T}(P^{\prime})}|\cdot|_{2,T}$
for $P^{\prime}\in\mathcal{M}$. Lemma 2.4, (4.3), and Lemma 4.1.d for
$P^{\prime}\in\mathcal{M}(E)$ lead to

$\displaystyle
C_{T}^{-1}h_{E}^{-3/2}\|[Gu_{h}]_{E}\|_{L^{2}(E)}\leq(2+C_{\mathrm{PF}})\sum_{P^{\prime}\in\mathcal{M}(E)}(|u_{h}-Gu_{h}|_{2,P^{\prime}}+|(1-J^{\prime})I_{\mathrm{M}}u_{h}|_{2,\mathcal{T}(P^{\prime})}).$
(6.20)

There exists a local version of Lemma 4.1.d established in [20, Lemma 5.1]
with a positive constant $C_{8}$ (that exclusively depends on the shape
regularity of $\mathcal{T}$) such that

$\displaystyle|(1-J^{\prime})I_{\mathrm{M}}u_{h}|_{2,\mathcal{T}(P^{\prime})}\leq
C_{8}\min_{v\in
V}\|D^{2}_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-v)\|_{L^{2}(\Omega(P^{\prime}))}\leq
C_{8}\|D^{2}_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-u)\|_{L^{2}(\Omega(P^{\prime}))}.$

A triangle inequality shows

$|(1-J^{\prime})I_{\mathrm{M}}u_{h}|_{2,\mathcal{T}(P^{\prime})}\leq
C_{8}(\|D^{2}_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-u_{h})\|_{L^{2}(\Omega(P^{\prime}))}+\|D^{2}_{\mathrm{pw}}(u_{h}-u)\|_{L^{2}(\Omega(P^{\prime}))}).$

This estimate for each $P^{\prime}\in\mathcal{M}(E)$,
$|u_{h}-I_{\mathrm{M}}u_{h}|_{2,\mathrm{pw}}\leq|u_{h}-Gu_{h}|_{2,\mathrm{pw}}$
from(4.2), (6.20), and a triangle inequality imply

$\displaystyle h_{E}^{-3/2}\|[Gu_{h}]_{E}\|_{L^{2}(E)}\leq
C_{T}(2+C_{\mathrm{PF}})(1+C_{8})\sum_{P^{\prime}\in\Omega(\mathcal{M}(E))}(|u-u_{h}|_{2,P^{\prime}}+|u-Gu_{h}|_{2,P^{\prime}}).$
(6.21)

It remains to control the term
$h_{E}^{-1/2}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}$ for each
$E\in\mathcal{E}(P)$. Since $u-u_{h}\in V_{\mathrm{nc}}$,
$\alpha_{E}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E}(u-u_{h})_{\textbf{n}}\,ds\in\mathbb{R}$ is
uniquely defined. Rewrite
$[(Gu_{h})_{\textbf{n}}]_{E}=[(Gu_{h}-u)_{\textbf{n}}+\alpha_{E}]_{E}$ for
$E\in\mathcal{E}(P)$. The triangle inequality
$\|[(Gu_{h}-u)_{\textbf{n}}+\alpha_{E}]_{E}\|_{L^{2}(E)}\leq\|[(Gu_{h}-u_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}+\|[(u_{h}-u)_{\textbf{n}}+\alpha_{E}]_{E}\|_{L^{2}(E)}$
and the trace inequality lead to

$\displaystyle h_{E}^{-1/2}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}$
$\displaystyle\lesssim(h_{E}^{-1}\|\nabla_{\mathrm{pw}}(Gu_{h}-u_{h})\|_{L^{2}(\omega(E))}+\|D^{2}_{\mathrm{pw}}(Gu_{h}-u_{h})\|_{L^{2}(\omega(E))})$
$\displaystyle\quad+(h_{E}^{-1}\|(u_{h}-u)_{\textbf{n}}+\alpha_{E}\|_{L^{2}(\omega(E))}+\|D^{2}_{\mathrm{pw}}(u-u_{h})\|_{L^{2}(\omega(E))}).$
(6.22)

Since $\int_{E}((u_{h}-u)_{\textbf{n}}+\alpha_{E})\,ds=0$, the Poincaré-
Friedrichs inequality in Theorem 2.2.a applies to
$\textbf{n}_{E}\cdot\nabla(u_{h}-u)+\alpha_{E}=f$ in each
$P^{\prime}\in\mathcal{M}(E)$ and asserts
$\|(u_{h}-u)_{\textbf{n}}+\alpha_{E}\|_{L^{2}(\omega(E))}\leq
C_{\mathrm{PF}}h_{P}\|D^{2}_{\mathrm{pw}}(u_{h}-u)\|_{L^{2}(\omega(E))}$.
This, $h_{E}\geq\rho h_{P^{\prime}}$ from (M2), and Lemma 2.4 show

$\displaystyle
h_{E}^{-1/2}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}\lesssim\|D^{2}_{\mathrm{pw}}(Gu_{h}-u_{h})\|_{L^{2}(\omega(E))}+\|D^{2}_{\mathrm{pw}}(u-u_{h})\|_{L^{2}(\omega(E))}.$

This and a triangle inequality result in

$h_{E}^{-1/2}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}\lesssim\sum_{P^{\prime}\in\mathcal{M}(E)}(|u-u_{h}|_{2,P^{\prime}}+|u-Gu_{h}|_{2,P^{\prime}}).\qed$

######  Remark 4 (efficiency of $H^{1}$ error control).

The upper bounds in (6.16)-(6.18) with a multiplication factor
$h_{P}^{2\sigma}$ for $0<\sigma\leq 1$ in front of $\mu_{P}^{2}$ show that the
error estimators in (6.1) for $m=1$ converge (at least) with the expected
convergence rate of the piecewise $H^{1}$ error.

######  Remark 5 (higher-order ncVEM and 3D).

The upper bound (6.1) for the error $|u-u_{h}|_{2,\mathrm{pw}}$ can be
generalized to ncVEM of higher order $r\geq 3$ (see [2] for the discrete
setting). The enrichment operator $E_{h}$ can be defined from the piecewise
polynomial space $\mathcal{P}_{r}(\mathcal{T})$ to $H^{2}_{0}(\Omega)$ [28]
and the arguments in this section hold in the three-dimensional case as well.

######  Remark 6 (conforming VEM).

There are papers on the a priori error estimates for the conforming VEM, but
there is no work on the a posteriori VE analysis for the biharmonic problem in
the current literature. The analysis in Section 6 applies to the conforming
case with $E_{h}\circ G=1$ and $J^{\prime}\circ I_{\mathrm{M}}=1$ in the proof
of (6.1) for $m=2$ and $m=1$. This establishes the reliable and efficient a
posteriori error estimator $\eta_{\mathcal{M}}+\zeta_{\mathcal{M}}$ for the
conforming VEM.

######  Remark 7 (extensions).

The source term $F$ is assumed to be an $L^{2}$ function in the main parts of
this paper for simplicity and brevity. A class of more general sources $F\in
H^{-2}(\Omega)$ is discussed in Theorem 5.3 only for the smoother $Q=J$ in the
discrete problem and then avoids the data oscillations. More examples on a
class of right-hand sides $F$ are discussed in [18] and, in particular, the a
posteriori error estimates can be generalized for this class of source terms
as well. The arguments of [18] apply here as well and further details are
omitted for brevity.

##  7 Numerical results

This section discusses two numerical experiments with uniform and adaptive
mesh-refinement.

###  7.1 Adaptive algorithm

A standard adaptive algorithm with the loop
$\text{Solve}\;\to\;\text{Estimate}\;\to\;\text{Mark}\;\to\text{Refine}$ from
[16, Sec. 6.1] is performed in two computational benchmarks.

Step 1 (SOLVE). Find the solution $u_{h}$ to (5.5) for $Q=G$ in the right-hand
side and compute the errors $H1e$ and $H2e$, $Hme:=|u-Gu_{h}|_{m,\mathrm{pw}}$
for $m=1,2$, using polygauss quadrature rule [31] for the input parameter
$n=10$.

Step 2 (ESTIMATE). Compute the local residuals in Theorem 6.1 and collect all
these contributions for $P\in\mathcal{M}$ to obtain the upper bound $H1\mu$
and $H2\mu$,
$Hm\mu^{2}:=\sum_{P\in\mathcal{M}}h_{P}^{2\sigma(2-m)}\mu_{P}^{2}$ for the
piecewise $H^{m}$ error for $m=1,2$. Abbreviate the number of degrees of
freedom by ndof.

Step 3 (MARK). The D$\ddot{o}$rfler marking strategy [33] detemines
$\mathcal{D}_{m}\subset\mathcal{M}$ for $m=1,2$ with

$\displaystyle Hm\mu^{2}\leq 0.5\sum_{P\in{\cal
D}_{m}}h_{P}^{2\sigma(2-m)}\mu_{P}^{2}.$

Step 4 (REFINE). A refinement strategy in VEM divides the marked polygonal
domains by connecting the mid-points of the edges to the centroid and allow at
most one hanging node per edge; cf. [34] for further details on a MATLAB
implementation.

###  7.2 Numerical example in L-shaped domain

This subsection considers an L-shaped domain of Figure 7.1 with the exact
solution of the model problem in polar co-ordinates $(r,\theta)$

$\displaystyle
u(r,\theta)=r^{5/3}\sin\Big{(}\frac{5\theta}{3}\Big{)}\quad\text{in}\;\Omega=(-1,1)^{2}\setminus[0,1)\times(-1,0].$

In this example, both $u$ and $u_{\textbf{n}}$ are not zero along the boundary
$\partial\Omega$ and $f=0$. The upper bound for inhomogeneous boundary data
can be established with minor modifications: The term $\Xi_{P}$ in the error
estimator for $P\in\mathcal{M}$, which share a boundary edge, changes to

$\displaystyle\Xi_{P}^{2}$
$\displaystyle=\sum_{E\in\mathcal{E}(P)\cap\mathcal{E}(\Omega)}\Big{(}\frac{1}{h_{E}^{3}}\|[Gu_{h}]_{E}\|_{L^{2}(E)}^{2}+\frac{1}{h_{E}}\|[(Gu_{h})_{\textbf{n}}]_{E}\|_{L^{2}(E)}^{2}\Big{)}$
$\displaystyle\quad+\sum_{E\in\mathcal{E}(P)\cap\mathcal{E}(\partial\Omega)}\Big{(}\frac{1}{h_{E}^{3}}\|[Gu_{h}-u]_{E}\|_{L^{2}(E)}^{2}+\frac{1}{h_{E}}\|[(Gu_{h}-u)_{\textbf{n}}]_{E}\|_{L^{2}(E)}^{2}\Big{)}.$

Figure 7.1 displays strong local mesh-refinement at the re-entry corner in the
adaptive mesh-refining. Figure 7.2 shows that uniform refinement yields the
sub-optimal convergence rate, whereas adaptive refinements recover the optimal
convergence rate.

Figure 7.1: Output $\mathcal{M}_{1},\mathcal{M}_{10},\mathcal{M}_{15}$ of the
adaptive algorithm in Subsection 7.2.

Figure 7.2: Convergence history plot of the errors resp. error estimators
$H2e$ resp. $H2\mu$ (left) and $H1e$ resp. $H1\mu$ (right) vs ndof for the
L-shaped domain in Subsection 7.2.

###  7.3 Numerical example in Z-shaped domain

The subsection considers the polygonal domain $\Omega$ with the vertices
$(0,0),(1,0),(1,1),(-1,1),$
$(-1,-1),(1,-1)$ of Figure 7.3. Define the right-hand side function $f$ in the
polar co-ordinates $(r,\theta)$ with the exact solution

$\displaystyle
u(r,\theta)=(1-r^{2}\cos^{2}(\theta))^{2}(1-r^{2}\sin^{2}(\theta))^{2}r^{(1+z)}g(\theta).$

Here $z=0.505009698896589$ is a noncharacteristic root of
$\sin^{2}(z\omega)=z^{2}\sin^{2}(\omega),\omega=7\pi/4$ and $g(\theta)$ is as
given in [29, p. 107]. Figure 7.3 and 7.4 display the numerical results.

Figure 7.3: Output $\mathcal{M}_{1},\mathcal{M}_{10},\mathcal{M}_{15}$ of the
adaptive algorithm in Subsection 7.3.

Figure 7.4: Convergence history plot of the errors resp. error estimators
$H2e$ resp. $H2\mu$ (left) and $H1e$ resp. $H1\mu$ (right) vs ndof for the
Z-shaped domain in Subsection 7.3.

###  7.4 Evaluation

Empirical convergence rates. The two domains from Subsection 7.2 resp. 7.3
have weak solutions $u\in H^{2+\sigma-\epsilon}(\Omega)$ for any $\epsilon>0$
with the typical corner singularity for $\sigma=2/3$ resp. $\sigma=z=0.505$
and hence we expect and observe the empirical convergence rates in the $H^{2}$
norm (resp. $H^{1}$ norm) $\sigma/2$ $(\text{resp.}\;\sigma)$ in terms of
$\text{ndof}^{-1/2}$ for uniform mesh-refinements. The adaptive mesh-refining
improves the empirical convergence rates to the optimal values $1/2$ (resp.
$(1+\sigma)/2$).

Efficiency indices. The ratio of the error estimator and the total error
(effectivity index) in the piecewise $H^{2}$ and $H^{1}$ norm remains bounded:
$3\leq H2\mu/H2e\leq 5$ and $5.5\leq H1\mu/H1e\leq 7$ in both examples. This
confirms empirically that the error estimator mimics the behaviour of the
total error and also validates Theorem 6.1.

Dominant error contributions. Figure 7.5 displays the individual components
$\eta_{\ell},\zeta_{\ell},\Xi^{1}_{\ell},\Xi^{2}_{\ell}$ in the error
estimator, which abbreviate
$\eta_{\mathcal{M}_{\ell}},\zeta_{\mathcal{M}_{\ell}},\Xi^{1}_{\mathcal{M}_{\ell}},\Xi^{2}_{\mathcal{M}_{\ell}}$
and shows the dominance of $\Xi_{\ell}^{1}$. The remaining part
$\Xi_{\ell}^{2}$ of $\Xi_{\ell}:=\Xi_{\ell}^{1}+\Xi_{\ell}^{2}$ and the volume
residual $\eta_{\ell}$ converge more rapidly. The error estimator components
for the $H^{1}$ error in the adaptive refinement behave similar.

Figure 7.5: Convergence history plot of the error estimator components
corresponding to $H2e$ for adaptive mesh-refinement vs ndof in Subsection 7.2
(left) and 7.3 (right).

Acknowledments. The research of the first author has been supported by the
German research foundation in the Priority Program 1748 foundation and
application of generalized mixed FEM towards nonlinear problems in solid
mechanics (CA 151/22-2) and SPARC project (id 235) the mathematics and
computation of plates and SERB POWER Fellowship SPF/2020/000019. The second
author acknowledges the financial support of the University Grants Commission
from the Government of India.

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##  Appendix A Proof of Lemma 2.6

The integral $a^{P}(v,w)=\int_{P}D^{2}v:D^{2}w\,dx$ for $v\in H^{4}(P)$ and
$w\in H^{2}(P)$ allows an integration by parts formula [2, Sec. 2.1] with the
boundary terms
${\cal{M}}_{\textbf{n}\textbf{n}}(v):=v_{\textbf{n}\textbf{n}},\;T(v):=(\Delta
v)_{\textbf{n}}+v_{\textbf{n}\bm{\tau}\bm{\tau}},\;{\cal{M}}_{\textbf{n}\bm{\tau}}(v):=v_{\textbf{n}\bm{\tau}}$
from [2] and the abbreviations
$[M_{\textbf{n}\bm{\tau}}(v)]_{z_{j}}:=M_{\textbf{n}\bm{\tau}}(v)|_{E(j-1)}(z_{j})-M_{\textbf{n}\bm{\tau}}(v)|_{E(j)}(z_{j})$
with $E(0):=E(N_{P})$ for a cyclic notation along $\partial P$. Those boundary
terms are well-defined as traces of a smooth function $v\in H^{4}(P)$ and then
the formula reads

$\displaystyle
a^{P}(v,w)=(\Delta^{2}v,w)_{L^{2}(P)}+(M_{\textbf{n}\textbf{n}}(v),w_{\textbf{n}})_{L^{2}(\partial
P)}-(T(v),w)_{L^{2}(\partial
P)}+\sum_{j=1}^{N_{P}}[M_{\textbf{n}\bm{\tau}}(v)]_{z_{j}}w(z_{j}).$ (A.1)

The boundary terms require a smooth function $v$ like the quadratic polynomial
$\chi\in\mathcal{P}_{2}(P)$ with $\Delta^{2}\chi=0=T(\chi)$ and with piecewise
constants
$M_{\textbf{n}\textbf{n}}(\chi)|_{E(k)}=\chi_{\textbf{n}\textbf{n}}|_{E(k)}$
and $M_{\textbf{n}\bm{\tau}}(\chi)|_{E(k)}=\chi_{\textbf{n}\bm{\tau}}|_{E(k)}$
for $k=1,\dots,N_{P}$, which are computable in terms of
$D^{2}\chi\in\mathbb{S}$ and of the geometry of $P$ from Figure 2.1.a. This
and the definition of $G$ from (2.15) lead, for $\chi\in\mathcal{P}_{2}(P)$
and $w\in H^{2}(P)$ in (A.1), to

$\displaystyle
a^{P}(Gw,\chi)=a^{P}(\chi,w)=(M_{\textbf{n}\textbf{n}}(\chi),w_{\textbf{n}})_{L^{2}(\partial
P)}+\sum_{j=1}^{N_{P}}[M_{\textbf{n}\bm{\tau}}(\chi)]_{z_{j}}w(z_{j}).$ (A.2)

The dofs of $w$ from (2.1) allow for a re-writing of the right-hand side of
(A.2), namely

$\displaystyle
a^{P}(Gw,\chi)=\sum_{k=1}^{N_{P}}\chi_{\textbf{n}\textbf{n}}|_{E(k)}\text{dof}_{N_{P}+k}(w)+\sum_{j=1}^{N_{P}}(\chi_{\textbf{n}\bm{\tau}}|_{E(j-1)}-\chi_{\textbf{n}\bm{\tau}}|_{E(j)})\text{dof}_{j}(w).$
(A.3)

This defines $Gw\in\mathcal{P}_{2}(P)$ up to an affine contribution fixed in
(2.16). The first condition in (2.16) reads

$\displaystyle
N_{P}^{-1}\sum_{j=1}^{N_{P}}Gw(z_{j})=N_{P}^{-1}\sum_{j=1}^{N_{P}}w(z_{j})=N_{P}^{-1}\sum_{j=1}^{N_{P}}\text{dof}_{j}(w).$
(A.4)

For the second condition in (2.16), the identity $\int_{E(k)}\nabla
w\,ds=\text{dof}_{N_{P}+k}(w)\textbf{n}_{E(k)}+(\bm{\tau}_{E(k-1)}-\bm{\tau}_{E(k)})\text{dof}_{k}(w)$
for $k=1,\dots,N_{P}$ from (2.11) shows

$\displaystyle\int_{\partial P}\nabla
w\,ds=\sum_{k=1}^{N_{P}}\Big{(}\int_{E(k)}\nabla
w\,ds\Big{)}=\sum_{k=1}^{N_{P}}(\text{dof}_{N_{P}+k}(w)\textbf{n}_{E(k)}+(\bm{\tau}_{E(k-1)}-\bm{\tau}_{E(k)})\text{dof}_{k}(w)).$
(A.5)

The equations (A.3)-(A.5) form a linear system of 6 equations for
$Gw\in\mathcal{P}_{2}(P)$ and the right-hand sides in (A.3)-(A.5) are
computable in terms of $\text{dof}_{1}(w),\dots,\text{dof}_{2N_{P}}(w)$ for
any $w\in H^{2}(P)$. Hence its solution $Gw$ is computable in terms of the
dofs of $w$. It is elementary to see that $Gw$ is uniquely determined and the
$6\times 6$ coefficient matrix in the resulting linear system of equations is
regular. (Another proof for this follows from the estimates in the second part
of the proof below.) This proves the first part of the lemma.

The second part part of the proof estimates $|Gw|_{m,P}$ for $m=0,1,2$ in
terms of $\text{Dof}(w)=(\text{dof}_{1}(w),\dots,\text{dof}_{2N_{P}}(w))$. The
definition of $G$ with $\chi=Gw\in\mathcal{P}_{2}(P)$ in (2.15) and (A.3)
imply

$\displaystyle|Gw|^{2}_{2,P}=a^{P}(w,Gw)$
$\displaystyle\leq\sum_{k=1}^{N_{P}}|\chi_{\textbf{n}\textbf{n}}|_{E(k)}||\text{dof}_{N_{P}+k}(w)|+\sum_{j=1}^{N_{P}}(|\chi_{\textbf{n}\bm{\tau}}|_{E(j-1)}-\chi_{\textbf{n}\bm{\tau}}|_{E(j)}|)|\text{dof}_{j}(w)|.$

Cauchy-Schwarz inequalities and
$|\chi_{\textbf{n}\textbf{n}}|_{E(k)}|,|\chi_{\textbf{n}\bm{\tau}}|_{E(k)}|\leq|D^{2}\chi|$
reveal

$\displaystyle|Gw|^{2}_{2,P}\leq
2|D^{2}\chi|\sum_{j=1}^{2N_{P}}|\text{dof}_{j}(w)|\leq\sqrt{8N_{P}}|P|^{-1/2}|Gw|_{2,P}|\text{Dof}(w)|_{\ell^{2}}.$

The bounds $|P|^{-1/2}\leq\pi^{-1/2}C_{\text{sr}}^{-2}h_{P}^{-1}$ (cf. [4,
Chap. 1]) with a shape regularity constant $C_{\text{sr}}$ of $\mathcal{T}(P)$
(that exclusively depends on $\rho$) and $N_{P}\leq M(\rho)$ from Subsection
2.1 show

$\displaystyle|Gw|_{2,P}\leq\sqrt{8\pi^{-1}C_{\text{sr}}^{-4}M(\rho)}h_{P}^{-1}|\text{Dof}(w)|_{\ell^{2}}.$
(A.6)

Define $a:=N_{P}^{-1}\sum_{j=1}^{N_{P}}w(z_{j})\in\mathbb{R}$, $B:=|\partial
P|^{-1}\int_{\partial P}\nabla w\,ds\in\mathbb{R}^{2}$, and the affine
function $g(x):=a+B(x-N_{P}^{-1}\sum_{j=1}^{N_{P}}z_{j})$ to control the
lower-order terms $\sum_{m=0}^{1}h_{P}^{m-1}|Gw|_{m,P}$. The definition (2.16)
of $G$ provides

$\displaystyle\sum_{j=1}^{N_{P}}(Gw-g)(z_{j})=\Big{(}\sum_{j=1}^{N_{P}}w(z_{j})\Big{)}-aN_{P}=0,\quad\int_{\partial
P}\nabla(Gw-g)\,ds=\int_{\partial P}\nabla w\,ds-|\partial P|B=0.$

The Poincaré-Friedrichs inequality from Theorem 2.2.b, therefore, shows for
$Gw-g$ that

$\displaystyle h_{P}^{-2}\|Gw-g\|_{L^{2}(P)}+h_{P}^{-1}|Gw-g|_{1,P}\leq
C_{\mathrm{PF}}|Gw|_{2,P}.$ (A.7)

The Cauchy-Schwarz inequality implies $|a|\leq
N_{P}^{-1/2}|\text{Dof}(w)|_{\ell^{2}}$. Since
$|\textbf{n}_{E(k)}|=1=|\bm{\tau}_{E(k)}|$, (A.5) shows that $|\int_{\partial
P}\nabla w\,ds|\leq 2\sum_{j=1}^{2N_{P}}|\text{dof}_{j}(w)|$. A Cauchy-Schwarz
inequality and $|\partial P|^{-1}\leq N_{P}^{-1}\rho^{-1}h_{P}^{-1}$ from (M2)
imply that $|B|\leq
2\sqrt{2}N_{P}^{-1/2}\rho^{-1}h_{P}^{-1}|\text{Dof}(w)|_{\ell^{2}}$. The
definition of $g$ and the previous two estimates for $|a|$ and $|B|$ result in

$\displaystyle\|g\|_{L^{2}(P)}\leq|P|^{1/2}(|a|+h_{P}|B|)\leq\big{(}\frac{\pi}{N_{P}}\big{)}^{1/2}(1+2\sqrt{2}\rho^{-1})h_{P}|\text{Dof}(w)|_{\ell^{2}}$
(A.8)

with the coarse bound $|P|\leq\pi h_{P}^{2}$ in the last step. This and the
inverse estimate for $|g|_{1,P}$ (collected from an inverse estimate in the
triangle $T\in\mathcal{T}(P)$ and so with a well-established bound for
$C_{\text{inv}}$) show

$\displaystyle|g|_{1,P}\leq
C_{\text{inv}}h_{P}^{-1}\|g\|_{L^{2}(P)}\leq\big{(}\frac{\pi}{N_{P}}\big{)}^{1/2}(1+2\sqrt{2}\rho^{-1})C_{\text{inv}}|\text{Dof}(w)|_{\ell^{2}}.$
(A.9)

Triangle inequalities
$\sum_{m=0}^{2}h_{P}^{m-2}|Gw|_{m,P}\leq\sum_{m=0}^{2}h_{P}^{m-2}(|Gw-g|_{m,P}+|g|_{m,P})$,
(A.7)-(A.9), and the abbreviation
$C_{9}:=1+C_{\mathrm{PF}}+(1+C_{\text{inv}})\pi^{1/2}(1+2\sqrt{2}\rho^{-1})$
prove

$\displaystyle\sum_{m=0}^{2}h_{P}^{m-1}|Gw|_{m,P}\leq
C_{9}(h_{P}|Gw|_{2,P}+|\text{Dof}(w)|_{\ell^{2}})\leq
C_{g}|\text{Dof}(w)|_{\ell^{2}}$

with (A.6) and $C_{g}:=C_{9}(1+\sqrt{8\pi^{-1}C_{\text{sr}}^{-4}M(\rho)})$ in
the last step. This concludes the proof. (Notice that, in particular,
$\text{Dof}(w)=0$ implies $Gw=0$ and this proves that the linear system of
equations (A.3)-(A.5) involves a regular coefficient matrix as announced.)∎

##  Appendix B Proof of Proposition 3.1

Step 1 defines an HCT finite element space. Recall the sub-triangulation
$\mathcal{T}(P)$ from Subsection 2.1 and decompose any triangle
$T\in\mathcal{T}(P)$ further into three sub-triangles $\mathcal{K}(T)$
depicted in Figure B.1.c. Then the Hsieh-Clough-Tocher (HCT) finite element
space [22] reads

$\mathrm{HCT}(\mathcal{T}(P)):=\\{\hat{v}\in
H^{2}(P):\forall\;T\in\mathcal{T}\quad\hat{v}|_{T}\in\mathcal{P}_{3}(\mathcal{K}(T))\\}.$

(a)

(b)

(c)

Figure B.1: (a) Triangle $T$, (b) Morley, (c) HCT.

The standard degrees of freedom (dofs) in the HCT finite element (cf. [22,
Chap. 6] or [27, Sec. 2.3]) are the nodal values of the function and its
first-order derivatives at each vertex and the mid-point values of the normal
derivatives along each edge of a triangle as depicted in Figure B.1.c. This
paper utilizes the integral means instead of the mid-point values of normal
derivatives along edges and all other dofs (i.e., nodal values) are unchanged.
Let $\psi_{1}^{\mathrm{HCT}},\dots,\psi_{2N_{P}}^{\mathrm{HCT}}$ be the
$2N_{P}$ nodal basis functions in $\mathrm{HCT}(\mathcal{T}(P))$ with
$\nabla\psi_{j}^{\mathrm{HCT}}(z_{\ell})=0$ for all $j=1,\dots,2N_{P}$ and,
for all $k,\ell=1,\dots,N_{P}$,

$\displaystyle\psi_{k}^{\mathrm{HCT}}(z_{\ell})=\delta_{k\ell},\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(\ell)}(\psi_{k}^{\mathrm{HCT}})_{\textbf{n}}\,ds=0,\;\text{and}\;\psi_{k+N_{P}}^{\mathrm{HCT}}(z_{\ell})=0,\;\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(\ell)}(\psi_{k+N_{P}}^{\mathrm{HCT}})_{\textbf{n}}\,ds=\delta_{k\ell}.$
(B.1)

In contrast to this, let
$\widetilde{\psi}_{\ell+N_{P}}^{\mathrm{HCT}}\in\mathrm{HCT}(\mathcal{T}(P))$
be a nodal basis function in the standard HCT finite element with
$\widetilde{\psi}_{\ell+N_{P}}^{\mathrm{HCT}}(\text{mid}(E(\ell)))=1$ for
$\ell=1,\dots,N_{P}$, while all other dofs vanish. Then
$4\int_{0}^{1}s(1-s)\,ds=2/3$ leads to
$\psi_{\ell+N_{P}}^{\mathrm{HCT}}=\frac{3}{2}\widetilde{\psi}_{\ell+N_{P}}^{\mathrm{HCT}}$
for $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(\ell)}(\psi_{\ell+N_{P}}^{\mathrm{HCT}})_{\textbf{n}}\,ds=1$
and $\ell=1,\dots,N_{P}$. This observation, the scaling of the standard HCT
basis functions from [27, Prop. 2.5], and the bound
$h_{T}^{-1}\leq\rho^{-1}h_{P}^{-1}$ from (M2) for all $T\in\mathcal{T}(P)$
provide a positive constant $C_{\mathrm{HCT}}$ (that exclusively depends on
$\rho$) in

$\displaystyle\max_{k=1}^{N_{P}}h_{P}|\psi_{k}^{\mathrm{HCT}}|_{2,P}+\max_{\ell=1}^{N_{P}}|\psi_{\ell+N_{P}}^{\mathrm{HCT}}|_{2,P}\leq
C_{\mathrm{HCT}}.$ (B.2)

Step 2 constructs an HCT interpolation. Recall the nodal basis functions
$\psi_{1}^{\mathrm{HCT}},\dots,\psi_{2N_{P}}^{\mathrm{HCT}}$ of
$\mathrm{HCT}(\mathcal{T}(P))$ selected in Step 1. The HCT interpolation
$w_{\mathrm{HCT}}\in\mathrm{HCT}(\mathcal{T}(P))$ of a given $w\in H^{2}(P)$
reads

$\displaystyle
w_{\mathrm{HCT}}:=\sum_{k=1}^{N_{P}}w(z_{k})\psi_{k}^{\mathrm{HCT}}+\sum_{\ell=1}^{N_{P}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(\ell)}w_{\textbf{n}}\,ds\right)\psi_{\ell+N_{P}}^{\mathrm{HCT}}.$
(B.3)

The duality relations (B.1) imply for $m=1,\dots,N_{P}$ in (B.3) that
$w_{\mathrm{HCT}}(z_{m})=w(z_{m})$ and
$\int_{E(m)}(w_{\mathrm{HCT}})_{\textbf{n}}\,ds=\int_{E(m)}w_{\textbf{n}}\,ds.$
In other words, $\text{Dof}(w_{\mathrm{HCT}})=\text{Dof}(w)$ for the vector
Dof with components from (2.1).

Step 3 defines the Hilbert space $(V_{0},a^{P})$. The kernel of the linear map
$\text{Dof}:H^{2}(P)\to\mathbb{R}^{2N_{P}}$ is the closed subspace

$V_{0}:=\left\\{v\in
H^{2}(P):v(z_{j})=0=\int_{E(j)}v_{\textbf{n}}\,ds\quad\text{for}\;j=1,\dots,N_{P}\right\\}$

of the Hilbert space $H^{2}(P)$ and so complete. Hence $(V_{0},a^{P})$ is a
Hilbert space. Notice that $w-w_{\mathrm{HCT}}\in V_{0}$ for any $w\in
H^{2}(P)$ from Step 2 and $Gw=0$ for $w\in V_{0}$ (from Lemma 2.6 with
$\text{Dof}(w)=0$ as explained in Appendix A).

Step 4 proves that $V_{0}^{\perp}:=\\{v\in H^{2}(P):a^{P}(v,w)=0\quad\text{for
all}\;w\in V_{0}\\}\subset\widehat{V}_{h}(P).$ Given any $v\in V_{0}^{\perp}$
and $w\in H^{2}(P)$, define the HCT interpolation $w_{\mathrm{HCT}}$ of $w$
from Step 2. Then $w-w_{\mathrm{HCT}}\in V_{0}$ from Step 3 implies that

$\displaystyle a^{P}(v,w)=a^{P}(v,w_{\mathrm{HCT}})$
$\displaystyle=\sum_{k=1}^{N_{P}}a^{P}(v,\psi_{k}^{\mathrm{HCT}})w(z_{k})+\sum_{\ell=1}^{N_{P}}a^{P}(v,\psi_{\ell+N_{P}}^{\mathrm{HCT}})h_{E(\ell)}^{-1}\int_{E(\ell)}w_{\textbf{n}}\,ds.$
(B.4)

Define $f:=0\in\mathcal{P}_{-1}(P)$,
$a_{\ell}:=a^{P}(v,\psi_{\ell}^{\mathrm{HCT}})$, and
$g\in\mathcal{P}_{0}(\mathcal{E}(P))$ by
$g|_{E(\ell)}:=a^{P}(v,\psi_{\ell+N_{P}}^{\mathrm{HCT}})h_{E(\ell)}^{-1}$ for
$\ell=1,\dots,N_{P}$ to rewrite the right-hand side in (B.4) as

$a^{P}(v,w)=(g,w_{\textbf{n}})_{L^{2}(\partial
P)}+\sum_{\ell=1}^{N_{P}}a_{\ell}w(z_{\ell})\quad\text{for all}\;w\in
H^{2}(P).$

This implies $v\in\widehat{V}_{h}(P)$ for $r=-1$ and concludes the proof of
Step 4. ∎

Step 5 proves that $\mathrm{Dof}:\widehat{V}_{h}(P)\to\mathbb{R}^{2N_{P}}$ is
surjective. Given any $x=(x_{1},\dots,x_{2N_{P}})\in\mathbb{R}^{2N_{P}}$ and
the functions $\psi_{1}^{\mathrm{HCT}},\dots,\psi_{2N_{P}}^{\mathrm{HCT}}$
selected in Step 1, define
$u_{BC}:=\sum_{k=1}^{2N_{P}}x_{k}\psi_{k}^{\mathrm{HCT}}\in H^{2}(P)$ with
$\text{Dof}(u_{BC})=x$ from Step 2. Let $u_{0}\in V_{0}$ denote the Riesz
representation of $a^{P}(u_{BC},\cdot)$ in $(V_{0},a^{P})$, i.e.,
$a^{P}(u_{0},\cdot)=a^{P}(u_{BC},\cdot)$ in $V_{0}$. Since
$a^{P}(u_{0}-u_{BC},\cdot)=0$ in $V_{0}$, Step 4 shows that
$\widehat{u}_{h}:=u_{BC}-u_{0}\in\widehat{V}_{h}(P)$. Recall
$\text{Dof}(u_{BC})=x$ and $\text{Dof}(u_{0})=0$ to deduce
$\text{Dof}(\widehat{u}_{h})=x$. ∎

Step 6 establishes an inclusion in $\widehat{V}_{h}(P)$ with a non-zero $f$.
Given $f\in\mathcal{P}_{r}(P)$, the Riesz-representation theorem guarantees
the unique existence of the weak solution $u(f)\in V_{0}$ to

$\displaystyle a^{P}(u(f),v)=(f,v)_{L^{2}(P)}\quad\text{for all}\;v\in V_{0}.$
(B.5)

Recall that $r$ is a fixed parameter in $\\{-1,0,1,2\\}$ and $r=-1$ is trivial
in this step. Since (B.5) implies $\Delta^{2}u(f)=f$ in $P$, it remains to
prove that $u(f)\in\widehat{V}_{h}(P)$. Given any $w\in H^{2}(P)$ with
$w_{\mathrm{HCT}}$ from (B.3), $w-w_{\mathrm{HCT}}\in V_{0}$ leads in (B.5) to
$a^{P}(u(f),w-w_{\mathrm{HCT}})=(f,w-w_{\mathrm{HCT}})_{L^{2}(P)}$. Hence

$\displaystyle
a^{P}(u(f),w)=(f,w-w_{\mathrm{HCT}})_{L^{2}(P)}+a^{P}(u(f),w_{\mathrm{HCT}})=(f,w)_{L^{2}(P)}+\Lambda(w_{\mathrm{HCT}})$
(B.6)

with the linear functional $\Lambda(v):=a^{P}(u(f),v)-(f,v)_{L^{2}(P)}$ for
any $v\in H^{2}(P)$. The representation (B.3) of $w_{\mathrm{HCT}}$ shows

$\displaystyle a^{P}(u(f),w)$
$\displaystyle=(f,w)_{L^{2}(P)}+\sum_{k=1}^{N_{P}}w(z_{k})\Lambda(\psi_{k}^{\mathrm{HCT}})+\sum_{\ell=1}^{N_{P}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(\ell)}w_{\textbf{n}}\,ds\right)\Lambda(\psi_{\ell+N_{P}}^{\mathrm{HCT}})$
$\displaystyle=(f,w)_{L^{2}(P)}+(g,w_{\textbf{n}})_{L^{2}(\partial
P)}+\sum_{k=1}^{N_{P}}a_{k}w(z_{k})\quad\text{for all}\;w\in H^{2}(P)$ (B.7)

with the definition of $a_{\ell}:=\Lambda(\psi_{\ell}^{\mathrm{HCT}})$ and
$g\in\mathcal{P}_{0}(\mathcal{E}(P))$ by
$g|_{E(\ell)}:=|E(\ell)|^{-1}\Lambda(\psi_{\ell+N_{P}}^{\mathrm{HCT}})$ for
$\ell=1,\dots,N_{P}$. This implies that $u(f)\in\widehat{V}_{h}(P)$. Notice
that $u(f)$ depends linearly on $f\in\mathcal{P}_{r}(P)$ and so
$\mathcal{P}_{r}(P)\to\widehat{V}_{h}(P),f\mapsto u(f)$ defines a linear map.

Step 7 proves that
$\mathcal{L}:\mathcal{P}_{r}(P)\to\mathcal{P}_{r}(P),f\mapsto\Pi_{r}u(f)$ is
an isomorphism. For any $f\in\mathcal{P}_{r}(P)$ with $\mathcal{L}f=0$, the
orthogonality $(1-\Pi_{r})u(f)\perp\mathcal{P}_{r}(P)$ in $L^{2}(P)$ shows
$0=\int_{P}(\mathcal{L}f)f\,dx=\int_{P}u(f)f\,dx$. This and $v=u(f)$ in (B.5)
result in $0=a^{P}(u(f),u(f))=|u(f)|_{2,P}.$ Consequently
$u(f)\in\mathcal{P}_{1}(P)\cap V_{0}$ and so $u(f)=0$. Thus
$f=\Delta^{2}u(f)=0$ and $\mathcal{L}$ is injective; whence bijective. ∎

Step 8 proves that $\mathrm{Dof}:V_{h}(P)\to\mathbb{R}^{2N_{P}}$ is an
isomorphism.

Proof of surjectivity. Given any $x\in\mathbb{R}^{2N_{P}}$, there exists
$\widehat{u}_{h}\in\widehat{V}_{h}(P)$ with $\text{Dof}(\widehat{u}_{h})=x$
from Step 4. This leads to
$g:=\Pi_{r}G\widehat{u}_{h}-\Pi_{r}\widehat{u}_{h}\in\mathcal{P}_{r}(P)$.
Since $\mathcal{L}$ is bijective in $\mathcal{P}_{r}(P)$ (from Step 7), there
exists $f\in\mathcal{P}_{r}(P)$ with $\Pi_{r}u(f)=g$. Recall that $u(f)\in
V_{0}$ implies $\mathrm{Dof}(u(f))=0$ and $Gu(f)=0$. Altogether,
$u_{h}^{P}:=u(f)+\widehat{u}_{h}\in\widehat{V}_{h}(P)$ satisfies
$\mathrm{Dof}(u^{P}_{h})=x$ and
$\Pi_{r}Gu^{P}_{h}=\Pi_{r}G\widehat{u}_{h}=g+\Pi_{r}\widehat{u}_{h}=\Pi_{r}u(f)+\Pi_{r}\widehat{u}_{h}=\Pi_{r}u_{h}^{P}$.
Hence $u^{P}_{h}\in V_{h}(P)$.∎

Proof of injectivity. Suppose $v_{h}\in V_{h}(P)$ satisfies
$\mathrm{Dof}(v_{h})=0$. Recall $v_{h}\in V_{h}(P)\cap V_{0}$ and $Gv_{h}=0$
from Step 3. The definition (3.1) (for $v=v_{h}$ with $\text{Dof}(v)=0$) leads
for some $f\in\mathcal{P}_{r}(P)$ such that
$a^{P}(v_{h},v_{h})=(f,v_{h})_{L^{2}(P)}$. This and (3.2) reveal

$|v_{h}|_{2,P}^{2}=a^{P}(v_{h},v_{h})=(f,v_{h})_{L^{2}(P)}=(f,\Pi_{r}v_{h})_{L^{2}(P)}=(f,\Pi_{r}Gv_{h})_{L^{2}(P)}=0.$

Hence $v_{h}\in V_{0}\cap\mathcal{P}_{1}(P)$ and so $v_{h}=0$.∎

Proof of (H1). The key observation from Step 8 is that $V_{h}(P)$ has the
dimension $2N_{P}$ and $\text{dof}_{1},\dots,\text{dof}_{2N_{P}}$ from (2.1)
are linear independent. Consequently
$(P,V_{h}(P),(\text{dof}_{1},\dots,\text{dof}_{2N_{P}}))$ is a finite element
in the sense of Ciarlet.∎

Step 9 provides the scaling of an HCT interpolation. Let
$\psi_{h}\equiv\psi_{p}$ be a nodal basis function of the finite element
$(P,V_{h}(P),\text{Dof})$ for some $p\in\\{1,\dots,2N_{P}\\}$ and let
$\psi_{\mathrm{HCT}}$ be its HCT interpolation from Step 2, namely

$\displaystyle\psi_{\mathrm{HCT}}=\sum_{k=1}^{N_{P}}\psi_{h}(z_{k})\psi_{k}^{\mathrm{HCT}}+\sum_{\ell=1}^{N_{P}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{E(\ell)}(\psi_{h})_{\textbf{n}}\,ds\right)\psi_{\ell+N_{P}}^{\mathrm{HCT}}.$
(B.8)

The definition of a nodal basis function $\psi_{h}$ shows that
$\psi_{h}(z_{k})$ and $\int_{E(\ell)}(\psi_{h})_{\textbf{n}}\,ds$ are zero or
one for $k,\ell=1,\dots,N_{P}$. The scaling of
$\psi_{1}^{\mathrm{HCT}},\dots,\psi_{2N_{P}}^{\mathrm{HCT}}$ from (B.2), and
the bound $h_{E}^{-1}\leq\rho^{-1}h_{P}^{-1}$ for all $E\in\mathcal{E}(P)$
from (M2) lead to

$\displaystyle|\psi_{\mathrm{HCT}}|_{2,P}\leq
C_{\mathrm{HCT}}(1+\rho^{-1})h_{P}^{-1}.$ (B.9)

Step 10 controls a nodal basis function in $V_{h}(P)$ by its HCT
interpolation. For a given nodal basis function $\psi_{h}\in V_{h}(P)$, its
HCT interpolation $\psi_{\mathrm{HCT}}$ from (B.8),
$\text{Dof}(\psi_{h}-\psi_{\mathrm{HCT}})=0$ from Step 3, and the test
function $w=\psi_{h}-\psi_{\mathrm{HCT}}$ lead in (3.1) to

$\displaystyle
a^{P}(\psi_{h},\psi_{h}-\psi_{\mathrm{HCT}})=(f,\psi_{h}-\psi_{\mathrm{HCT}})_{L^{2}(P)}$
(B.10)

for some $f\in\mathcal{P}_{r}(P)$. The definition of the $L^{2}$ projection
$\Pi_{r}$ and the relation $\Pi_{r}G\psi_{h}=\Pi_{r}\psi_{h}$ from (3.2) for
$r=-1,0,1,2$ show that

$(f,\psi_{h})_{L^{2}(P)}=(f,\Pi_{r}G\psi_{h})_{L^{2}(P)}=(f,\Pi_{r}G\psi_{\mathrm{HCT}})_{L^{2}(P)}$

with $G\psi_{h}=G\psi_{\mathrm{HCT}}$ from
$\text{Dof}(\psi_{h})=\text{Dof}(\psi_{\mathrm{HCT}})$ and Lemma 2.6 in the
last step. This, (B.10), and a Cauchy-Schwarz inequality imply that

$\displaystyle|\psi_{h}|^{2}_{2,P}$
$\displaystyle\leq|\psi_{h}|_{2,P}|\psi_{\mathrm{HCT}}|_{2,P}+\|f\|_{L^{2}(P)}\|\Pi_{r}G\psi_{\mathrm{HCT}}-\psi_{\mathrm{HCT}}\|_{L^{2}(P)}.$
(B.11)

Recall $f=0$ for $r=-1$ so suppose $r=0,1,2$ for the time being. Since
$\int_{P}(1-\Pi_{r})G\psi_{\mathrm{HCT}}\,dx=0$ from the definition of the
$L^{2}$ projection $\Pi_{r}$ for $r=0,1,2$, the Poincaré inequality from [7,
Subsec. 2.1.5] shows that $\|(1-\Pi_{r})G\psi_{\mathrm{HCT}}\|_{L^{2}(P)}\leq
C_{\text{P}}h_{P}|G\psi_{\mathrm{HCT}}|_{1,P}$ with a positive constant
$C_{\text{P}}$ (that exclusively depends on $\rho$). Since $\psi_{h}$ is a
nodal basis function,
$\int_{E}(\psi_{\mathrm{HCT}})_{\textbf{n}}\,ds=\int_{E}(\psi_{h})_{\textbf{n}}\,ds$
for all $E\in\mathcal{E}(P)$ imply that
$\int_{E}(\psi_{\mathrm{HCT}})_{\textbf{n}}\,ds=0$ for all but at most one
$E\in\mathcal{E}(P)$ and the Poincaré-Friedrichs inequality from Theorem 2.2.a
shows $|\psi_{\mathrm{HCT}}|_{1,P}\leq
C_{\mathrm{PF}}h_{P}|\psi_{\mathrm{HCT}}|_{2,P}$. This, Lemma 2.4, and a
triangle inequality result in
$|G\psi_{\mathrm{HCT}}|_{1,P}\leq|G\psi_{\mathrm{HCT}}-\psi_{\mathrm{HCT}}|_{1,P}+|\psi_{\mathrm{HCT}}|_{1,P}\leq
2C_{\mathrm{PF}}h_{P}|\psi_{\mathrm{HCT}}|_{2,P}$. Hence the previous
estimates lead to

$\displaystyle\|(1-\Pi_{r})G\psi_{\mathrm{HCT}}\|_{L^{2}(P)}\leq
2C_{\text{P}}C_{\mathrm{PF}}h_{P}^{2}|\psi_{\mathrm{HCT}}|_{2,P}.$

This, Lemma 2.4, and
$\|(1-\Pi_{r}G)\psi_{\mathrm{HCT}}\|_{L^{2}(P)}\leq\|(1-\Pi_{r})G\psi_{\mathrm{HCT}}\|_{L^{2}(P)}+\|(1-G)\psi_{\mathrm{HCT}}\|_{L^{2}(P)}$
result in

$\displaystyle\|(1-\Pi_{r}G)\psi_{\mathrm{HCT}}\|_{L^{2}(P)}\leq
C_{\mathrm{PF}}(1+2C_{\text{P}})h_{P}^{2}|\psi_{\mathrm{HCT}}|_{2,P}.$ (B.12)