# Virtual element methods for Biot–Kirchhoff poroelasticity

Rekha Khot  School of Mathematics, Monash University, 9 Rainforest Walk,
Melbourne, Victoria 3800, Australia. Inria 75012, Paris, France.
<EMAIL_ADDRESS>,  David Mora  GIMNAP, Departamento de Matemática,
Universidad del Bío-Bío, Concepción, Chile; and CI2MA, Universidad de
Concepción, Concepción, Chile<EMAIL_ADDRESS>and  Ricardo Ruiz-Baier
School of Mathematics, Monash University, 9 Rainforest Walk, Melbourne,
Victoria 3800, Australia; and Sechenov First Moscow State Medical University,
Moscow, Russia; and Universidad Adventista de Chile, Casilla 7-D, Chillán,
Chile<EMAIL_ADDRESS>
###### Abstract.

This paper analyses conforming and nonconforming virtual element formulations
of arbitrary polynomial degrees on general polygonal meshes for the coupling
of solid and fluid phases in deformable porous plates. The governing equations
consist of one fourth-order equation for the transverse displacement of the
middle surface coupled with a second-order equation for the pressure head
relative to the solid with mixed boundary conditions. We propose novel
enrichment operators that connect nonconforming virtual element spaces of
general degree to continuous Sobolev spaces. These operators satisfy
additional orthogonal and best-approximation properties (referred to as
conforming companion operators in the context of finite element methods),
which play an important role in the nonconforming methods. This paper proves a
priori error estimates in the best-approximation form, and derives
residual–based reliable and efficient a posteriori error estimates in
appropriate norms, and shows that these error bounds are robust with respect
to the main model parameters. The computational examples illustrate the
numerical behaviour of the suggested virtual element discretisations and
confirm the theoretical findings on different polygonal meshes with mixed
boundary conditions.

###### 2010 Mathematics Subject Classification:

Primary 65N30, 65N12, 65N15

This research was supported by the Monash Mathematics Research Fund
S05802-3951284; by the National Agency for Research and Development, ANID-
Chile through FONDECYT project 1220881 and by project Centro de Modelamiento
Matemático (CMM), FB210005, BASAL funds for centers of excellence; by the
Australian Research Council through the Discovery Project grant DP220103160
and the Future Fellowship grant FT220100496; and by the Ministry of Science
and Higher Education of the Russian Federation within the framework of state
support for the creation and development of World-Class Research Centres
“Digital biodesign and personalised healthcare” No. 075-15-2022-304.

##  1\. Introduction

Scope. Fluid-saturated porous media that deform are an essential ingredient in
many engineering, biophysical and environmental applications. From these
materials, a family featuring interesting properties is compressible thin
plates. Porosity and permeability characteristics through the thickness can be
averaged, leading to a different scaling of poromechanical properties from the
typical structure exhibited in Biot’s consolidation systems (see, for example
[18, Chapter 8]).

A number of works have addressed the rigorous derivation of poroelastic plate
effective equations [33, 34, 37, 39, 40]. The well-posedness analysis has been
conducted, for a slightly different model, in the recent paper [29]. Regarding
numerical methods, a discontinuous Galerkin formulation has been proposed in
[31] (following [38]) and splitting algorithms have been analysed. High-order
finite element methods have been used for layer-wise poroelastic shells in
[26].

The virtual element method (VEM) has gained popularity in recent years due to
its ability to handle complex geometries and provide high accuracy numerical
solutions for partial differential equations (PDEs). Another important feature
of VEM is the possibility of easily implementing highly regular discrete
spaces. This idea is initiated in [12], where spaces of high global regularity
(such as $C^{1}$, $C^{2}$ or more) are easily built in a very efficient way.
This has been applied and tested in some biharmonic models of thin plates. The
literature contains error analysis of VEM for biharmonic problems for thin
plates models (Kirchhoff plates), with a particular emphasis on conforming and
nonconforming approximations, including eigenvalue problems [2, 4, 12, 16, 22,
35, 36]. Other VE discretisations for biharmonic problems in plate models
provide a detailed error analysis of the particular type of method and
demonstrate its effectiveness through numerical experiments, and the analysis
also includes a posteriori error estimates, the time dependent case, and the
extension to 3D, among others. See for example [1, 7, 8, 21, 43, 42].

Since nonconforming spaces might not be a subset of the continuous space, the
enrichment (averaging) operator introduced in [9] is adapted to prove a priori
error estimates utilising a posteriori error bounds in [28] (known as medius
analysis). These averaging operators essentially assist in connecting the
nonconforming discrete space to the continuous space with the required
approximation properties. Such operators are then modified in [17] (referred
to as conforming companion operators) satisfying, in addition, orthogonality
properties and best approximation estimates. A similar strategy can be carried
out in VEM. Indeed, enrichment operators for VEM were recently proposed in
[30], and companion operators for VEM were analysed in [15, 16]. Since VE
functions are not known explicitly, the computable or non-computable
enrichment/companion operators acting on VE functions can be constructed
depending on the specific purpose. In particular, computable conforming
companion operators can be exploited to define the discrete problem and
allowing rough sources [16]. In turn, the non-computable ones can be used for
purposes of the analysis (not necessarily knowing the structure explicitly).
So far, the design of such computable companion operators requires a shape-
regular sub-triangulation of the polygonal decomposition together with lowest-
order finite element spaces such as Crouzeix–Raviart for second-order and
Morley for fourth–order problems. One of the primal advantages of computable
companion maps is that they permit to approximate rough sources in
$H^{-2}(\Omega)$. In this paper we consider sources in $L^{2}(\Omega)$, and
hence we only need to present the construction of non-computable companion
operators. This is enough for the required analysis and most importantly, we
can avoid to go down on the sub-triangular mesh as done in the construction of
computable companion operators. We stress that the proposed enrichment
operator maps nonconforming VE spaces to conforming VE spaces of one degree
higher (which is different from the construction in [30]), and in addition, it
satisfies $H^{2}$-orthogonality and best approximation estimates. We then
modify this enrichment operator through a variety of bubble-functions to
design new companion operators having a lower-order orthogonality property (in
$H^{1}$ and $L^{2}$). The treatment of general boundary conditions is
carefully addressed in this paper, for which the definition and thorough
analysis of the new companion operators is key to establish well-posedness and
to obtain error estimates.

This paper presents an extension of nonconforming VE formulations for the
coupling of biharmonic problems and second-order elliptic equations (see the
similar methods introduced for each of the component sub-problems of second-
order linear elliptic and fourth-order elliptic in the recent contributions
[15, 16], respectively). The model encodes the interaction with a fluid phase,
and the study of this type of problems has gained significant attention due to
its relevance in various physical applications. More generally, the proposed
framework offers a unified approach to solve coupled problems with mixed
boundary conditions on polygonal domains, even when they are non-convex. For
conforming cases, we combine $C^{1}-C^{0}$ types of VEMs with various
polynomial degrees. The error estimates, measured in the weighted
${H}^{2}\times{H}^{1}$ energy norm (for deflection and moment pressure),
demonstrate robustness with respect to material parameters. Additionally, we
introduce a reliable and efficient a posteriori error estimator of residual
type. Leveraging the flexibility of VEMs in utilising polygonal meshes, we
employ the error estimator to drive an adaptive scheme. Notably, the proposed
a posteriori analysis is novel for high-order nonconforming VEMs and can be
applied to tackle more complex coupled problems: we emphasise that the models
presented in this work can serve as a fundamental building block for
establishing a comprehensive framework for complex mixed-dimensional
poroelastic models. These models can also be extended to incorporate
interaction with multi-layered structures, such as thermostats and micro-
actuators, offering broad applicability and versatility.

The main contributions of this work can be summarised as follows:

  * •

Application of the proposed conforming and nonconforming VEMs to the plate
Biot equations.

  * •

Design of new companion operators with the orthogonal properties and the best-
approximation estimates.

  * •

A priori error estimates in the energy norm for both conforming and
nonconforming formulations in the best-approximation form that remain robust
with respect to material parameters.

  * •

The detailed proofs of the inverse estimate and the norm equivalence for the
nonconforming VE functions.

  * •

Introduction and analysis of a residual-based a posteriori error estimator.

  * •

Presentation of numerical results validating the theoretical estimates and
demonstrating the competitive performance of the proposed schemes.

Content and structure. The remainder of the paper has been organised in the
following manner. In the rest of this section we provide preliminary
notational conventions and definitions to be used throughout the paper.
Section 2 contains the model description and defines the weak formulation of
the governing equations. The local and global VE spaces, the degrees of
freedom and the computable polynomial projection operators are addressed in
Subsection 3.1, and the derivations for both conforming and nonconforming
approximations and the analysis of existence and uniqueness of discrete
solution are conducted in Subsection 3.2. The a priori error analysis for the
conforming VE methods in the best-approximation form is carried out in Section
4. For the nonconforming case, the companion operators are defined in
Subsection 5.1 along with the proofs of the properties and the best
approximation estimates followed by the a priori error estimates in Subsection
5.2. Subsections 6.1 recalls the preliminary estimates and 6.2 contains the
detailed proofs of standard estimates such as the inverse estimate and the
norm equivalence for the nonconforming VE functions, and a Poincaré-type
inequality for $H^{2}$ functions. The reliability and efficiency of an a
posteriori error estimator are included in Subsection 6.4-6.5. Finally, a
collection of illustrative numerical tests is presented in Section 7.

Recurrent notation and domain configuration. Consider an open, bounded,
connected Lipschitz domain of $\mathbb{R}^{3}$, denoted
$\widehat{\Omega}=\Omega\times(-\zeta,\zeta)\subset\mathbb{R}^{3}$ and
occupied by an undeformed thin poroelastic plate (a deformable solid matrix or
an array of solid particles) of characteristic thickness $2\zeta$, and where
$\Omega\subset\mathbb{R}^{2}$ represents the mid-surface of the undeformed
poroelastic plate. The plate is assumed to be isotropic in the plate plane and
to follow the Kirchhoff law. In particular, it is assumed that the plate
fibers remain orthogonal to the deflected mid-surface [24]. An appropriate
modification of Biot constitutive poroelasticity equations is adopted in
combination with Darcy flow in deforming pores (see [32]). Following the model
presented in [31], we assume that the equations governing the balance of
momentum and mass of the solid and fluid phases can be written in terms of the
averaged-through-thickness deflection $u$ (vertical displacement of the solid
phase) and the first moment of the pressure of the fluid phase $p$. We will
denote by $\boldsymbol{n}$ the unit normal vector on the undeformed boundary
$\partial\Omega$. The boundary $\partial\Omega$ is disjointly split between a
closed set $\Gamma^{c}$ and an open set $\Gamma^{s}$ where we impose,
respectively, homogeneous deflections and homogeneous normal derivatives of
deflections and of pressure moment (clamped sub-boundary with zero-flux) and
homogeneous pressures with normal deflections and bending moments (simply
supported sub-boundary).

For a subdomain $S\subseteq\Omega$ we will adopt the notation
$(\cdot,\cdot)_{m,S}$ for the inner product, and $\|\cdot\|_{m,S}$ (resp.
$|\cdot|_{m,S}$) for the norm (resp. seminorm) in the Sobolev space $H^{m}(S)$
(or in its vector counterpart $\mathbf{H}^{m}(S)$) with $m\geq 0$. We
sometimes drop $0$ from the subscript in $L^{2}$ inner product and norms for
convenience. In view of the boundary conditions mentioned above, we also
define the Sobolev spaces $H^{2}_{\Gamma^{c}}(\Omega):=\\{v\in
H^{2}(\Omega):v=\partial_{\boldsymbol{n}}v=0\;\text{on}\;\Gamma^{c}\\}$, where
$\partial_{\boldsymbol{n}}v$ denotes the normal derivative of $v$, and
$H^{1}_{\Gamma^{s}}(\Omega):=\\{q\in
H^{1}(\Omega):q=0\;\text{on}\;\Gamma^{s}\\}$. Also, given an integer $k\geq 1$
and $S\subset\mathbb{R}^{d}$, $d=1,2$, by $\mathbb{P}_{k}(S)$ we will denote
the space of polynomial functions defined locally in $S$ and being of total
degree up to $k$. Given a barycentre $x_{S}$ and diameter $h_{S}$ of a domain
$S$, we define the set of scaled monomials $\mathbb{M}_{k}(S)$ of total degree
up to $k$ and $\mathbb{M}_{k}^{*}(S)$ of degree equal to $k$ by

$\mathbb{M}_{k}(S)=\Big{\\{}\Big{(}\frac{x-x_{S}}{h_{S}}\Big{)}^{\ell}:|\ell|\leq
k\Big{\\}},\text{and}\mathbb{M}^{*}_{k}(S)=\Big{\\{}\Big{(}\frac{x-x_{S}}{h_{S}}\Big{)}^{\ell}:|\ell|=k\Big{\\}}.$

Throughout the paper we use $C$ to denote a generic positive constant
independent of the mesh size $h$ and of the main model parameters, that might
take different values at its different occurrences. Moreover, given any
positive expressions $X$ and $Y$, the notation $X\,\lesssim\,Y$ means that
$X\,\leq\,C\,Y$ (similarly for $X\,\gtrsim\,Y$).

##  2\. Plate Biot equations and solvability analysis

The two-dimensional poroelasticity problem arising when a fluid flows through
a deformable porous plate of thickness $d$ can be written in the following
form

(2.1a) $\displaystyle\hat{\rho}\frac{\partial^{2}u}{\partial
t^{2}}+\text{div}\mathop{\mathbf{div}}\nolimits\mathbb{A}\nabla^{2}u+\alpha
b\Delta p$ $\displaystyle=f$ $\displaystyle\text{in $\Omega\times(0,T]$},$
(2.1b)
$\displaystyle\bigg{(}c_{0}+\frac{\alpha^{2}}{\lambda+2\mu}\bigg{)}\frac{\partial
p}{\partial t}-\alpha b\frac{d^{3}}{12}\frac{\partial(\Delta u)}{\partial
t}-\frac{\kappa}{\eta}\Delta p$ $\displaystyle=g$ $\displaystyle\text{in
$\Omega\times(0,T]$},$ (2.1c) $\displaystyle
u=\partial_{\boldsymbol{n}}u=\partial_{\boldsymbol{n}}p$ $\displaystyle=0$
$\displaystyle\text{on $\Gamma^{c}\times(0,T]$},$ (2.1d) $\displaystyle
u=\partial_{\boldsymbol{n}\boldsymbol{n}}u=p$ $\displaystyle=0$
$\displaystyle\text{on $\Gamma^{s}\times(0,T]$},$

with appropriate initial conditions and where $\hat{\rho}$ is the reduced
density (taking into account a volume-to-surface scaling),
$\mathbb{A}:=D((1-\nu)\mathbb{I}+\nu\mathbb{I}\otimes\mathbb{I})$ for the
flexural rigidity
$D:=\frac{Ed^{3}}{12(1-\nu^{2})}=\frac{\mu(\lambda+\mu)d^{3}}{3(\lambda+2\mu)}$
of the plate for the Young modulus $E$, the Poisson ratio $\nu\in(0,0.5)$, the
Lamé parameters $\lambda,\mu$ and
$\partial_{\boldsymbol{n}\boldsymbol{n}}u:=(\nabla^{2}u)\boldsymbol{n}\cdot\boldsymbol{n}$.
Also $\alpha$ is the Biot–Willis poroelastic coefficient,
$b=2\mu(\lambda+2\mu)^{-1}$, $c_{0}$ is the total storage capacity, $\kappa$
is the absolute permeability, and $\eta$ is the viscosity of the fluid. We
concentrate our attention in the following modification of this Biot–Kirchhoff
system in the case of a specific adimensionalisation proposed in [31, eq.
(7)-(9)]:

(2.2a) $\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}+\Delta^{2}u+\alpha\Delta p$ $\displaystyle=f$ $\displaystyle\text{in
$\Omega\times(0,T]$},$ (2.2b) $\displaystyle\beta\frac{\partial p}{\partial
t}-\alpha\frac{\partial(\Delta u)}{\partial t}-\gamma\Delta p$
$\displaystyle=g$ $\displaystyle\text{in $\Omega\times(0,T]$},$

where with an abuse of notation we have kept the same notation for the
dimensionless problem. Here $f\in{L}^{2}(0,T;\Omega)$ is the normal vertical
loading and $g\in{L}^{2}(0,T;\Omega)$ is a prescribed mass source/sink.
Henceforth, we treat $\alpha,\beta,\gamma$ as the main model parameters, where
$\alpha\leq 1\leq\gamma$ and

$\beta=\bigl{(}c_{0}[\lambda+2\mu]+\alpha^{2}\bigr{)}\gamma,\quad\gamma=\frac{\lambda+\mu}{\mu}.$

Note that the reduced density $\hat{\rho}$ and the rigidity $D$ of the plate
are absorbed in the load $f$ and the model parameter $\alpha$. In plate
problems, the Poisson ratio $\nu$ tending to $0.5$ is one of the interesting
limiting phenomena and that, in turn, leads to the case of the first Lame
parameter $\lambda\to\infty$. In our new normalised problem, it essentially
implies that the model parameters $\beta,\gamma$ will go to infinity. The aim
in this paper is to analyse the normalised problem (2.2) and show that its
robust with respect to the model parameters $\alpha,\beta,\gamma$.

System (2.2) is similar to the non-inertial problem in [31] which accommodates
fluid-saturated plates where diffusion is possible in the in-plane direction
(see also the set of problems recently analysed in [29]), here extended to the
case of mixed boundary conditions. In order to fix ideas, we will focus first
on a simplified system, resulting from applying a centred and backward Euler
semi-discretisation in time to (2.2a)-(2.2b), with a conveniently rescaled
final time $T$ and rescaled time step to $\Delta t=1$. Define the displacement
and pressure space

$V:=H^{2}_{\Gamma^{c}}(\Omega)\cap H^{1}_{0}(\Omega),\quad
Q:=H^{1}_{\Gamma^{s}}(\Omega).$

Owing to the specification of boundary conditions (taken homogeneous for sake
of simplicity of the presentation), a weak formulation is obtained, which
reads: Find $(u,p)\in V\times Q$ such that

(2.3a)
$\displaystyle(u,v)_{\Omega}+(\nabla^{2}u,\nabla^{2}v)_{\Omega}-\alpha(\nabla
p,\nabla v)_{\Omega}$ $\displaystyle=(\tilde{f},v)_{\Omega}\qquad\forall\;v\in
V,$ (2.3b) $\displaystyle\beta(p,q)_{\Omega}+\alpha(\nabla q,\nabla
u)_{\Omega}+\gamma(\nabla p,\nabla q)_{\Omega}$
$\displaystyle=(\tilde{g},q)_{\Omega}\qquad\forall\;q\in Q,$

with $\nabla^{2}v:=\begin{pmatrix}v_{xx}&v_{xy}\\\ v_{yx}&v_{yy}\end{pmatrix}$
being the Hessian matrix (of second-order derivatives) for a given $v\in
H^{2}(\Omega)$. The right-hand side terms also include the value of deflection
and pressure moments in the previous backward Euler time steps, denoted as
$\widehat{u}^{n},\widehat{u}^{n-1}$ and $\widehat{p}^{n}$, respectively:

$\tilde{f}=f+2\widehat{u}^{n}-\widehat{u}^{n-1},\qquad\tilde{g}=g+\widehat{p}^{n},$

where the index $n\geq 0$ indicates the time step.

The product space $\mathbf{H}_{\epsilon}$ contains all
$\vec{\boldsymbol{u}}\in[H^{2}_{\Gamma^{c}}(\Omega)\cap
H^{1}_{0}(\Omega)]\times H^{1}_{\Gamma^{s}}(\Omega)$ which are bounded in the
norm

(2.4)
$\|\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}}^{2}:=\|u\|_{\Omega}^{2}+|u|_{2,\Omega}^{2}+\beta\|{p}\|_{\Omega}^{2}+\gamma|{p}|_{1,\Omega}^{2}.$

The subscript $\epsilon$ denotes the weighting parameters (in our case,
$\beta,\gamma$). Let us now group the trial and test fields as
$\vec{\boldsymbol{u}}=(u,p)$ and $\vec{\boldsymbol{v}}=(v,q)$, respectively;
and introduce the operator
$\mathcal{A}:\mathbf{H}_{\epsilon}\to\mathbf{H}_{\epsilon}$ defined as

$\displaystyle\langle\mathcal{A}(\vec{\boldsymbol{u}}),\vec{\boldsymbol{v}}\rangle$
$\displaystyle:=(u,v)_{\Omega}+(\nabla^{2}u,\nabla^{2}v)_{\Omega}-\alpha(\nabla
p,\nabla v)_{\Omega}+\beta(p,q)_{\Omega}+\alpha(\nabla q,\nabla u)_{\Omega}$
$\displaystyle\quad+\gamma(\nabla p,\nabla q)_{\Omega},$

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between
$\mathbf{H}_{\epsilon}$ and $\mathbf{H}_{\epsilon}^{\prime}$. Note that
$|u|_{2,\Omega}$ defines a norm on $V$, which is equivalent to $H^{2}$-norm
[23, pp. 34]. In particular, this implies for any $v\in V$ that

(2.5) $\displaystyle\|v\|_{2,\Omega}\lesssim|v|_{2,\Omega}.$

We also define the linear and bounded operator
$\mathcal{F}:\mathbf{H}_{\epsilon}\to\mathbb{R}$ as

$\vec{\boldsymbol{v}}\mapsto\mathcal{F}(\vec{\boldsymbol{v}}):=(\tilde{f},v)_{\Omega}+(\tilde{g},q)_{\Omega},$

and therefore Problem (2.3) is recast as: Find
$\vec{\boldsymbol{u}}\in\mathbf{H}_{\epsilon}$ such that

(2.6)
$\langle\mathcal{A}(\vec{\boldsymbol{u}}),\vec{\boldsymbol{v}}\rangle=\mathcal{F}(\vec{\boldsymbol{v}})\qquad\forall\;\vec{\boldsymbol{v}}\in\mathbf{H}_{\epsilon}.$

We are now in a position to state the solvability of the continuous problem
(2.6).

######  Theorem 2.1.

Problem (2.6) is well-posed in the space $\mathbf{H}_{\epsilon}$ equipped with
the norm (2.4).

###### Proof.

It follows from the Lax–Milgram lemma (see, e.g., [25, Lemma 25.2]), requiring
the boundedness of $\mathcal{A}$ over the space $\mathbf{H}_{\epsilon}$

$\langle\mathcal{A}(\vec{\boldsymbol{u}}),\vec{\boldsymbol{v}}\rangle\lesssim\|\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}}\|\vec{\boldsymbol{v}}\|_{\mathbf{H}_{\epsilon}}\qquad\forall\;\vec{\boldsymbol{u}},\vec{\boldsymbol{v}}\in\mathbf{H}_{\epsilon},$

and the boundedness of $\mathcal{F}$, as well as the coercivity condition

$\langle\mathcal{A}(\vec{\boldsymbol{u}}),\vec{\boldsymbol{u}}\rangle=\|\vec{\boldsymbol{u}}\|^{2}_{\mathbf{H}_{\epsilon}}\qquad\forall\;\vec{\boldsymbol{u}}\in\mathbf{H}_{\epsilon}.$

For the continuity it suffices to apply the Cauchy–Schwarz inequality while
the coercivity is a direct consequence of the definition of the solution
operator (whose off-diagonal terms cancel out). ∎

Now, we state an additional regularity result for the solution of problem
(2.6).

Regularity estimates [27]. Given $\tilde{f}\in H^{s-4}(\Omega)$ and
$\tilde{g}\in H^{r-2}(\Omega)$ with $s\geq 2$ and $r\geq 1$, there exists a
unique solution $\vec{\boldsymbol{u}}=(u,p)\in(H^{s}(\Omega)\cap
V)\times(H^{r}(\Omega)\cap Q)$ to (2.6) such that

(2.7)
$\displaystyle\|u\|_{s,\Omega}+\|p\|_{r,\Omega}\lesssim\|\tilde{f}\|_{s-4,\Omega}+\|\tilde{g}\|_{r-2,\Omega}.$

######  Remark 2.2 (Simply supported boundary condition).

The boundary condition $u=\partial_{\boldsymbol{n}\boldsymbol{n}}u=0$ on the
simply supported part and an integration by parts

$(\Delta^{2}u,v)_{\Omega}=(\nabla^{2}u,\nabla^{2}v)_{\Omega}+(\partial_{\boldsymbol{n}}(\Delta
u),v)_{\partial\Omega}-(\partial_{\boldsymbol{n}\boldsymbol{n}}u,\partial_{\boldsymbol{n}}v)_{\partial\Omega}-(\partial_{\boldsymbol{n}\boldsymbol{t}}u,\partial_{\boldsymbol{t}}v)_{\partial\Omega}$

for $v\in V$ allow the hessian term
$(\nabla^{2}u,\nabla^{2}v)_{\Omega}=(\Delta^{2}u,v)_{\Omega}$ in the weak
form. However to be consistent with plate mechanics, a zero bending moment
$M_{\boldsymbol{n}\boldsymbol{n}}(u):=\nu\Delta
u+(1-\nu)\partial_{\boldsymbol{n}\boldsymbol{n}}u=0$ should be prescribed in
place of $\partial_{\boldsymbol{n}\boldsymbol{n}}u=0$ and instead one can
consider the integration by parts

$\displaystyle\widetilde{a}(u,v)$ $\displaystyle:=\int_{\Omega}\nu\Delta
u\Delta v+(1-\nu)u_{ij}v_{ij}$
$\displaystyle=\int_{\Omega}\Delta^{2}uv+\int_{\partial\Omega}M_{\boldsymbol{n}\boldsymbol{n}}(u)\partial_{\boldsymbol{n}}v-\int_{\partial\Omega}(\partial_{\boldsymbol{n}}(\Delta
u)v+(1-\nu)\partial_{\boldsymbol{n}\boldsymbol{t}}u\partial_{\boldsymbol{t}}v).$

This replaces the hessian term by the above bilinear form $\widetilde{a}(u,v)$
and the weak formulation becomes

(2.8) $\displaystyle(u,v)_{\Omega}+\widetilde{a}(u,v)-\alpha(\nabla p,\nabla
v)_{\Omega}$ $\displaystyle=(\tilde{f},v)_{\Omega}\qquad\forall\;v\in V,$
(2.9) $\displaystyle\beta(p,q)_{\Omega}+\alpha(\nabla q,\nabla
u)_{\Omega}+\gamma(\nabla p,\nabla q)_{\Omega}$
$\displaystyle=(\tilde{g},q)_{\Omega}\qquad\forall\;q\in Q.$

The boundedness of modified $\mathcal{A}$ is straightforward and the
coercivity follows from the observation $\widetilde{a}(v,v)=(\nu\|\Delta
v\|^{2}_{\Omega}+(1-\nu)|v|^{2}_{2,\Omega})\gtrsim|v|^{2}_{2,\Omega}$ with the
equivalence (2.5) in the last inequality.

##  3\. Virtual element formulation and unique solvability of the discrete
problem

Let us denote by $\\{\mathcal{T}_{h}\\}_{h>0}$ a shape-regular family of
partitions of $\bar{\Omega}$, conformed by polygons $K$ of diameter $h_{K}$,
and we denote the mesh size by $h:=\max\\{h_{K}:K\in\mathcal{T}_{h}\\}$. Let
$\mathcal{V}=\mathcal{V}^{i}\cup\mathcal{V}^{c}\cup\mathcal{V}^{s}$ and
$\mathcal{E}=\mathcal{E}^{i}\cup\mathcal{E}^{c}\cup\mathcal{E}^{s}$ be the set
of interior vertices $\mathcal{V}^{i}$ and boundary vertices
$\mathcal{V}^{c}\cup\mathcal{V}^{s}$, and the set of interior edges
$\mathcal{E}^{i}$ and boundary edges $\mathcal{E}^{c}\cup\mathcal{E}^{s}$. By
$N_{K}$ we will denote the number of vertices/edges in the generic polygon
$K$. For all edges $e\in\partial K$, we denote by $\boldsymbol{n}_{K}^{e}$ the
unit normal pointing outwards $K$, $\boldsymbol{t}^{e}_{K}$ the unit tangent
vector along $e$ on $K$, and $V_{i}$ represents the $i^{th}$ vertex of the
polygon $K$. We suppose that there exists a universal positive constant $\rho$
such that

  1. (M1)

every polygon $K\in\mathcal{T}_{h}$ of diameter $h_{K}$ is star-shaped with
respect to every point of a ball of radius greater than or equal to $\rho
h_{K}$,

  2. (M2)

every edge $e$ of $K$ has a length $h_{e}$ greater than or equal to $\rho
h_{K}$.

Throughout this section we will construct and analyse a conforming and a
nonconforming family of VE methods.

###  3.1. Virtual element spaces

VE spaces for displacement approximation. First we define the bilinear form
$a^{K}$ as the restriction to $K$ of

$a(v,w):=\int_{\Omega}\nabla^{2}v:\nabla^{2}w\,\mbox{d}\boldsymbol{x}.$

For $K\in\mathcal{T}_{h}$ and $k\geq 2$, define the projection operator
$\Pi^{\nabla^{2\\!}}_{k}:H^{2}(K)\to\mathbb{P}_{k}(K)$, for $v\in H^{2}(K)$,
by

(3.1) $\displaystyle
a^{K}(\Pi^{\nabla^{2\\!}}_{k}v,\chi_{k})=a^{K}(v,\chi_{k})$
$\displaystyle\qquad\forall\chi_{k}\in\mathbb{P}_{k}(K),$

with the additional conditions

(3.2a) $\displaystyle\overline{\Pi^{\nabla^{2\\!}}_{k}v}=\overline{v}\quad$
$\displaystyle\text{and}\quad\overline{\nabla\Pi^{\nabla^{2\\!}}_{k}v}=\overline{\nabla
v}$ $\displaystyle\text{for conforming VEM},$ (3.2b)
$\displaystyle\overline{\Pi^{\nabla^{2\\!}}_{k}v}=\overline{v}\quad$
$\displaystyle\text{and}\quad\int_{\partial
K}\nabla\Pi^{\nabla^{2\\!}}_{k}v\,\mbox{d}s=\int_{\partial K}\nabla
v\,\mbox{d}s$ $\displaystyle\text{for nonconforming VEM},$

where $\overline{v}$ is the average
$\frac{1}{N_{K}}\sum_{i=1}^{N_{K}}v(V_{i})$ of the values of $v$ at the
vertices $V_{i}$ of $K$. Since the linear polynomials
$\chi_{k}\in\mathbb{P}_{1}(K)\subset\mathbb{P}_{k}(K)$ lead to the identity
$0=0$ in (3.1), it follows that the two conditions in (3.2a) for conforming
and (3.2b) for nonconforming fix the affine contribution and define
$\Pi^{\nabla^{2\\!}}_{k}v$ uniquely for a given $v$. Furthermore, the
Poincaré–Friedrichs inequality implies

(3.3) $\displaystyle\|v-\Pi^{\nabla^{2\\!}}_{k}v\|_{K}\lesssim
h_{K}|v-\Pi^{\nabla^{2\\!}}_{k}v|_{1,K}\lesssim
h_{K}^{2}|v-\Pi^{\nabla^{2\\!}}_{k}v|_{2,K}.$

The local conforming VE space $V_{h}^{k,\mathrm{c}}(K)$ [12] is a set of
solutions to a biharmonic problem over $K$ with clamped boundary conditions on
$\partial K$, and it is defined, for $k\geq 2$ and $r=\max\\{k,3\\}$, as

$V_{h}^{k,\mathrm{c}}(K):=\begin{dcases}\begin{rcases}&v_{h}\in H^{2}(K)\cap
C^{1}(\partial
K):\Delta^{2}v_{h}\in\mathbb{P}_{k}(K),\;v_{h}|_{e}\in\mathbb{P}_{r}(e)\;\text{and}\\\
&\;\nabla
v_{h}|_{e}\cdot\boldsymbol{n}^{e}_{K}\in\mathbb{P}_{k-1}(e)\quad\forall\;e\in\partial
K,\;\text{and}\\\
&\;(v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h},\chi)_{K}=0\quad\forall\;\chi\in\mathbb{P}_{k}(K)\setminus\mathbb{P}_{k-4}(K)\end{rcases}.\end{dcases}$

On the other hand, the local nonconforming VE space is a set of solutions to a
biharmonic problem with simply supported boundary conditions and was first
introduced in [43]. However, it is pointed out in [16] that the definition in
[43] works for a polygon $K$ without hanging nodes, and that new work provides
an alternative definition for the lowest-order case ($k=2$) with possibly
hanging nodes in $K$. In this paper, we extend such a definition of the
nonconforming VE space for a general degree $k$. First we need some
preliminary geometrical notations. Let $K\in\mathcal{T}_{h}$ be a polygonal
element, and $\mathcal{E}_{K}:=\\{e_{1},\dots,e_{N_{K}}\\}$ and
$V_{1},\dots,V_{N_{K}}$ be the edges and vertices of $K$. Suppose that
$z_{1},\dots,z_{\tilde{N}_{K}}$ denote the corner points of $K$ for some
$\tilde{N}_{K}\leq N_{K}$, where the angle at each $z_{j}$ is different from
$0,\pi,2\pi$. The boundary $\partial K=e_{1}\cup\dots\cup e_{N_{K}}$ can also
be viewed as a union of the sides $s_{1},\dots,s_{\tilde{N}_{K}}$, where
$s_{j}:=\text{conv}\\{z_{j},z_{j+1}\\}$ (convex hull of $z_{j}$ and $z_{j+1}$)
for $z_{j}=V_{m_{j}}$ and $z_{j+1}=V_{m_{j}+n_{j}}$ with
$z_{\tilde{N}_{K}+1}=z_{1}$. Note that $m_{j}$ is the index in the vertex
numbering corresponding to the $j^{\text{th}}$ index of the corners of $K$ and
$n_{j}$ is the total number of straight edges on the sides $s_{j}$. See a
sketch in Figure 1 (e.g., $m_{1}=1$ and $n_{1}=2$, $m_{2}=3$ and $n_{2}=1$,
and so on.)

Figure 1. Sample of pentagonal element with vertices $V_{1},\dots,V_{5}$,
edges $e_{1},\dots,e_{5}$, corners $z_{1},\dots,z_{4}$, and sides
$s_{1},\dots,s_{4}$.

With these notations, we are in a position to define the local nonconforming
VE space $V_{h}^{k,\mathrm{nc}}(K)$ for $k\geq 2$ by

$V_{h}^{k,\mathrm{nc}}(K):=\begin{dcases}\begin{rcases}&v_{h}\in
H^{2}(K):\Delta^{2}v_{h}\in\mathbb{P}_{k}(K),\;v_{h}|_{e}\in\mathbb{P}_{k}(e)\;\text{and}\;\Delta
v_{h}|_{e}\in\mathbb{P}_{k-2}(e)\\\
&\quad\forall\;e\in\mathcal{E}_{K},\;v_{h}|_{s_{j}}\in
C^{1}(s_{j}),\;\int_{e_{m_{j}}}v_{h}\chi\,\mbox{d}s=\int_{e_{m_{j}}}\Pi^{\nabla^{2\\!}}_{k}v_{h}\chi\,\mbox{d}s\\\
&\quad\forall\;\chi\in\mathbb{P}_{k-2}(e_{m_{j}}),\;\text{and}\;\int_{e_{m_{j}+i}}v_{h}\chi\,\mbox{d}s=\int_{e_{m_{j}+i}}\Pi^{\nabla^{2\\!}}_{k}v_{h}\chi\,\mbox{d}s\\\
&\quad\forall\;\chi\in\mathbb{P}_{k-3}(e_{m_{j}+i})\;\text{for}\;i=1,\dots,n_{j},\;\text{and}\;j=1,\dots,\tilde{N}_{K},\\\
&\quad(v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h},\chi)_{K}=0\quad\forall\;\chi\in\mathbb{P}_{k}(K)\setminus\mathbb{P}_{k-4}(K)\end{rcases}.\end{dcases}$

The local degrees of freedom (DoFs) for both conforming and nonconforming VE
spaces are summarised in Table 1.

degree | DoFs of $v_{h}\in V_{h}^{k,\mathrm{c}}(K)$ | DoFs of $v_{h}\in V_{h}^{k,\mathrm{nc}}(K)$ $\displaystyle{\int_{K}^{K}}$
---|---|---
$k\geq 2$ | ($\mathbb{D}$1) $v_{h}(V_{i})\quad\forall\;i=1,\dots,N_{K}$ | ($\mathbb{D}$1⋆) $v_{h}(V_{i})\quad\forall\;i=1,\dots,N_{K}$$\displaystyle{\int_{K}^{K}}$
| ($\mathbb{D}$2) $h_{V_{i}}\nabla v_{h}(V_{i})\quad\forall\;i=1,\dots,N_{K}$ | ($\mathbb{D}$2⋆) $\int_{e}\partial_{\boldsymbol{n}}v_{h}\chi\,\mbox{d}s\quad\forall\;\chi\in\mathbb{M}_{k-2}(e)$$\displaystyle{\int_{K}^{K}}$
$k\geq 3$ | ($\mathbb{D}$3) $\int_{e}\partial_{\boldsymbol{n}}v_{h}\chi\,\mbox{d}s\quad\forall\;\chi\in\mathbb{M}_{k-3}(e)$ | ($\mathbb{D}$3⋆) $\mathchoice{{\vbox{\hbox{$\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}\chi\,\mbox{d}s\quad\forall\;\chi\in\mathbb{M}_{k-3}(e)$$\displaystyle{\int_{K}^{K}}$
$k\geq 4$ | ($\mathbb{D}$4) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{e}v_{h}\chi\,\mbox{d}s\quad\forall\;\chi\in\mathbb{M}_{k-4}(e)$ | ($\mathbb{D}$4⋆) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{K}v_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\;\chi\in\mathbb{M}_{k-4}(K)$$\displaystyle{\int_{K}^{K}}$
| ($\mathbb{D}$5) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{K}v_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\;\chi\in\mathbb{M}_{k-4}(K)$ |
Table 1. The left panel describes the DoFs of $V_{h}^{k,\mathrm{c}}(K)$ with
the characteristic length (see [12], for example) $h_{V_{i}}$ associated with
each vertex $V_{i}$ for all $i=1,\dots,N_{K}$ and the right column lists the
DoFs of $V_{h}^{k,\mathrm{nc}}(K)$. The edge $e$ belongs to the set
$\mathcal{E}_{K}$ of all edges of $K$.

It can be shown that the triplets
$(K,V_{h}^{k,\mathrm{c}}(K),\\{(\mathbb{D}1)-(\mathbb{D}5)\\})$ and
$(K,V_{h}^{k,\mathrm{nc}}(K),\\{(\mathbb{D}1^{\star})-(\mathbb{D}4^{\star})\\})$
form a finite element in the sense of Ciarlet [23], and the projection
operator $\Pi^{\nabla^{2\\!}}_{k}v_{h}$ for $v_{h}\in V_{h}^{k,\mathrm{c}}(K)$
(resp. $v_{h}\in V_{h}^{k,\mathrm{c}}(K)$) is computable in terms of the DoFs
($\mathbb{D}$1)-($\mathbb{D}$5) (resp. ($\mathbb{D}$1⋆)-($\mathbb{D}$4⋆)). We
refer to [12] (resp. [16]) for a proof.

Let $\Pi_{k}$ denote the $L^{2}$-projection onto the polynomial space
$\mathbb{P}_{k}(K)$. That is,

$(\Pi_{k}v,\chi)_{K}=(v,\chi)_{K}\qquad\forall\;\chi\in\mathbb{P}_{k}(K).$

The orthogonality condition in the definition of the local VE spaces
$V_{h}^{k,\mathrm{c}}(K)$ and $V_{h}^{k,\mathrm{nc}}(K)$ implies that
$\Pi_{k}$ is also computable in terms of the DoFs.

For $v\in H^{1}(K)$ and $\vec{\chi}\in(\mathbb{P}_{k-1}(K))^{2}$, an
integration by parts leads to the expressions

(3.4) $\displaystyle(\Pi_{k-1}\nabla
v,\vec{\chi})_{K}=-(v,\mathop{\mathbf{div}}\nolimits\vec{\chi})_{K}+(v,\chi\cdot\boldsymbol{n}^{e}_{K})_{\partial
K}=-(\Pi_{k}v,\mathop{\mathbf{div}}\nolimits\vec{\chi})_{K}+(v,\chi\cdot\boldsymbol{n}^{e}_{K})_{\partial
K},$

owing to the definition of $\Pi_{k}$ in the last step. Observe that the DoFs
($\mathbb{D}$1)-($\mathbb{D}$2) and ($\mathbb{D}$4) determine
$v_{h}\in\mathbb{P}_{r}(e)$ explicitly for all $e\in\partial K$. This and the
computability of $\Pi_{k}$ imply that $\Pi_{k-1}\nabla v_{h}$ for $v_{h}\in
V_{h}^{k,\mathrm{c}}(K)$ is computable in terms of the DoFs. Since
$\Pi^{\nabla^{2\\!}}_{k}v_{h}$ is computable, the values
$\int_{e_{m_{j}}}v_{h}\chi\,\mbox{d}s$ for
$\chi\in\mathbb{M}_{k-2}(e_{m_{j}})$ are computable from the definition of
$V_{h}^{k,\mathrm{nc}}(K)$. If $n_{j}=0$, these $(k-1)$ estimates, and the
values at the vertices $V_{m_{j}}$ and $V_{m_{j}+1}$ uniquely determine
$v_{h}\in\mathbb{P}_{k}(e_{m_{j}})$. If $n_{j}>0$, the point values
$v_{h}(V_{m_{j}+i}),v_{h}(V_{m_{j}+i+1}),\partial_{\mathbf{\tau}}v_{h}(V_{m_{j}+i})$
and $\int_{e_{m_{j}+i}}v_{h}\chi\,\mbox{d}s$ for
$\chi\in\mathbb{M}_{k-3}(e_{m_{j}+i})$ evaluate $v_{h}$ on each edge
$e_{m_{j}+i}$ for $i=1,\dots,n_{j},j=1,\dots,\tilde{N}_{K}$, and consequently
$v_{h}$ is known on the boundary $\partial K$. Similarly as above, this step
and the computability of $\Pi_{k}$ imply that $\Pi_{k-1}\nabla v_{h}$ is
computable in terms of the DoFs for $v_{h}\in V_{h}^{k,\mathrm{nc}}(K)$.

######  Proposition 3.1 (Polynomial approximation [11]).

Under the assumption (M1), for every $v\in H^{s}(K)$, there exists
$\chi_{k}\in\mathbb{P}_{k}(K)$ with $k\in\mathbb{N}_{0}$ such that

$\displaystyle|v-\chi_{k}|_{m,K}\lesssim
h_{K}^{s-m}|v|_{s,K}\quad\text{for}\;0\leq m\leq s\leq k+1.$

The global VE spaces $V_{h}^{k,\mathrm{c}}$ and $V_{h}^{k,\mathrm{nc}}$ are
defined, respectively, as

$\displaystyle V_{h}^{k,\mathrm{c}}:=\\{v_{h}\in V:v_{h}|_{K}\in
V_{h}^{k,\mathrm{c}}(K)\quad\forall\;K\in\mathcal{T}_{h}\\},$

and

$\displaystyle V_{h}^{k,\mathrm{nc}}:=\begin{dcases}\begin{rcases}&v_{h}\in
L^{2}(\Omega):v_{h}|_{K}\in
V_{h}^{k,\mathrm{nc}}(K)\quad\forall\;K\in\mathcal{T}_{h},\;v_{h}\;\text{is
continuous at}\\\ &\text{interior vertices and zero at boundary
vertices},\;\int_{e}[\partial_{\boldsymbol{n}}v_{h}]\chi\,\mbox{d}s=0\\\
&\forall\;\chi\in\mathbb{P}_{k-2}(e),\;e\in\mathcal{E}^{i}\cup\mathcal{E}^{c}\;\text{and}\;\int_{e}[v_{h}]\chi\,\mbox{d}s=0\quad\forall\;\chi\in\mathbb{P}_{k-3}(e),\;e\in\mathcal{E}\end{rcases}.\end{dcases}$

VE spaces for pressure approximation. We define the projection operator
$\Pi^{\nabla}_{\ell}:H^{1}(K)\to\mathbb{P}_{\ell}(K)$ for $\ell\geq 1$ and
$q\in H^{1}(K)$ through the following equation

(3.5) $\displaystyle(\nabla\Pi^{\nabla}_{\ell}q,\nabla\chi_{\ell})_{K}$
$\displaystyle=(\nabla
q,\nabla\chi_{\ell})_{K}\qquad\forall\;\chi_{\ell}\in\mathbb{P}_{\ell}(K),$

with the additional condition needed to fix the constant

(3.6a) $\displaystyle\overline{\Pi^{\nabla}_{\ell}q}$
$\displaystyle=\overline{q}$ $\displaystyle\text{for conforming VEM},$ (3.6b)
$\displaystyle\int_{\partial K}\Pi^{\nabla}_{\ell}q\,\mbox{d}s$
$\displaystyle=\int_{\partial K}q\,\mbox{d}s$ $\displaystyle\text{for
nonconforming VEM}.$

This defines $\Pi^{\nabla}_{\ell}q$ uniquely for a given $q$. To approximate
the pressure space $Q$, we introduce the local conforming VE space
$Q_{h}^{\ell,\mathrm{c}}(K)$ for $\ell\geq 1$ and $K\in\mathcal{T}_{h}$ as the
set of solutions to a Poisson problem with Dirichlet boundary conditions [6].
In particular,

$Q_{h}^{\ell,\mathrm{c}}(K):=\begin{dcases}\begin{rcases}&q_{h}\in
H^{1}(K)\cap C^{0}(\partial K):\Delta
q_{h}\in\mathbb{P}_{\ell}(K),\;q_{h}|_{e}\in\mathbb{P}_{\ell}(e)\quad\forall\;e\in\partial
K,\\\
&\qquad\text{and}\;(q_{h}-\Pi^{\nabla}_{\ell}q_{h},\chi)_{K}=0\quad\forall\;\chi\in\mathbb{P}_{\ell}(K)\setminus\mathbb{P}_{\ell-2}(K)\end{rcases}.\end{dcases}$

In turn, the local nonconforming VE space $Q_{h}^{\ell,\mathrm{nc}}(K)$ is the
set of solutions to a Poisson problem with Neumann boundary condition [5] and
is defined for $\ell\geq 1$ as

$Q_{h}^{\ell,\mathrm{nc}}(K):=\begin{dcases}\begin{rcases}&q_{h}\in
H^{1}(K)\cap C^{0}(\partial K):\Delta
q_{h}\in\mathbb{P}_{\ell}(K),\;\partial_{\boldsymbol{n}}q_{h}|_{e}\in\mathbb{P}_{\ell-1}(e)\\\
&\quad\forall\;e\in\partial
K,\;\text{and}\;(q_{h}-\Pi^{\nabla}_{\ell}q_{h},\chi)_{K}=0\quad\forall\;\chi\in\mathbb{P}_{\ell}(K)\setminus\mathbb{P}_{\ell-2}(K)\end{rcases}.\end{dcases}$

The DoFs for $Q_{h}^{\ell,\mathrm{c}}(K)$ and $Q_{h}^{\ell,\mathrm{nc}}(K)$
are provided in Table 2.

degree | DoFs of $q_{h}\in Q_{h}^{\ell,\mathrm{c}}(K)$ | DoFs of $q_{h}\in Q_{h}^{\ell,\mathrm{nc}}(K)$ $\displaystyle{\int_{K}^{K}}$
---|---|---
$\ell\geq 1$ | ($\mathbb{F}$1) $q_{h}(V_{i})\quad\forall\;i=1,\dots,N_{K}$ | ($\mathbb{F}$1⋆) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{e}q_{h}\chi\,\mbox{d}s\quad\forall\chi\in\mathbb{M}_{\ell-1}(e)$$\displaystyle{\int_{K}^{K}}$
$\ell\geq 2$ | ($\mathbb{F}$2) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{e}q_{h}\chi\,\mbox{d}s\quad\forall\chi\in\mathbb{M}_{\ell-2}(e)$ | ($\mathbb{F}$2⋆) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{K}q_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\chi\in\mathbb{M}_{\ell-2}(K)$$\displaystyle{\int_{K}^{K}}$
| ($\mathbb{F}$3) $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{K}q_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\chi\in\mathbb{M}_{\ell-2}(K)$ |
Table 2. The left (resp. right) panel describes the DoFs of
$Q_{h}^{\ell,\mathrm{c}}(K)$ (resp. $Q_{h}^{\ell,\mathrm{nc}}(K)$). The edge
$e$ belongs to the set $\mathcal{E}_{K}$ of all edges of $K$.

The triplets
$(K,Q_{h}^{\ell,\mathrm{c}}(K),\\{(\mathbb{F}1)-(\mathbb{F}3)\\})$ and
$(K,Q_{h}^{\ell,\mathrm{nc}}(K),\\{(\mathbb{F}1^{\star})-(\mathbb{F}2^{\star})\\})$
form a finite element in the sense of Ciarlet [23] (see, e.g. [6]). Note that
$\Pi^{\nabla}_{\ell}q_{h}$ can be computed from DoFs of
($\mathbb{F}$1)-($\mathbb{F}$3) (resp. ($\mathbb{F}$1⋆)-($\mathbb{F}$2⋆)) for
$q_{h}\in Q_{h}^{\ell,\mathrm{c}}(K)$ (resp. $q_{h}\in
Q_{h}^{\ell,\mathrm{nc}}(K)$). Refer to [6] (resp. [15]) for a proof.
Consequently, the $L^{2}$-projection $\Pi_{\ell}$ is also computable from the
orthogonality condition in the definition of the spaces
$Q_{h}^{\ell,\mathrm{c}}(K)$ and $Q_{h}^{\ell,\mathrm{nc}}(K)$. This and the
explicit expression of $q_{h}$ on the boundary $\partial K$ in (3.4) show that
$\Pi_{\ell-1}\nabla q_{h}$ is computable for $q_{h}\in
Q_{h}^{\ell,\mathrm{c}}(K)$. The computability of $\Pi_{\ell}$ and
($\mathbb{F}$1⋆) in (3.4) imply that of $\Pi_{\ell-1}\nabla q_{h}$ for
$q_{h}\in Q_{h}^{\ell,\mathrm{nc}}(K)$.

Next we define the global VE spaces for conforming and nonconforming pressure
approximation, for $\ell\geq 1$, as

$Q_{h}^{\ell,\mathrm{c}}:=\\{q_{h}\in Q:q_{h}|_{K}\in
Q_{h}^{\ell}(K)\quad\forall\;K\in\mathcal{T}_{h}\\},$

and

$\displaystyle Q_{h}^{\ell,\mathrm{nc}}:=\begin{dcases}\begin{rcases}q_{h}\in
L^{2}(\Omega):&q_{h}|_{K}\in
Q_{h}^{\ell,\mathrm{nc}}(K)\quad\forall\;K\in\mathcal{T}_{h}\;\text{and}\\\
&\int_{e}[q_{h}]\chi\,\mbox{d}s=0\quad\forall\;\chi\in\mathbb{P}_{\ell-1}(e),\;\forall\;e\in\mathcal{E}^{i}\cup\mathcal{E}^{s}\end{rcases},\end{dcases}$

respectively.

###  3.2. Discrete problem and well-posedness

Let us first set the continuous bilinear forms $a_{1}:V\times V,a_{2}:Q\times
V$ and $a_{3}:Q\times Q$ as

$\displaystyle a_{1}(u,v)$ $\displaystyle:=(u,v)_{\Omega}+a(u,v)$
$\displaystyle\forall\;u,v\in V,$ $\displaystyle a_{2}(p,v)$
$\displaystyle:=\alpha(\nabla p,\nabla v)_{\Omega}$
$\displaystyle\forall\;p\in Q\;\text{and}\;\forall\;v\in V,$ $\displaystyle
a_{3}(p,q)$ $\displaystyle:=\beta(p,q)_{\Omega}+\gamma(\nabla p,\nabla
q)_{\Omega}$ $\displaystyle\forall\;p,q\in Q$

with the local counterparts $a_{1}^{K},a_{2}^{K}$ and $a_{3}^{K}$ for
$K\in\mathcal{T}_{h}$ and the piecewise versions
$a_{1}^{\mathrm{pw}}:=\sum_{K}a_{1}^{K},a_{2}^{\mathrm{pw}}:=\sum_{K}a_{2}^{K}$
and $a_{3}^{\mathrm{pw}}:=\sum_{K}a_{3}^{K}$ respectively. For all
$u_{h},v_{h}\in V_{h}^{k,\mathrm{c}}(K)$ or $V_{h}^{k,\mathrm{nc}}(K)$ and
$p_{h},q_{h}\in Q_{h}^{\ell,\mathrm{c}}(K)$ or $Q_{h}^{\ell,\mathrm{nc}}(K)$
with $k\geq 2$ and $\ell\geq 1$, define the discrete counterparts by

$\displaystyle a_{1}^{h}(u_{h},v_{h})|_{K}$
$\displaystyle:=(\Pi_{k}u_{h},\Pi_{k}v_{h})_{K}+S_{1,0}^{K}((1-\Pi_{k})u_{h},(1-\Pi_{k})v_{h})$
(3.7a)
$\displaystyle\quad+(\Pi_{k-2}(\nabla^{2}u_{h}),\Pi_{k-2}(\nabla^{2}v_{h}))_{K}+S_{\nabla^{2}}^{K}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})v_{h}),$
(3.7b) $\displaystyle a_{2}^{h}(p_{h},v_{h})|_{K}$
$\displaystyle:=\alpha(\Pi_{\ell-1}\nabla p_{h},\Pi_{k-1}\nabla v_{h})_{K},$
$\displaystyle a_{3}^{h}(p_{h},q_{h})|_{K}$
$\displaystyle:=\beta(\Pi_{\ell}p_{h},\Pi_{\ell}q_{h})_{K}+S_{2,0}^{K}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})q_{h})$
(3.7c) $\displaystyle\quad+\gamma(\Pi_{\ell-1}(\nabla
p_{h}),\Pi_{\ell-1}(\nabla
q_{h}))_{K}+S_{\nabla}^{K}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})q_{h}).$

The stabilisation terms $S_{\nabla^{2}}^{K}$ and $S_{1,0}^{K}$ on
$V_{h}^{k,\mathrm{c}}(K)$ or $V_{h}^{k,\mathrm{nc}}$, and $S_{\nabla}^{K}$ and
$S_{2,0}^{K}$ on $Q_{h}^{\ell,\mathrm{c}}(K)$ or $Q_{h}^{\ell,\mathrm{nc}}(K)$
are positive definite bilinear forms and there exist positive constants
$C_{\nabla^{2}},C_{1,0},C_{\nabla},C_{2,0}$ such that

(3.8a) $\displaystyle C_{\nabla^{2}}^{-1}|v_{h}|_{2,K}^{2}$ $\displaystyle\leq
S_{\nabla^{2}}^{K}(v_{h},v_{h})\leq C_{\nabla^{2}}|v_{h}|_{2,K}^{2}$
$\displaystyle\forall\;v_{h}\in\text{Ker$(\Pi^{\nabla^{2\\!}}_{k})$},$ (3.8b)
$\displaystyle C_{1,0}^{-1}\|v_{h}\|_{K}^{2}$ $\displaystyle\leq
S_{1,0}^{K}(v_{h},v_{h})\leq C_{1,0}\|v_{h}\|_{K}^{2}$
$\displaystyle\forall\;v_{h}\in\text{Ker$(\Pi_{k})$},$ (3.8c) $\displaystyle
C_{\nabla}^{-1}\gamma|q_{h}|_{1,K}^{2}$ $\displaystyle\leq
S_{\nabla}^{K}(q_{h},q_{h})\leq C_{\nabla}\gamma|q_{h}|_{1,K}^{2}$
$\displaystyle\forall\;q_{h}\in\text{Ker$(\Pi^{\nabla}_{\ell})$},$ (3.8d)
$\displaystyle C_{2,\nabla}^{-1}\beta\|q_{h}\|_{K}^{2}$ $\displaystyle\leq
S_{2,0}^{K}(q_{h},q_{h})\leq C_{2,0}\beta\|q_{h}\|_{K}^{2}$
$\displaystyle\forall\;q_{h}\in\text{Ker$(\Pi_{\ell})$}.$

Let $\text{dof}_{i}$ denote the $i^{\text{th}}$ degree of freedom. Standard
examples for stabilisation terms satisfying (3.8a)-(3.8d) respectively are

$\displaystyle
S_{\nabla^{2}}^{K}(v_{h},w_{h})=h_{K}^{-2}\sum_{i}\text{dof}_{i}(v_{h})\text{dof}_{i}(w_{h}),\quad
S_{1,0}^{K}(v_{h},w_{h})=h_{K}^{4}S_{\nabla^{2}}^{K}(v_{h},w_{h}),$
$\displaystyle
S_{\nabla}^{K}(p_{h},q_{h})=\gamma\sum_{j}\text{dof}_{j}(p_{h})\text{dof}_{j}(q_{h}),\quad
S_{2,0}^{K}(p_{h},q_{h})=\beta
h_{K}^{2}\sum_{j}\text{dof}_{j}(p_{h})\text{dof}_{j}(q_{h}),$

for all $v_{h},w_{h}\in
V_{h}^{k,\mathrm{c}}(K)\;\text{or}\;V_{h}^{k,\mathrm{nc}}(K)$ and
$p_{h},q_{h}\in
Q_{h}^{\ell,\mathrm{c}}(K)\;\text{or}\;Q_{h}^{\ell,\mathrm{nc}}(K)$. The
global discrete bilinear forms $a_{1}^{h}:V_{h}^{k,\mathrm{c}}\times
V_{h}^{k,\mathrm{c}}$ (resp. $V_{h}^{k,\mathrm{nc}}\times
V_{h}^{k,\mathrm{nc}}$), $a_{2}^{h}:Q_{h}^{\ell,\mathrm{c}}\times
V_{h}^{k,\mathrm{c}}$ (resp. $Q_{h}^{\ell,\mathrm{nc}}\times
V_{h}^{k,\mathrm{nc}}$) and $a_{3}^{h}:Q_{h}^{\ell,\mathrm{c}}\times
Q_{h}^{\ell,\mathrm{c}}$ (resp. $Q_{h}^{\ell,\mathrm{nc}}\times
Q_{h}^{\ell,\mathrm{nc}}$) are defined by
$a_{1}^{h}(\cdot,\cdot):=\sum_{K\in\mathcal{T}_{h}}a_{1}^{h}(\cdot,\cdot)|_{K},a_{2}^{h}(\cdot,\cdot):=\sum_{K\in\mathcal{T}_{h}}a_{2}^{h}(\cdot,\cdot)|_{K}$
and
$a_{3}^{h}(\cdot,\cdot):=\sum_{K\in\mathcal{T}_{h}}a_{3}^{h}(\cdot,\cdot)|_{K}$
for conforming (resp. nonconforming) VEM. We assume that $\ell\leq k$, and
then the discrete problem is to find $(u_{h},p_{h})\in
V_{h}^{k,\mathrm{c}}\times Q_{h}^{\ell,\mathrm{c}}$ (resp.
$V_{h}^{k,\mathrm{nc}}\times Q_{h}^{\ell,\mathrm{nc}}$) such that

(3.9a) $\displaystyle a_{1}^{h}(u_{h},v_{h})-a_{2}^{h}(p_{h},v_{h})$
$\displaystyle=(\tilde{f}_{h},v_{h})_{\Omega}$ $\displaystyle\forall\;v_{h}\in
V_{h}^{k,\mathrm{c}}\;(\text{resp.}\;V_{h}^{k,\mathrm{nc}}),$ (3.9b)
$\displaystyle a_{2}^{h}(q_{h},u_{h})+a_{3}^{h}(p_{h},q_{h})$
$\displaystyle=(\tilde{g}_{h},q_{h})_{\Omega}$ $\displaystyle\forall\;q_{h}\in
Q_{h}^{\ell,\mathrm{c}}\;(\text{resp.}\;Q_{h}^{\ell,\mathrm{nc}}),$

with the discrete right-hand sides
$(\tilde{f}_{h},v_{h})_{\Omega}:=(\tilde{f},\Pi_{k}v_{h})_{\Omega}$ and
$(\tilde{g}_{h},q_{h})_{\Omega}:=(\tilde{g},\Pi_{\ell}q_{h})_{\Omega}$. To
rewrite the above discrete problem, define the discrete product space
$\mathbf{H}_{\epsilon}^{h,\mathrm{c}}:=V_{h}^{k,\mathrm{c}}\times
Q_{h}^{\ell,\mathrm{c}}$ and the discrete operator
$\mathcal{A}_{h}^{c}:\mathbf{H}_{\epsilon}^{h,\mathrm{c}}\to\mathbf{H}_{\epsilon}^{h,\mathrm{c}}$
as

(3.10)
$\displaystyle\langle\mathcal{A}_{h}^{c}(\vec{\boldsymbol{u}}_{h}),\vec{\boldsymbol{v}}_{h}\rangle:=a_{1}^{h}(u_{h},v_{h})-a_{2}^{h}(p_{h},v_{h})+a_{2}^{h}(q_{h},u_{h})+a_{3}^{h}(p_{h},q_{h})$

for
$\vec{\boldsymbol{u}}_{h}=(u_{h},p_{h}),\vec{\boldsymbol{v}}_{h}=(v_{h},q_{h})\in\mathbf{H}_{\epsilon}^{h,\mathrm{c}}$.
We also define the linear and bounded functional
$\mathcal{F}_{h}^{c}:\mathbf{H}_{\epsilon}^{h,\mathrm{c}}\to\mathbb{R}$ as

$\vec{\boldsymbol{v}}_{h}\mapsto\mathcal{F}_{h}^{c}(\vec{\boldsymbol{v}}_{h}):=(\tilde{f}_{h},v_{h})_{\Omega}+(\tilde{g}_{h},q_{h})_{\Omega},$

and therefore problem (3.9) is recast as: Find
$\vec{\boldsymbol{u}}_{h}^{c}\in\mathbf{H}_{\epsilon}^{h,\mathrm{c}}$ such
that

(3.11)
$\langle\mathcal{A}_{h}^{c}(\vec{\boldsymbol{u}}_{h}^{c}),\vec{\boldsymbol{v}}_{h}\rangle=\mathcal{F}_{h}^{c}(\vec{\boldsymbol{v}}_{h})\qquad\forall\;\vec{\boldsymbol{v}}_{h}\in\mathbf{H}_{\epsilon}^{h,\mathrm{c}}.$

Similarly we define
$\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}:=V_{h}^{k,\mathrm{nc}}\times
Q_{h}^{\ell,\mathrm{nc}}$, the discrete operators
$\mathcal{A}_{h}^{\mathrm{nc}}$ and $\mathcal{F}_{h}^{\mathrm{nc}}$, and seek
$\vec{\boldsymbol{u}}_{h}^{\mathrm{nc}}\in\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}$
such that

(3.12)
$\langle\mathcal{A}_{h}^{\mathrm{nc}}(\vec{\boldsymbol{u}}_{h}),\vec{\boldsymbol{v}}_{h}\rangle=\mathcal{F}_{h}^{\mathrm{nc}}(\vec{\boldsymbol{v}}_{h})\qquad\forall\;\vec{\boldsymbol{v}}_{h}\in\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}.$

Define the piecewise version $\|\cdot\|_{\mathbf{H}_{\epsilon}^{h}}$ of the
norm $\|\cdot\|_{\mathbf{H}_{\epsilon}}$ for $\vec{\boldsymbol{u}}=(u,p)\in
H^{2}(\mathcal{T}_{h})\times H^{1}(\mathcal{T}_{h})$ as

$\|\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}^{2}:=\|u\|_{\Omega}^{2}+|u|_{2,h}^{2}+\beta\|{p}\|_{\Omega}^{2}+\gamma|{p}|_{1,h}^{2}:=\sum_{K\in\mathcal{T}_{h}}(\|u\|_{K}^{2}+|u|_{2,K}^{2}+\beta\|{p}\|_{K}^{2}+\gamma|{p}|_{1,K}^{2}).$

The following result yields the solvability of the discrete problems.

######  Theorem 3.1.

Problem (3.11) (resp. (3.12)) is well-posed in the space
$\mathbf{H}_{\epsilon}^{h,c}$ (resp. $\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}$)
equipped with the norm (2.4) (resp. $\|\cdot\|_{\mathbf{H}_{\epsilon}^{h}}$).

###### Proof.

The boundedness of $\mathcal{A}_{h}^{c}$ and $\mathcal{A}_{h}^{\mathrm{nc}}$
clearly follows from the stability of the $L^{2}$-projection operators
$\Pi_{k-2},\Pi_{k},\Pi_{\ell-1},$ and $\Pi_{\ell}$ for $k\geq 2$ and $\ell\geq
1$, and from (3.8a)-(3.8d). For
$\vec{\boldsymbol{v}}_{h}\in\mathbf{H}_{\epsilon}^{h,c}$ or
$\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}$, the definition (3.10) implies
$\langle\mathcal{A}_{h}(\vec{\boldsymbol{v}}_{h}),\vec{\boldsymbol{v}}_{h}\rangle=a_{1}^{h}(v_{h},v_{h})+a_{3}^{h}(q_{h},q_{h})$.
The definition (3.7a) of $a_{1}^{h}$ and the lower bounds of stabilisation
terms (3.8a)-(3.8b) lead to

$\displaystyle a_{1}^{h}(v_{h},v_{h})$
$\displaystyle\gtrsim\|\Pi_{k}v_{h}\|_{\Omega}^{2}+\|(1-\Pi_{k})v_{h}\|_{\Omega}^{2}+\|\Pi_{k-2}(\nabla^{2}v_{h})\|_{\Omega}^{2}+|(1-\Pi^{\nabla^{2\\!}}_{k})v_{h}|_{2,h}^{2}$
$\displaystyle\gtrsim\|v_{h}\|_{\Omega}^{2}+|v_{h}|_{2,\Omega}^{2},$

where we have employed
$\|(1-\Pi_{k-2})(\nabla^{2}v_{h})\|_{\Omega}\leq|(1-\Pi^{\nabla^{2\\!}}_{k})v_{h}|_{2,h}$
and triangle inequalities in the last step. Analogously we can prove that
$a_{3}^{h}$ is coercive, and consequently $\mathcal{A}_{h}^{c}$ (also
$\mathcal{A}_{h}^{\mathrm{nc}}$) is coercive with respect to the weighted norm
$\|\cdot\|_{\mathbf{H}_{\epsilon}^{h}}$. Hence the Lax–Milgram lemma concludes
the proof. ∎

##  4\. Error analysis for conforming VEM

This section recalls the standard conforming interpolation estimates and
establishes the a priori error estimates in the energy norm
$\|\cdot\|_{\mathbf{H}_{\epsilon}}$ (cf. Theorem 4.1).

######  Proposition 4.1 (Conforming interpolation [13, 21]).

There exists an interpolation operator $\vec{I}_{h}^{c}:(V\cap
H^{s}(\Omega))\times(Q\cap H^{r}(\Omega))\to V_{h}^{k,\mathrm{c}}\times
Q_{h}^{\ell,\mathrm{c}}$ such that, for $v\in V\cap H^{s}(\Omega)$ with $2\leq
s\leq k+1$ and $q\in Q\cap H^{r}(\Omega)$ with $1\leq r\leq\ell+1$,
$\vec{I}_{h}^{c}v:=(v_{I}^{c},q_{I}^{c})$ and

$|v-v_{I}^{c}|_{j,h}\lesssim h^{s-j}|v|_{s,\Omega}\quad\text{for}\;0\leq
j\leq{2}\quad\text{and}\quad|q-q_{I}^{c}|_{j,h}\lesssim
h^{r-j}|q|_{r,\Omega}\quad\text{for}\;0\leq j\leq{1}.$

Throughout this paper, the oscillations of $\tilde{f},\tilde{g}\in
L^{2}(\Omega)$ for $k\geq 2$ and $\ell\geq 1$ are defined as

$\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h}):=\Big{(}\sum_{K\in\mathcal{T}_{h}}\|h_{K}^{2}(1-\Pi_{k})\tilde{f}\|_{K}^{2}\Big{)}^{1/2},\;\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h}):=\Big{(}\sum_{K\in\mathcal{T}_{h}}\|h_{K}(1-\Pi_{\ell})\tilde{g}\|_{K}^{2}\Big{)}^{1/2}.$

######  Theorem 4.1.

Given $\vec{\boldsymbol{u}}:=(u,p)\in(V\cap H^{s}(\Omega))\times(Q\cap
H^{r}(\Omega))$ for $s\geq 2$ and $r\geq 1$, the unique solution
$\vec{\boldsymbol{u}}_{h}^{c}=(u_{h}^{c},p_{h}^{c})\in\mathbf{H}_{\epsilon}^{h,c}=V_{h}^{k,\mathrm{c}}\times
Q_{h}^{\ell,\mathrm{c}}$ for $k\geq 2$ and $1\leq\ell\leq k$ to (3.11)
satisfies

$\displaystyle\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}^{c}\|_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\|\vec{\boldsymbol{u}}-\vec{I}_{h}^{c}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\alpha^{1/2}|u-\Pi^{\nabla}_{\ell}u|_{1,h}+\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h})$
$\displaystyle\quad+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h})\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(\|\tilde{f}\|_{s-4,\Omega}+\|\tilde{g}\|_{r-2,\Omega}),$

for
$\vec{\Pi}_{h}\vec{\boldsymbol{u}}=(\Pi^{\nabla^{2\\!}}_{k}u,\Pi^{\nabla}_{\ell}p)$.

###### Proof.

We drop the superscript $c$ (denoting the conforming case) in the proof just
for the sake of notational simplicity. Let
$\vec{\boldsymbol{e}}_{h}:=(e_{h}^{u},e_{h}^{p})=(u_{I}-u_{h},p_{I}-p_{h})=\vec{\boldsymbol{u}}_{I}-\vec{\boldsymbol{u}}_{h}\in\mathbf{H}_{\epsilon}^{h,c}$
for $\vec{\boldsymbol{u}}_{I}=(u_{I},p_{I})$. The coercivity of
$\mathcal{A}_{h}$ from Theorem 3.1 and the discrete problem (3.11) in the
first step, and an elementary algebra in the second step lead to

$\displaystyle\|\vec{\boldsymbol{e}}_{h}\|^{2}_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\mathcal{A}_{h}(\vec{\boldsymbol{u}}_{I},\vec{\boldsymbol{e}}_{h})-\mathcal{F}_{h}(\vec{\boldsymbol{e}}_{h})=(a_{1}^{h}(u_{I}-\Pi^{\nabla^{2\\!}}_{k}u,e_{h}^{u})+a_{1}^{\mathrm{pw}}(\Pi^{\nabla^{2\\!}}_{k}u-u,e_{h}^{u}))$
$\displaystyle\quad+(a_{2}(p,e_{h}^{u})-a_{2}^{h}(p_{I},e_{h}^{u}))+(a_{3}^{h}(p_{I}-\Pi^{\nabla}_{\ell}p,e_{h}^{p})+a_{3}^{\mathrm{pw}}(\Pi^{\nabla}_{\ell}p-p,e_{h}^{p}))$
$\displaystyle\quad+(a_{2}^{h}(e_{h}^{p},u_{I})-a_{2}(e_{h}^{p},u))+((\tilde{f}-\tilde{f}_{h},e_{h}^{u})_{\Omega}+(\tilde{g}-\tilde{g}_{h},e_{h}^{p})_{\Omega})$
(4.1) $\displaystyle\quad=:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}.$

The continuity of $a_{1}^{h}$ and $a_{3}^{h}$ from Theorem 3.1, and the
Cauchy–Schwarz inequality for $a_{1}^{\mathrm{pw}}$ and $a_{3}^{\mathrm{pw}}$
show

$\displaystyle T_{1}+T_{3}$
$\displaystyle\lesssim(\|u_{I}-\Pi^{\nabla^{2\\!}}_{k}u\|_{\Omega}+\|u-\Pi^{\nabla^{2\\!}}_{k}u\|_{\Omega})\|e_{h}^{u}\|_{\Omega}+(|u_{I}-\Pi^{\nabla^{2\\!}}_{k}u|_{2,h}$
$\displaystyle\qquad+|u-\Pi^{\nabla^{2\\!}}_{k}u|_{2,h})|e_{h}^{u}|_{2,\Omega}+\beta^{1/2}(\|p_{I}-\Pi^{\nabla}_{\ell}p\|_{\Omega}+\|p-\Pi^{\nabla}_{\ell}p\|_{\Omega})\beta^{1/2}\|e_{h}^{p}\|_{\Omega}$
$\displaystyle\qquad+\gamma^{1/2}(\|p_{I}-\Pi^{\nabla}_{\ell}p\|_{1,h}+\|p-\Pi^{\nabla}_{\ell}p\|_{1,h})\gamma^{1/2}|e_{h}^{p}|_{1,h}$
$\displaystyle\lesssim(\|\vec{\boldsymbol{u}}-\vec{I}_{h}^{c}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}})\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}}$
(4.2) $\displaystyle\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega})\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}},$

with triangle inequalities in the second step, and Propositions 3.1-4.1 in the
last step. Algebraic manipulations and the $L^{2}$-orthogonality of
$\Pi_{\ell-1}$ imply that

$\displaystyle\alpha^{-1}(T_{2}+T_{4})$ $\displaystyle=(\nabla
p-\Pi_{\ell-1}\nabla p_{I},\nabla e_{h}^{u})_{\Omega}+(\Pi_{\ell-1}\nabla
p_{I},(1-\Pi_{k-1})\nabla e_{h}^{u})_{\Omega}$ (4.3)
$\displaystyle\quad+(\Pi_{\ell-1}\nabla e_{h}^{p},\Pi_{k-1}\nabla u_{I}-\nabla
u)_{\Omega}+((\Pi_{\ell-1}-1)\nabla e_{h}^{p},(1-\Pi_{\ell-1})\nabla
u)_{\Omega}.$

In addition, triangle inequalities and the $L^{2}$-orthogonality of
$\Pi_{\ell-1}$ provide

$\|\nabla p-\Pi_{\ell-1}\nabla
p_{I}\|_{\Omega}\leq|p-p_{I}|_{1,\Omega}+|p_{I}-\Pi^{\nabla}_{\ell}p|_{1,h}\lesssim|p-p_{I}|_{1,\Omega}+|p-\Pi^{\nabla}_{\ell}p|_{1,h}.$

The second term in (4.3) vanishes because of the $L^{2}$-orthogonality of
$\Pi_{k-1}$ and assumption $\ell\leq k$. Similarly, the third term in (4.3)
reduces to $(\Pi_{\ell-1}\nabla e_{h}^{p},\Pi_{k-1}\nabla u_{I}-\nabla
u)_{\Omega}=(\Pi_{\ell-1}\nabla e_{h}^{p},\nabla u_{I}-\nabla u)_{\Omega}$.
The Cauchy–Schwarz inequality in combination with the previous bounds in (4.3)
and $\alpha^{1/2}\leq 1\leq\gamma^{1/2}$ result in

$\displaystyle T_{2}+T_{4}$
$\displaystyle\lesssim\gamma^{1/2}(|p-p_{I}|_{1,\Omega}+|p-\Pi^{\nabla}_{\ell}p|_{1,h})|e_{h}^{u}|_{1,\Omega}$
$\displaystyle\quad+\alpha^{1/2}(|u-u_{I}|_{1,\Omega}+|u-\Pi^{\nabla}_{\ell}u|_{1,h})\gamma^{1/2}|e_{h}^{p}|_{1,\Omega}$
$\displaystyle\lesssim(\|\vec{\boldsymbol{u}}-\vec{I}_{h}^{c}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\alpha^{1/2}|u-\Pi^{\nabla}_{\ell}u|_{1,h})(|e_{h}^{u}|_{2,\Omega}+\gamma^{1/2}|e_{h}^{p}|_{1,\Omega})$
$\displaystyle\lesssim
h^{\min\\{\ell,r-1,k,s-1\\}}(|e_{h}^{u}|_{2,\Omega}+\gamma^{1/2}|e_{h}^{p}|_{1,\Omega}),$

where $|u-u_{I}|_{1,\Omega}\lesssim|u-u_{I}|_{2,\Omega}$ and
$|e_{h}^{u}|_{1,\Omega}\lesssim|e_{h}^{u}|_{2,\Omega}$ from (2.5), and
Propositions 3.1-4.1 were used for the last two inequalities. The
$L^{2}$-orthogonality of $\Pi_{k}$ and $\Pi_{\ell}$ together with Proposition
3.1 and $\gamma\geq 1$ allow us to assert that

$\displaystyle T_{5}$
$\displaystyle=(h_{\mathcal{T}_{h}}^{2}(\tilde{f}-\Pi_{k}\tilde{f}),h_{\mathcal{T}_{h}}^{-2}(1-\Pi_{k})e_{h}^{u})_{\Omega}+(h_{\mathcal{T}_{h}}(\tilde{g}-\Pi_{\ell}\tilde{g}),h_{\mathcal{T}_{h}}^{-1}(1-\Pi_{\ell})e_{h}^{p})_{\Omega}$
$\displaystyle\lesssim\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h})|e_{h}^{u}|_{2,\Omega}+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h})|e_{h}^{p}|_{1,\Omega}$
(4.4) $\displaystyle\lesssim
h^{\min\\{k+3,s-2,\ell+2,r-1\\}}(\|\tilde{f}\|_{s-4,\Omega}+\|\tilde{g}\|_{r-2,\Omega})(|e_{h}^{u}|_{2,\Omega}+\gamma^{1/2}|e_{h}^{p}|_{1,\Omega}).$

The estimates (4.2)-(4.4) in (4.1) show that

$\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}}\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega}+\|\tilde{f}\|_{s-4,\Omega}+\|\tilde{g}\|_{r-2,\Omega}).$

This and Proposition 4.1 in the triangle inequality
$\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}^{c}\|_{\mathbf{H}_{\epsilon}}\leq\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{I}\|_{\mathbf{H}_{\epsilon}}+\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}}$
followed by the regularity estimates conclude the proof of the theorem. ∎

##  5\. Error analysis for nonconforming VEM

Since the nonconforming discrete spaces $V_{h}^{k,\mathrm{nc}}$ and
$Q_{h}^{\ell,\mathrm{nc}}$ need not be subsets of continuous spaces $V$ and
$Q$, this section explains the different constructions (at least two) of
conforming companion operators which connect nonconforming VE spaces to
continuous Sobolev spaces. The two crucial ideas in the design are

  * •

first to map a nonconforming VE space to a conforming VE space of one degree
higher, and

  * •

second to modify the linear operator constructed in the first step through
standard bubble-function techniques to achieve additional orthogonal
properties (in particular, $L^{2}$-orthogonality).

###  5.1. Construction of companion operators

Let $\mathrm{dof}_{i}^{\ell,\mathrm{c}}$ for
$i=1,\dots,\mathrm{N}_{1}^{\ell,\mathrm{c}}$ and
$\mathrm{dof}_{j}^{\ell,\mathrm{nc}}$ for
$j=1,\dots,\mathrm{N}_{1}^{\ell,\mathrm{nc}}$ be the linear functionals
associated with DoFs of the VE spaces $Q_{h}^{\ell,\mathrm{c}}$ and
$Q_{h}^{\ell,\mathrm{nc}}$ of dimensions $\mathrm{N}_{1}^{\ell,\mathrm{c}}$
and $\mathrm{N}_{1}^{\ell,\mathrm{nc}}$ for $\ell\geq 1$. Let
$\mathrm{dof}_{i}^{k,\mathrm{c}}$ for
$i=1,\dots,\mathrm{N}_{2}^{k,\mathrm{c}}$ and $\mathrm{dof}_{j}^{k,\rm{nc}}$
for $j=1,\dots,\mathrm{N}_{2}^{k,\mathrm{nc}}$ be the linear functionals
associated with DoFs of the VE spaces $V_{h}^{k,\mathrm{c}}$ and
$V_{h}^{k,\mathrm{nc}}$ of dimensions $\mathrm{N}_{2}^{k,\mathrm{c}}$ and
$\mathrm{N}_{2}^{k,\mathrm{nc}}$ for $k\geq 2$.

######  Theorem 5.1.

There exists a linear operator $J_{1}:V_{h}^{k,\mathrm{nc}}\to
V_{h}^{k+1,\mathrm{c}}$ satisfying the following properties:

  1. (a)

$\mathrm{dof}_{j}^{k,\mathrm{nc}}(J_{1}v_{h})=\mathrm{dof}_{j}^{k,\mathrm{nc}}(v_{h})$
for all $j=1,\dots,{\mathrm{N}_{2}^{k,\mathrm{nc}}}$,

  2. (b)

$a^{\mathrm{pw}}(v_{h}-J_{1}v_{h},\chi)=0$ for all
$\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})$,

  3. (c)

$\nabla(v_{h}-J_{1}v_{h})\perp(\mathbb{P}_{k-3}(\mathcal{T}_{h}))^{2}$ in
$(L^{2}(\Omega))^{2}$ for $k\geq 3$,

  4. (d)

$\displaystyle\sum_{j=0}^{2}h^{j-2}|v_{h}-J_{1}v_{h}|_{j,h}\lesssim\inf_{\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})}|v_{h}-\chi|_{2,h}+\inf_{v\in
V}|v_{h}-v|_{2,h}.$

Construction of $J_{1}$. First we observe that DoFs of $V_{h}^{k,\mathrm{nc}}$
is a subset of DoFs of $V_{h}^{k+1,c}$. Next we define a linear operator
$J_{1}:V_{h}^{k,\mathrm{nc}}\to V_{h}^{k+1,c}$ through DoFs of
$V_{h}^{k+1,c}$, for $v_{h}\in V_{h}^{k,\mathrm{nc}}$, by

$\displaystyle\mathrm{dof}_{j}^{k,\mathrm{nc}}(J_{1}v_{h})$
$\displaystyle=\mathrm{dof}_{j}^{k,\mathrm{nc}}(v_{h})\quad\forall\;j=1,\dots,\mathrm{N}_{2}^{k,\mathrm{nc}},$
$\displaystyle\nabla J_{1}v_{h}(z)$
$\displaystyle=\frac{1}{|\mathcal{T}_{z}|}\sum_{K\in\mathcal{T}_{z}}\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{K}(z)\quad\forall\;z\in\mathcal{V}^{i},$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}J_{1}v_{h}\chi\,\mbox{d}\boldsymbol{x}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}\Pi^{\nabla^{2\\!}}_{k}v_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\;\chi\in\mathbb{M}^{*}_{k-3}(K),$

where the set $\mathcal{T}_{z}:=\\{K\in\mathcal{T}_{h}:z\in K\\}$ of
cardinality $|\mathcal{T}_{z}|$ contains the neighbouring polygons $K$ sharing
the vertex $z$. We assign $\nabla J_{1}v_{h}(z)=\boldsymbol{0}$ for the
boundary vertices $z\in\mathcal{V}^{s}$ if $z$ is a corner (the angle at $z$
is not equal to $0,\pi,2\pi$) and for all $z\in\mathcal{V}^{c}$. If the angle
at $z\in\mathcal{V}^{s}$ is equal to $0,\pi,2\pi$, then we assign

$\partial_{\boldsymbol{t}}(J_{1}v_{h})(z)=0\quad\text{and}\quad\partial_{\boldsymbol{n}}(J_{1}v_{h})(z)=\frac{1}{|\mathcal{T}_{z}|}\sum_{K\in\mathcal{T}_{z}}\partial_{\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}v_{h})|_{K}(z).$

###### Proof of Theorem 5.1(a).

This is an immediate consequence of the definition of $J_{1}$. ∎

###### Proof of Theorem 5.1(b).

Let $\chi\in\mathbb{P}_{k}(K)$ and set the notation
$T(\chi):=\partial_{\boldsymbol{n}}(\Delta\chi+\partial_{\boldsymbol{\tau}\boldsymbol{\tau}}\chi)$,
and
$[M_{\boldsymbol{n}\boldsymbol{\tau}}(\chi)]_{z_{j}}:=\partial_{\boldsymbol{n}\tau}(\chi)|_{e_{j-1}}(z_{j})-\partial_{\boldsymbol{n}\tau}(\chi)|_{e_{j}}(z_{j})$
for $j=1,\dots,N_{K}$ with $e_{0}=e_{N_{K}}$. Since $\chi\in H^{4}(K)$ and
$v_{h}-J_{1}v_{h}\in H^{2}(K)$, an integration by parts leads to

$\displaystyle a^{K}(v_{h}-J_{1}v_{h},\chi)$
$\displaystyle=\int_{K}\Delta^{2}\chi(v_{h}-J_{1}v_{h})\,\mbox{d}\boldsymbol{x}+\int_{\partial
K}\partial_{\boldsymbol{n}\boldsymbol{n}}\chi\partial_{\boldsymbol{n}}(v_{h}-J_{1}v_{h})\,\mbox{d}s$
$\displaystyle\qquad-\int_{\partial
K}T(\chi)(v_{h}-J_{1}v_{h})\,\mbox{d}s+\sum_{j=1}^{N_{K}}[M_{\boldsymbol{n}\boldsymbol{\tau}}\chi]_{z_{j}}(v_{h}-J_{1}v_{h})(z_{j})=0,$

with part 5.1(a) being used in the last step. This holds for any
$K\in\mathcal{T}_{h}$ and concludes the proof of Theorem 5.1(b). ∎

###### Proof of Theorem 5.1(c).

For any $v_{h}\in V_{h}^{k,\mathrm{nc}},\chi\in\mathbb{P}_{k-2}(K)$ with
$k\geq 3$ and $K\in\mathcal{T}_{h}$, an integration by parts and Theorem
5.1(a) show that

$(\nabla(v_{h}-J_{1}v_{h}),\nabla\chi)_{K}=-(v_{h}-J_{1}v_{h},\Delta\chi)_{K}+(v_{h}-J_{1}v_{h},\partial_{\boldsymbol{n}}\chi)_{\partial
K}=0.$

This proves 5.1(c). ∎

###### Proof of Theorem 5.1(d).

Since $(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})|_{K}\in
V_{h}^{k+1,\rm{c}}(K)$, the norm equivalence found in, e.g., [30, Lemma 3.6]
shows that

(5.1) $\displaystyle|\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h}|_{2,K}\simeq
h_{K}^{-1}\|\mathrm{Dof}^{k+1,\mathrm{c}}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})\|_{\ell^{2}},$

for the vector $\mathrm{Dof}^{k+1,\mathrm{c}}$ with arguments as the local
DoFs of $V_{h}^{k+1,\rm{c}}(K)$. Let $z$ be an interior vertex in
$\mathcal{V}^{i}\cap\mathcal{V}_{K}$ belonging to an edge
$e\in\mathcal{E}_{K}$. The equality $J_{1}v_{h}(z)=v_{h}(z)$ from Theorem
5.1(a) and the inverse estimate for polynomials imply

$\displaystyle|(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})|_{K}(z)|\leq\|(\Pi^{\nabla^{2\\!}}_{k}v_{h}-v_{h})\|_{\infty,e}\lesssim
h_{e}^{-1/2}\|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}\|_{e}$
$\displaystyle\qquad\lesssim
h_{e}^{-1}\|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}\|_{K}+|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{1,K}\lesssim
h_{K}|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,K}.$

The third step follows from the trace inequality, and the last step from (3.3)
and (M2). Let $z$ be an interior vertex in
$\mathcal{V}^{i}\cap\mathcal{V}_{K}$ or a boundary vertex in
$\mathcal{V}^{s}\cap\mathcal{V}_{K}$ with angle at $z$ equal to $\pi$, and
polygons $K_{1}=K,\dots,K_{|\mathcal{T}_{z}|}$ share the node $z$. Suppose
$(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{i}=\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{K_{i}},$
and $K_{i}$ and $K_{i+1}$ are two neighbouring polygons. Then

$\displaystyle\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})_{1}(z)$
$\displaystyle=\frac{1}{|\mathcal{T}_{z}|}\sum_{j=2}^{|\mathcal{T}_{z}|}(\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{1}-\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{j})(z)$
(5.2)
$\displaystyle=\frac{1}{|\mathcal{T}_{z}|}\sum_{j=2}^{|\mathcal{T}_{z}|}\sum_{i=1}^{j-1}(\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{i}-\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{i+1})(z).$

A consequence of the mesh regularity assumptions (M1)-(M2) is that
$|\mathcal{T}_{z}|$ is uniformly bounded for any $z\in\mathcal{V}$. Hence it
suffices to bound the term
$(\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{1}-\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{2})(z)$
for $z\in e$ and an edge $e\in K_{1}\cap K_{2}$. In addition, the inverse
estimate for polynomials leads to

$\displaystyle|(\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{1}-\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{2})(z)|$
$\displaystyle\leq\|\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{1}-\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h})_{2}\|_{\infty,e}$
(5.3) $\displaystyle\lesssim
h_{e}^{-1/2}\|[\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}]_{e}\|_{e}.$

Let $v\in V$ be an arbitrary function and
$a_{e}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\nabla(v-v_{h})\,\mbox{d}s$. Since $a_{e}$ is
uniquely defined from the definition of $v_{h}\in V_{h}^{k,\mathrm{nc}}$,
rewrite
$h_{e}^{-1/2}\|[\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}]_{e}\|_{e}=h_{e}^{-1/2}\|[\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}-\nabla
v+a_{e}]_{e}\|_{e}$. Let $\omega_{e}$ denote the edge patch of $e$. Then the
trace inequality and the triangle inequality show

$\displaystyle h_{e}^{-1/2}\|[\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}-\nabla
v+a_{e}]_{e}\|_{e}\lesssim
h_{e}^{-1}\|\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}-\nabla
v+a_{e}\|_{\omega_{e}}+|\Pi^{\nabla^{2\\!}}_{k}v_{h}-v|_{2,\omega_{e}}$
$\displaystyle\lesssim
h_{e}^{-1}(\|\Pi^{\nabla^{2\\!}}_{k}v_{h}-v_{h}\|_{1,\omega_{e}}+\|\nabla(v_{h}-v)+a_{e}\|_{\omega_{e}})+|\Pi^{\nabla^{2\\!}}_{k}v_{h}-v_{h}|_{2,\omega_{e}}+|v_{h}-v|_{2,\omega_{e}}.$

Since $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\nabla(v_{h}-v)+a_{e}\,\mbox{d}s=0$, the
Poincaré–Friedrichs inequality and (M2) imply
$\|\nabla(v_{h}-v)+a_{e}\|_{\omega_{e}}\lesssim
h_{e}|v_{h}-v|_{2,\omega_{e}}$. This and (3.3) in the above displayed estimate
provide

(5.4) $\displaystyle
h_{e}^{-1/2}\|[\nabla\Pi^{\nabla^{2\\!}}_{k}v_{h}]_{e}\|_{e}\lesssim|\Pi^{\nabla^{2\\!}}_{k}v_{h}-v_{h}|_{2,\omega_{e}}+|v_{h}-v|_{2,\omega_{e}}.$

The combination (5.2)-(5.4) results in

$|h_{z}\nabla(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})|_{K}(z)|\lesssim
h_{K}(|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,\omega_{e}}+|v-v_{h}|_{2,\omega_{e}}).$

Theorem 5.1(a), the Cauchy–Schwarz inequality, the trace inequality lead for
any $\chi\in\mathbb{M}_{k-2}(e)$ and
$e\in\mathcal{E}_{K}\setminus\mathcal{E}^{c}$ to

$\displaystyle\int_{e}\partial_{\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})\chi\,\mbox{d}s\leq
h_{e}^{1/2}\|\partial_{\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-v_{h})\|_{e}$
$\displaystyle\qquad\lesssim|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{1,\omega_{e}}+h_{e}|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,\omega_{e}}\lesssim
h_{e}|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,\omega_{e}},$

with (3.3) in the end. Analogously we can prove for any
$\chi\in\mathbb{M}_{k-3}(e)$ and $e\in\mathcal{E}_{K}\cap\mathcal{E}^{i}$ that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})\chi\,\mbox{d}s\lesssim
h_{e}|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,\omega_{e}}.$

Again Theorem 5.1(a), the Cauchy–Schwarz inequality, and (3.3) show for any
$\chi\in\mathbb{M}_{k-4}(K)$ that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})\chi\,\mbox{d}\boldsymbol{x}\lesssim
h_{K}|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,K},$

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_{K}(\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h})\chi\,\mbox{d}\boldsymbol{x}=0$
for $\chi\in\mathbb{M}^{*}_{k-3}(K)$. The definition of
$\Pi^{\nabla^{2\\!}}_{k}$ from (3.1) implies
$\|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}\|_{2,h}\leq\inf_{\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})}|v_{h}-\chi|_{2,h}$.
The previous estimates in (5.1) prove that

$|\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h}|_{2,h}\lesssim\inf_{\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})}|v_{h}-\chi|_{2,h}+\inf_{v\in
V}|v_{h}-v|_{2,h}.$

Hence the triangle inequality
$|v_{h}-J_{1}v_{h}|_{2,h}\leq|v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h}|_{2,h}+|\Pi^{\nabla^{2\\!}}_{k}v_{h}-J_{1}v_{h}|_{2,h}$
and (3.3) prove the estimate in Theorem 5.1(d) for the term
$|v_{h}-J_{1}v_{h}|_{2,h}$. The Poincaré–Friedrichs inequality implies
$\sum_{j=0}^{1}h^{j-2}|v_{h}-J_{1}v_{h}|_{j,h}\lesssim|v_{h}-J_{1}v_{h}|_{2,h}$.
∎

The following theorem establishes the construction of the second companion
operator which will be used in the sequel.

######  Theorem 5.2.

There exists a linear operator $J_{2}:V_{h}^{k,\mathrm{nc}}\to V$ such that it
satisfies Theorem 5.1(a)-5.1(d) and in addition the $L^{2}$-orthogonality
property. In particular,

  1. (a)

$\mathrm{dof}_{j}^{k,\mathrm{nc}}(J_{2}v_{h})=\mathrm{dof}_{j}^{k,\mathrm{nc}}(v_{h})$
for all $j=1,\dots,{\mathrm{N}_{2}^{k,\mathrm{nc}}}$,

  2. (b)

$a_{\rm{pw}}(v_{h}-J_{2}v_{h},\chi)=0$ for all
$\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})$,

  3. (c)

$\nabla(v_{h}-J_{2}v_{h})\perp(\mathbb{P}_{k-3}(\mathcal{T}_{h}))^{2}$ in
$(L^{2}(\Omega))^{2}$ for $k\geq 3$,

  4. (d)

$v_{h}-J_{2}v_{h}\perp\mathbb{P}_{k}(\Omega)$ in $L^{2}(\Omega)$,

  5. (e)

$\displaystyle\sum_{j=0}^{2}h^{j-2}|v_{h}-J_{2}v_{h}|_{j,h}\lesssim\inf_{\chi\in\mathbb{P}_{k}(\mathcal{T}_{h})}|v_{h}-\chi|_{2,\mathrm{pw}}+\inf_{v\in
V}|v_{h}-v|_{2,h}.$

Construction of $J_{2}$. Let $b_{K}\in H^{2}_{0}(K)$ be a bubble-function
supported in $K$ and $v_{K}\in\mathbb{P}_{k}(K)$ be the Riesz representative
of the linear functional $\mathbb{P}_{k}(K)\to\mathbb{R},$ defined by,
$w_{k}\mapsto(v_{h}-J_{1}v_{h},w_{k})_{K}$, for $w_{k}\in\mathbb{P}_{k}(K)$ in
the Hilbert space $\mathbb{P}_{k}(K)$ endowed with the weighted scalar product
$(b_{K}\bullet,\bullet)_{K}$. Given $v_{h}\in V_{h}^{k,\mathrm{nc}}$, the
function $\tilde{v}_{h}\in\mathbb{P}_{k}(\mathcal{T}_{h})$ with
$\tilde{v}_{h}|_{K}:=v_{K}$ and the bubble-function $b_{h}|_{K}:=b_{K}\in
H^{2}_{0}(\Omega)$ satisfy

(5.5)
$\displaystyle(b_{h}\tilde{v}_{h},w_{k})_{\Omega}=(v_{h}-J_{1}v_{h},w_{k})_{\Omega}\qquad\forall\;w_{k}\in\mathbb{P}_{k}(\mathcal{T}_{h}),$

and define

(5.6) $\displaystyle J_{2}v_{h}:=J_{1}v_{h}+b_{h}\tilde{v}_{h}\in V.$

###### Proof of Theorem 5.2(a).

Since $b_{K}=0=\partial_{\boldsymbol{n}}(b_{K})$ on $\partial K$ for any
$K\in\mathcal{T}_{h}$, there holds, for any $v_{h}\in V_{h}^{k,\mathrm{nc}}$,

$\displaystyle J_{2}v_{h}(z)$ $\displaystyle=J_{1}v_{h}(z)=v_{h}(z)$
$\displaystyle\text{for any }\;z\in\mathcal{V},$
$\displaystyle\int_{e}\partial_{\boldsymbol{n}}(J_{2}v_{h})\chi\,\mbox{d}s$
$\displaystyle=\int_{e}\partial_{\boldsymbol{n}}(J_{1}v_{h})\chi\,\mbox{d}s=\int_{e}\partial_{\boldsymbol{n}}(v_{h})\chi\,\mbox{d}s$
$\displaystyle\text{for}\;\chi\in\mathbb{M}_{k-2}(e)\;\text{and}\;e\in\mathcal{E},$
$\displaystyle\mathchoice{{\vbox{\hbox{$\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_{2}(v_{h})\chi\,\mbox{d}s$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\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_{1}(v_{h})\chi\,\mbox{d}s=\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}\chi\,\mbox{d}s$
$\displaystyle\text{for}\;\chi\in\mathbb{M}_{k-3}(e)\;\text{and}\;e\in\mathcal{E}.$

For $\chi\in\mathcal{M}_{k-4}(K)$ and $K\in\mathcal{T}_{h}$, the definition
(5.6) of $J_{2}$ and (5.5) show $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}J_{2}v_{h}\chi\,\mbox{d}\boldsymbol{x}=\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_{K}(J_{1}v_{h}+b_{K}v_{K})\chi\,\mbox{d}\boldsymbol{x}=\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_{K}v_{h}\chi\,\mbox{d}\boldsymbol{x}$. This
concludes the proof of Theorem 5.2(a). ∎

###### Proof of Theorem 5.2(b)-5.2(c).

This results from Theorem 5.2(a) and it follows as in the proof of Theorem
5.1(b)-5.1(c). ∎

###### Proof of Theorem 5.2(d).

This is an immediate consequence of the definition (5.6) of $J_{2}$ and (5.5).
∎

###### Proof of Theorem 5.2(e).

The Poincaré–Friedrichs inequality implies
$\sum_{j=0}^{1}h^{j-2}|v_{h}-J_{2}v_{h}|_{j,h}\lesssim|v_{h}-J_{2}v_{h}|_{2,h}$.
Hence it remains to bound the term $|v_{h}-J_{2}v_{h}|_{2,h}$. The triangle
inequality and (5.6) lead to

(5.7)
$\displaystyle|v_{h}-J_{2}v_{h}|_{2,h}\leq|v_{h}-J_{1}v_{h}|_{2,h}+|b_{h}\tilde{v}_{h}|_{2,h}.$

For any $\chi\in\mathbb{P}_{k}(K)$ and $K\in\mathcal{T}_{h}$, there exist
inverse estimates

(5.8)
$\displaystyle\|\chi\|^{2}_{K}\lesssim(b_{K},\chi^{2})_{K}\lesssim\|\chi\|^{2}_{K}\quad\text{and}\quad\|\chi\|_{K}\lesssim$
$\displaystyle\sum_{m=0}^{2}h_{K}^{m}|b_{K}\chi|_{m,K}\lesssim\|\chi\|_{K}.$

This implies

(5.9) $\displaystyle|b_{K}v_{K}|_{2,K}\lesssim h_{K}^{-2}\|v_{K}\|_{K}.$

The first inequality in (5.8), and (5.5) with
$w_{k}=v_{K}\in\mathbb{P}_{k}(K)$ result in

$\displaystyle\|v_{K}\|_{K}^{2}\lesssim(b_{K}v_{K},v_{K})_{K}=(v_{h}-J_{1}v_{h},v_{K})_{K}.$

Hence $\|v_{K}\|_{K}\lesssim\|v_{h}-J_{1}v_{h}\|_{K}$. This, the estimates
(5.7) and (5.9), and Theorem 5.1(d) conclude the proof of Theorem 5.2(e). ∎

######  Corollary 5.1.

The piecewise $H^{2}$-seminorm forms a norm on $V_{h}^{k,\mathrm{nc}}$, and it
is in turn equivalent to the piecewise $H^{2}$-norm. That is, for any
$v_{h}\in V_{h}^{k,\mathrm{nc}}$, there holds

$\|v_{h}\|_{2,h}\lesssim|v_{h}|_{2,h}.$

###### Proof.

Recall that $J_{2}v_{h}\in V$ for $v_{h}\in V_{h}^{k,\mathrm{nc}}$. Then the
triangle inequality leads to

$\displaystyle\|v_{h}\|_{2,h}$
$\displaystyle\leq\|v_{h}-J_{2}v_{h}\|_{2,h}+\|J_{2}v_{h}\|_{2,\Omega}\lesssim|v_{h}|_{2,h}+|J_{2}v_{h}|_{2,\Omega}$
$\displaystyle\lesssim|v_{h}|_{2,h}+|J_{2}v_{h}-v_{h}|_{2,h}+|v_{h}|_{2,h}\lesssim|v_{h}|_{2,h},$

with Theorem 5.2(e) and (2.5) in the second step, again the triangle
inequality and Theorem 5.2(e) in the last two steps. ∎

The same idea follows for the second-order VE space $Q_{h}^{\ell,\mathrm{nc}}$
and the following two theorems similarly construct $J_{3}$ (as $J_{1}$) and
modify $J_{3}$ to obtain $J_{4}$ (as $J_{2}$) with the $L^{2}$-orthogonality.
We prefer to highlight only the main steps in the construction of $J_{3}$ to
avoid the repetition of the arguments.

######  Theorem 5.3.

There exists a linear operator $J_{3}:Q_{h}^{\ell,\mathrm{nc}}\to
Q_{h}^{\ell+1,c}$ satisfying the following properties:

  1. (a)

$\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(J_{3}q_{h})=\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(q_{h})$
for all $j=1,\dots,{\mathrm{N}_{1}^{\ell,\mathrm{nc}}}$,

  2. (b)

$(\nabla_{\mathrm{pw}}(q_{h}-J_{3}q_{h}),\nabla_{\mathrm{pw}}\chi)_{\Omega}=0$
for all $\chi\in\mathbb{P}_{\ell}(\mathcal{T}_{h})$,

  3. (c)

$\displaystyle\sum_{j=0}^{1}h^{j-1}|q_{h}-J_{3}q_{h}|_{j,h}\lesssim\inf_{\chi\in\mathbb{P}_{\ell}(\mathcal{T}_{h})}|q_{h}-\chi|_{1,h}+\inf_{q\in
Q}|q_{h}-q|_{1,h}.$

Construction of $J_{3}$. First we observe that the DoFs of
$Q_{h}^{\ell,\mathrm{nc}}$ constitute a subset of the DoFs of
$Q_{h}^{\ell+1,\mathrm{c}}$. We define a linear operator
$J_{3}:Q_{h}^{\ell,\mathrm{nc}}\to Q_{h}^{\ell+1,\mathrm{c}}$ through DoFs of
$Q_{h}^{\ell+1,\mathrm{c}}$, for $q_{h}\in Q_{h}^{\ell,\mathrm{nc}}$, by

$\displaystyle\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(J_{3}q_{h})$
$\displaystyle=\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(q_{h})\quad\forall\;j=1,\dots,\mathrm{N}_{1}^{\ell,\mathrm{nc}},$
$\displaystyle J_{3}q_{h}(z)$
$\displaystyle=\frac{1}{|\mathcal{T}_{z}|}\sum_{K\in\mathcal{T}_{z}}\Pi^{\nabla}_{\ell}q_{h}|_{K}(z)\quad\forall\;z\in\mathcal{V}^{i}\cup\mathcal{V}^{c},$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}J_{3}v_{h}\chi\,\mbox{d}\boldsymbol{x}$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}\Pi^{\nabla}_{\ell}q_{h}\chi\,\mbox{d}\boldsymbol{x}\quad\forall\;\chi\in\mathbb{M}^{*}_{\ell-1}(K).$

###### Proof of Theorem 5.3(a).

This is an immediate consequence of the definition of $J_{3}$. ∎

###### Proof of Theorem 5.3(b).

An integration by parts and Theorem 5.3(a) prove, for any
$\chi\in\mathbb{P}_{\ell}(K)$ and $K\in\mathcal{T}_{h}$, that

$\displaystyle(\nabla(q_{h}-J_{3}q_{h}),\nabla\chi)_{K}=-(q_{h}-J_{3}q_{h},\Delta\chi)_{K}+(q_{h}-J_{3}q_{h},\partial_{\boldsymbol{n}}\chi)_{\partial
K}=0.$

This concludes the proof of Theorem 5.3(b). ∎

###### Proof of Theorem 5.3(c).

This follows analogously as the proof of Theorem 5.1(d) with obvious
modifications. ∎

######  Theorem 5.4.

There exists a linear operator $J_{4}:Q_{h}^{\ell,\mathrm{nc}}\to Q$ such that
it satisfies Theorem 5.3(a)-5.3(c) and in addition the $L^{2}$-orthogonality
property. In particular,

  1. (a)

$\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(J_{4}q_{h})=\mathrm{dof}_{j}^{\ell,\mathrm{nc}}(q_{h})$
for all $j=1,\dots,{\mathrm{N}_{1}^{\ell,\mathrm{nc}}}$,

  2. (b)

$(\nabla_{\mathrm{pw}}(q_{h}-J_{3}q_{h}),\nabla_{\mathrm{pw}}\chi)_{\Omega}=0$
for all $\chi\in\mathbb{P}_{\ell}(\mathcal{T}_{h})$,

  3. (c)

$q_{h}-J_{4}q_{h}\perp\mathbb{P}_{\ell}(\Omega)$ in $L^{2}(\Omega)$,

  4. (d)

$\displaystyle\sum_{j=0}^{1}h^{j-1}|q_{h}-J_{4}q_{h}|_{j,h}\lesssim\inf_{\chi\in\mathbb{P}_{\ell}(\mathcal{T}_{h})}|q_{h}-\chi|_{1,h}+\inf_{q\in
Q}|q_{h}-q|_{1,h}.$

###  5.2. Energy error estimate

This section proves the energy error estimate for the nonconforming case
invoking the companion operators constructed in Subsection 5.1.

######  Proposition 5.1 (Nonconforming interpolation).

There exists an interpolation operator $\vec{I}_{h}^{\mathrm{nc}}:(V\cap
H^{s}(\Omega))\times(Q\cap H^{r}(\Omega))\to V_{h}^{k,\mathrm{nc}}\times
Q_{h}^{\ell,\mathrm{nc}}$ such that, for $v\in V\cap H^{s}(\Omega)$ with
$2\leq s\leq k+1$ and $q\in Q\cap H^{r}(\Omega)$ with $1\leq r\leq\ell+1$,
$\vec{I}_{h}^{\mathrm{nc}}v:=(v_{I}^{\mathrm{nc}},q_{I}^{\mathrm{nc}})$
satisfies

$|v-v_{I}^{\mathrm{nc}}|_{j,h}\lesssim
h^{s-j}|v|_{s,\Omega}\;\text{for}\;0\leq j\leq
2\;\text{and}\;|q-q_{I}^{\mathrm{nc}}|_{j,h}\lesssim
h^{r-j}|q|_{r,\Omega}\;\text{for}\;0\leq j\leq 1.$

######  Theorem 5.5.

Given $\vec{\boldsymbol{u}}:=(u,p)\in(V\cap H^{s}(\Omega))\times(Q\cap
H^{r}(\Omega))$ for $s\geq 2$ and for $r\geq 1$, the unique solution
$\vec{\boldsymbol{u}}_{h}^{\mathrm{nc}}=(u_{h}^{\mathrm{nc}},p_{h}^{\mathrm{nc}})\in\mathbf{H}_{\epsilon}^{h,\mathrm{nc}}=V_{h}^{k,\mathrm{nc}}\times
Q_{h}^{\ell,\mathrm{nc}}$ for $k\geq 2$ and $\ell\geq 1$ to (3.12) satisfies

$\displaystyle\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}^{\mathrm{nc}}\|_{\mathbf{H}_{\epsilon}^{h}}$
$\displaystyle\lesssim\|\vec{\boldsymbol{u}}-\vec{I}_{h}^{\mathrm{nc}}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\alpha^{1/2}(|u-\Pi^{\nabla}_{\ell}u|_{1,h}$
$\displaystyle\quad+h|p-\Pi^{\nabla}_{k-2}p|_{1,h})+\beta
h\|p-\Pi^{\nabla}_{\ell}p\|_{\Omega}+\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h})+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h})$
$\displaystyle\lesssim
h^{\min\\{k-1,\ell,s-2,r-1\\}}(\|\tilde{f}\|_{s-4,\Omega}+\|\tilde{g}\|_{r-2,\Omega}).$

###### Proof.

Let $\vec{\boldsymbol{u}}_{I}:=(u_{I},p_{I})\in V_{h}^{k,\mathrm{nc}}\times
Q_{h}^{\ell,\mathrm{nc}}$ be an interpolation of $\vec{\boldsymbol{u}}$ and
$\vec{\boldsymbol{e}}_{h}=(e_{h}^{u},e_{h}^{p}):=(u_{I}-u_{h},p_{I}-p_{h})$.
The coercivity of $\mathcal{A}_{h}$ from Theorem 3.1 and the discrete problem
(3.12) lead to

$\displaystyle\|\vec{\boldsymbol{e}}_{h}\|^{2}_{\mathbf{H}_{\epsilon}^{h}}$
$\displaystyle\lesssim\mathcal{A}_{h}(\vec{\boldsymbol{e}}_{h},\vec{\boldsymbol{e}}_{h})=\mathcal{A}_{h}(\vec{\boldsymbol{u}}_{I},\vec{\boldsymbol{e}}_{h})-\mathcal{F}_{h}(\vec{\boldsymbol{e}}_{h})$
$\displaystyle=a_{1}^{h}(u_{I},e_{h}^{u})-a_{2}^{h}(p_{I},e_{h}^{u})+a_{2}^{h}(e_{h}^{p},u_{I})+a_{3}^{h}(p_{I},e_{h}^{p})-(\tilde{f}_{h},e_{h}^{u})_{\Omega}-(\tilde{g}_{h},e_{h}^{p})_{\Omega}$
$\displaystyle=(a_{1}^{h}(u_{I}-\Pi^{\nabla^{2\\!}}_{k}u,e_{h}^{u})+a_{1}^{\mathrm{pw}}(\Pi^{\nabla^{2\\!}}_{k}u-u,e_{h}^{u}))+(a_{1}^{\mathrm{pw}}(u,e_{h}^{u})-(\tilde{f},e_{h}^{u})_{\Omega})$
$\displaystyle\quad+(\tilde{f}-\tilde{f}_{h},e_{h}^{u})_{\Omega}+(-a_{2}^{h}(p_{I},e_{h}^{u})+a_{2}^{h}(e_{h}^{p},u_{I}))+(a_{3}^{h}(p_{I}-\Pi^{\nabla}_{\ell}p,e_{h}^{p})$
$\displaystyle\quad+a_{3}^{\mathrm{pw}}(\Pi^{\nabla}_{\ell}p-p,e_{h}^{p}))+(a_{3}^{\mathrm{pw}}(p,e_{h}^{p})-(\tilde{g},e_{h}^{p})_{\Omega})+(\tilde{g}-\tilde{g}_{h},e_{h}^{p})_{\Omega}$
$\displaystyle=:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}+T_{6}+T_{7},$

with an elementary algebra in the last two steps. The boundedness of
$\mathcal{A}_{h}$ from Theorem 3.1 and the Cauchy–Schwarz inequality for
$a_{1}^{\mathrm{pw}}$ and $a_{3}^{\mathrm{pw}}$ show the chain of bounds

$\displaystyle T_{1}+T_{5}$
$\displaystyle\lesssim(\|u_{I}-\Pi^{\nabla^{2\\!}}_{k}u\|_{\Omega}+\|u-\Pi^{\nabla^{2\\!}}_{k}u\|_{\Omega})\|e_{h}^{u}\|_{\Omega}+(|u_{I}-\Pi^{\nabla^{2\\!}}_{k}u|_{2,h}$
$\displaystyle\quad+|u-\Pi^{\nabla^{2\\!}}_{k}u|_{2,h})|e_{h}^{u}|_{2,h}+\beta^{1/2}(\|p_{I}-\Pi^{\nabla}_{\ell}p\|_{\Omega}+\|p-\Pi^{\nabla}_{\ell}p\|_{\Omega})\beta^{1/2}\|e_{h}^{p}\|_{\Omega}$
$\displaystyle\quad+\gamma^{1/2}(|p_{I}-\Pi^{\nabla}_{\ell}p|_{1,h}+|p-\Pi^{\nabla}_{\ell}p|_{1,h})\gamma^{1/2}|e_{h}^{p}|_{1,h}$
$\displaystyle\lesssim(\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{I}\|_{\mathbf{H}_{\epsilon}^{h}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}})\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}$
(5.10) $\displaystyle\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega})\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}},$

with the triangle inequality in the second step, and Propositions 3.1-5.1 in
the last step. Taking $\vec{\boldsymbol{v}}=(J_{2}e_{h}^{u},J_{4}e_{h}^{p})$
in the continuous problem (2.6) allows us to assert that

$\displaystyle T_{2}+T_{4}+T_{6}$
$\displaystyle\quad=a_{1}^{\mathrm{pw}}(u,e_{h}^{u}-J_{2}e_{h}^{u})+a_{2}(p,J_{2}e_{h}^{u})+(\tilde{f},J_{2}e_{h}^{u}-e_{h}^{u})_{\Omega}+a_{3}^{\mathrm{pw}}(p,e_{h}^{p}-J_{4}e_{h}^{p})$
$\displaystyle\qquad-
a_{2}(J_{4}e_{h}^{p},u)+(\tilde{g},J_{4}e_{h}^{p}-e_{h}^{p})_{\Omega}-(a_{2}^{h}(p_{I},e_{h}^{u})-a_{2}^{h}(e_{h}^{p},u_{I}))$
$\displaystyle\quad=\biggl{(}a_{1}^{\mathrm{pw}}(u-\Pi^{\nabla^{2\\!}}_{k}u,e_{h}^{u}-J_{2}e_{h}^{u})+(\tilde{f}-\Pi_{k}\tilde{f},J_{2}e_{h}^{u}-e_{h}^{u})_{\Omega}+a_{3}^{\mathrm{pw}}(p-\Pi^{\nabla}_{\ell}p,e_{h}^{p}-J_{4}e_{h}^{p})$
$\displaystyle\qquad+(\tilde{g}-\Pi_{\ell}\tilde{g},J_{4}e_{h}^{p}-e_{h}^{p})_{\Omega}\biggr{)}+\biggl{(}a_{2}(p,J_{2}e_{h}^{u})-a_{2}^{h}(p_{I},e_{h}^{u})+a_{2}^{h}(e_{h}^{p},u_{I})-a_{2}(J_{4}e_{h}^{p},u)\biggr{)}$
$\displaystyle\quad=:T_{8}+T_{9}.$

The last step follows from Theorems 5.2(b)-5.2(d) and 5.4(b)-5.4(c). The
Cauchy–Schwarz inequality and Theorems 5.2(e) and 5.4(d) show for $T_{8}$ that

$\displaystyle T_{8}$
$\displaystyle\lesssim(|u-\Pi^{\nabla^{2\\!}}_{k}u|_{2,h}+\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h}))|e_{h}^{u}|_{2,h}+(\beta
h\|p-\Pi^{\nabla}_{\ell}p\|_{\Omega}+\gamma^{1/2}|p-\Pi^{\nabla}_{\ell}p|_{1,h}$
$\displaystyle\quad+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h}))\gamma^{1/2}|e_{h}^{p}|_{1,h}$
$\displaystyle\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega}+|\tilde{f}|_{s-4,\Omega}+|\tilde{g}|_{r-2,\Omega})\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}.$

Next, an elementary algebraic manipulation for $T_{9}$ provides

$\displaystyle\alpha^{-1}T_{9}$ $\displaystyle=(\nabla p-\Pi_{\ell-1}\nabla
p_{I},\nabla J_{2}e_{h}^{u})_{\Omega}+(\Pi_{\ell-1}\nabla p_{I},\nabla
J_{2}e_{h}^{u}-\Pi_{k-1}\nabla e_{h}^{u})_{\Omega}$ (5.11)
$\displaystyle\quad+(\Pi_{\ell-1}\nabla e_{h}^{p},\Pi_{k-1}\nabla u_{I}-\nabla
u)_{\Omega}+(\Pi_{\ell-1}\nabla e_{h}^{p}-\nabla J_{4}e_{h}^{p},\nabla
u-\nabla\Pi^{\nabla}_{\ell}u)_{\Omega},$

with the $L^{2}$-orthogonality of $\Pi_{\ell-1}$ and Theorem 5.4(b) in the
last term. The Cauchy–Schwarz inequality, the triangle inequality $\|\nabla
p-\Pi_{\ell-1}\nabla
p_{I}\|_{\Omega}\leq|p-p_{I}|_{1,h}+\|(1-\Pi_{\ell-1})\nabla_{\mathrm{pw}}p_{I}\|_{\Omega}$
for the first term in (5.11) lead to

$\displaystyle(\nabla p-\Pi_{\ell-1}\nabla p_{I},\nabla
J_{2}e_{h}^{u})_{\Omega}$
$\displaystyle\leq(|p-p_{I}|_{1,h}+\|(1-\Pi_{\ell-1})\nabla_{\mathrm{pw}}p_{I}\|_{\Omega})|J_{2}e_{h}^{u}|_{1,\Omega}$
$\displaystyle\lesssim(|p-p_{I}|_{1,h}+\|\nabla p-\Pi_{\ell-1}\nabla
p\|_{\Omega})|J_{2}e_{h}^{u}|_{2,\Omega}$ (5.12) $\displaystyle\lesssim
h^{\min\\{\ell,r-1\\}}|p|_{r,\Omega}|e_{h}^{u}|_{2,h},$

having employed $\|\nabla p_{I}-\Pi_{\ell-1}\nabla p_{I}\|_{K}\leq\|\nabla
p_{I}-\Pi_{\ell-1}\nabla p\|_{K}$ for any $K\in\mathcal{T}_{h}$ followed by
the triangle inequality in the second step, and Propositions 3.1-5.1 and the
stability of $J_{2}$ from Theorem 5.2(e) in the last step. For $k=2$, the
Cauchy–Schwarz inequality and the $L^{2}$-stability of $\Pi_{\ell-1}$ for the
second term in (5.11) imply

$\displaystyle(\Pi_{\ell-1}\nabla p_{I},\nabla J_{2}e_{h}^{u}-\Pi_{1}\nabla
e_{h}^{u})_{\Omega}\leq|p_{I}|_{1,h}\|\nabla J_{2}e_{h}^{u}-\Pi_{1}\nabla
e_{h}^{u}\|_{\Omega}$
$\displaystyle\qquad\leq(|p_{I}-p|_{1,h}+|p|_{1,\Omega})(|e_{h}^{u}-J_{2}e_{h}^{u}|_{1,h}+|e_{h}^{u}-\Pi_{1}e_{h}^{u}|_{1,h})\lesssim
h|p|_{1,\Omega}|e_{h}^{u}|_{2,h}.$

The second step results from the triangle inequality, and the last step from
Propositions 3.1-5.1, and Theorem 5.2(e). Theorem 5.2(c) in the second term of
(5.11) for $k\geq 3$ leads to

$\displaystyle(\Pi_{\ell-1}\nabla p_{I},\nabla J_{2}e_{h}^{u}-\Pi_{k-1}\nabla
e_{h}^{u})_{\Omega}=(\Pi_{\ell-1}\nabla
p_{I}-\nabla_{\mathrm{pw}}(\Pi^{\nabla}_{k-2}p),\nabla
J_{2}e_{h}^{u}-\Pi_{k-1}\nabla e_{h}^{u})_{\Omega}$
$\displaystyle\leq(\|\nabla p-\Pi_{\ell-1}\nabla
p\|_{\Omega}+|p-p_{I}|_{1,h}+|p-\Pi^{\nabla}_{k-2}p|_{1,h})$
$\displaystyle\quad\times(|e_{h}^{u}-J_{2}e_{h}^{u}|_{1,h}+\|(1-\Pi_{k-1})\nabla_{\mathrm{pw}}e_{h}^{u}\|_{\Omega})\lesssim
h^{\min\\{\ell,r,k-1\\}}|p|_{r,\Omega}|e_{h}^{u}|_{2,h},$

where we have used the bound $\|\nabla p_{I}-\Pi_{\ell-1}\nabla
p_{I}\|_{K}\leq\|\nabla p_{I}-\Pi_{\ell-1}\nabla p\|_{K}$ for any
$K\in\mathcal{T}_{h}$ and the triangle inequality in the second step, and
Propositions 3.1-5.1 and Theorem 5.2(e) in the last step. Similarly the
remaining two terms in (5.11) are handled as

(5.13a) $\displaystyle(\Pi_{\ell-1}\nabla e_{h}^{p},\Pi_{k-1}\nabla
u_{I}-\nabla u)_{\Omega}$ $\displaystyle\lesssim
h^{\min\\{k-1,s-1\\}}|u|_{s,\Omega}|e_{h}^{p}|_{1,h},$ (5.13b)
$\displaystyle(\Pi_{\ell-1}\nabla e_{h}^{p}-\nabla J_{4}e_{h}^{p},\nabla
u-\nabla_{\mathrm{pw}}(\Pi^{\nabla}_{\ell}u))_{\Omega}$ $\displaystyle\lesssim
h^{\min\\{\ell,s-1\\}}|u|_{s,\Omega}|e_{h}^{p}|_{1,h}.$

The combination (5.12)-(5.13b) and the observation $\alpha\leq 1\leq\gamma$ in
(5.11) prove that

$\displaystyle T_{9}$
$\displaystyle\lesssim(\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{I}\|_{\mathbf{H}_{\epsilon}^{h}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}})|e_{h}^{u}|_{2,h}$
$\displaystyle\quad+\alpha^{1/2}(|u-\Pi^{\nabla}_{\ell}u|_{1,h}+h|p-\Pi^{\nabla}_{k-2}p|_{1,h})\gamma^{1/2}|e_{h}^{p}|_{1,h}$
$\displaystyle\lesssim
h^{\min\\{k-1,s-1,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega})(|e_{h}^{u}|_{2,h}+\gamma^{1/2}|e_{h}^{p}|_{1,h}).$

The $L^{2}$-orthogonality of $\Pi_{k}$ and $\Pi_{\ell}$, and Proposition 3.1
result in

$\displaystyle T_{3}+T_{7}$
$\displaystyle=(h_{\mathcal{T}_{h}}^{2}(1-\Pi_{k})\tilde{f},h_{\mathcal{T}_{h}}^{-2}(1-\Pi_{k})e_{h}^{u})_{\Omega}+(h_{\mathcal{T}_{h}}(1-\Pi_{\ell})\tilde{g},h_{\mathcal{T}_{h}}^{-1}(1-\Pi_{\ell})e_{h}^{p})_{\Omega}$
$\displaystyle\lesssim\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h})|e_{h}^{u}|_{2,h}+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h})|e_{h}^{p}|_{1,h}$
$\displaystyle\lesssim
h^{\min\\{k+1,s-2,\ell+1,r-1\\}}(|\tilde{f}|_{s-4,\Omega}+|\tilde{g}|_{r-2,\Omega})(|e_{h}^{u}|_{2,h}+\gamma^{1/2}|e_{h}^{p}|_{1,h}),$

with $\gamma\geq 1$ in the end. The previous estimates in (5.2) readily prove
that

$\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}\lesssim
h^{\min\\{k-1,s-2,\ell,r-1\\}}(|u|_{s,\Omega}+|p|_{r,\Omega}+|\tilde{f}|_{s-4,\Omega}+|\tilde{g}|_{r-2,\Omega}).$

This and Proposition 5.1 in the triangle inequality
$\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}\leq\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{I}\|_{\mathbf{H}_{\epsilon}^{h}}+\|\vec{\boldsymbol{e}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}$
followed by regularity estimates conclude the proof of the theorem. ∎

######  Remark 5.6 (Best-approximation for lowest-order case).

If we reconstruct $J_{2}$ for $k=2$ with an additional $H^{1}$-orthogonality
in Theorem 5.2(c) as
$\nabla(v_{h}-J_{2}v_{h})\perp(\mathbb{P}_{0}(\mathcal{T}_{h}))^{2}$ in
$(L^{2}(\Omega))^{2}$ (see Subsection 6.3 for a definition), then the error
estimate in Theorem 5.5 for $k=2$ and $\ell=1$ can be written in the best-
approximation form

$\displaystyle\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}^{\mathrm{nc}}\|_{\mathbf{H}_{\epsilon}^{h}}$
$\displaystyle\lesssim\|\vec{\boldsymbol{u}}-\vec{I}_{h}^{\mathrm{nc}}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\|\vec{\boldsymbol{u}}-\vec{\Pi}_{h}\vec{\boldsymbol{u}}\|_{\mathbf{H}_{\epsilon}^{h}}+\alpha^{1/2}|u-\Pi^{\nabla}_{\ell}u|_{1,h}+\beta
h\|p-\Pi^{\nabla}_{\ell}p\|_{\Omega}$
$\displaystyle\quad+\mathrm{osc}_{2}(\tilde{f},\mathcal{T}_{h})+\mathrm{osc}_{1}(\tilde{g},\mathcal{T}_{h}).$

##  6\. A posteriori error analysis

This section contains the derivation of a posteriori error indicators and the
proof of their robustness. We provide the details for the nonconforming case
and a remark for the conforming case to avoid the repetition of arguments.

###  6.1. Preliminaries

We collect here the following local estimates, proven in [30, Lemma 3.2] and
[30, Lemmas 3.3-3.4], respectively.

######  Lemma 6.1.

For any $\epsilon>0$, there exists a positive constant $c(\epsilon)$ such that

$|v|_{1,K}\lesssim\epsilon
h_{K}|v|_{2,K}+c(\epsilon)h_{K}^{-1}\|v\|_{K}\qquad\forall v\in H^{2}(K).$

######  Lemma 6.2.

For every $v\in H^{2}(K)$ such that $\Delta^{2}v\in\mathbb{P}_{k-4}(K)$, there
exists a polynomial $p\in\mathbb{P}_{k}(K)$ satisfying

$\Delta^{2}v=\Delta^{2}p\qquad\text{in }K.$

Moreover, the following estimates hold

$|p|_{2,K}\lesssim
h_{K}^{2}\|\Delta^{2}v\|_{K}\lesssim|v|_{2,K},\qquad|p|_{2,K}\lesssim
h_{K}^{-2}\|v\|_{K},\qquad\|p\|_{K}\lesssim\|v\|_{K}.$

###  6.2. Standard estimates

We start with technical results (inverse estimates and norm equivalences) that
are required in the analysis of the nonconforming formulations. These tools
are available in the literature only for the conforming case. There are two
terminologies, namely original and enhanced VE spaces, in the VE literature
(see [3] for more details). This paper utilises the enhanced versions, but we
first prove the results for the original space and then build for the enhanced
space. Let us denote the original local nonconforming VE space for deflections
by $\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$ and define by

$\widetilde{V}_{h}^{k,\mathrm{nc}}(K):=\begin{dcases}\begin{rcases}&v_{h}\in
H^{2}(K)\cap C^{0}(\partial
K):\Delta^{2}v_{h}\in\mathbb{P}_{k-4}(K),\;v_{h}|_{e}\in\mathbb{P}_{k}(e)\;\text{and}\\\
&\;\Delta
v_{h}|_{e}\in\mathbb{P}_{k-2}(e)\quad\forall\;e\in\mathcal{E}_{K},\;v_{h}|_{s_{j}}\in
C^{1}(s_{j}),\\\
&\;\int_{e_{m_{j}}}(v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h})\chi\,\mbox{d}s=0\quad\forall\;\chi\in\mathbb{P}_{k-2}(e_{m_{j}}),\;\text{and}\\\
&\int_{e_{m_{j}+i}}(v_{h}-\Pi^{\nabla^{2\\!}}_{k}v_{h})\chi\,\mbox{d}s=0\quad\forall\;\chi\in\mathbb{P}_{k-3}(e_{m_{j}+i})\\\
&\;\text{for}\;i=1,\dots,n_{j};\;j=1,\dots,\tilde{N}_{K}\end{rcases}.\end{dcases}$

######  Lemma 6.3 (Inverse estimates).

For any $\tilde{v}\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$ and
$K\in\mathcal{T}_{h}$, there holds

(6.1) $|\tilde{v}|_{2,K}\lesssim
h_{K}^{-2}\|\tilde{v}\|_{K}\quad\text{and}\quad|\tilde{v}|_{1,K}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}.$

###### Proof.

Given $\tilde{v}\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$,
$\Delta^{2}\tilde{v}\in\mathbb{P}_{k-4}(K)$ and consequently, we can choose a
polynomial $p\in\mathbb{P}_{k}(K)$ from Lemma 6.2. The triangle inequality and
the second bound in Lemma 6.2 assert that

$|\tilde{v}|_{2,K}\leq|\tilde{v}-p|_{2,K}+|p|_{2,K}\lesssim|\tilde{v}-p|_{2,K}+h_{K}^{-2}\|\tilde{v}\|_{K}.$

If we prove $|\tilde{v}-p|_{2,K}\lesssim h_{K}^{-2}\|\tilde{v}-p\|_{K}$, then
the triangle inequality together with the third bound in Lemma 6.2 will
provide

$|\tilde{v}-p|_{2,K}\lesssim h_{K}^{-2}\|\tilde{v}-p\|_{K}\lesssim
h_{K}^{-2}(\|\tilde{v}\|_{K}+\|p\|_{K})\lesssim h_{K}^{-2}\|\tilde{v}\|_{K}.$

Hence we concentrate on showing $|\tilde{v}-p|_{2,K}\lesssim
h_{K}^{-2}\|\tilde{v}-p\|_{K}$. First we note that since
$\Delta^{2}\tilde{v}=\Delta^{2}p$ and
$\tilde{v}-p\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$, without loss of
generality we can assume that $\Delta^{2}\tilde{v}=0$. Then we define, for a
fixed $\tilde{v}$, the following set

(6.2) $S(K):=\\{w\in H^{2}(K):w=\tilde{v}\text{on $\partial
K$,}\int_{e}\partial_{\boldsymbol{n}}(\tilde{v}-w)\chi\,\mbox{d}s=0\quad\forall\chi\in\mathbb{P}_{k-2}(e),\,e\in\mathcal{E}_{K}\\}$

and the fact $a^{K}(\tilde{v},\tilde{v}-w)=0$ for $w\in S(K)$ leads to

(6.3) $\displaystyle|\tilde{v}|_{2,K}\leq|w|_{2,K}\quad\forall w\in S(K).$

Next, we define $Q_{K}\tilde{v}\in\widetilde{V}_{h}^{k+1,\rm{c}}(K)$ for
$\tilde{v}\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$ through the DoFs as

(6.4) $\mathrm{Dof}_{\partial
K}^{k+1,\mathrm{c}}(Q_{K}\tilde{v})=\mathrm{Dof}_{\partial
K}^{k+1,\mathrm{c}}(\tilde{v}),\quad\text{and
}\mathrm{Dof}_{K}^{k+1,\mathrm{c}}(Q_{K}\tilde{v})=0,$

where

$\mathrm{Dof}^{k+1,\mathrm{c}}(\bullet)=\underbrace{\mathrm{Dof}_{\partial
K}^{k+1,\mathrm{c}}(\bullet)}_{\text{boundary
DoFs}}\cup\underbrace{\mathrm{Dof}_{K}^{k+1,\mathrm{c}}(\bullet)}_{\text{interior
DoFs}}.$

Observe that, for $\tilde{v}\in H^{2}(K)$, its tangential derivative
$\partial_{\boldsymbol{t}}\tilde{v}$ is well-defined along $\partial K$. If
$z$ is not a corner in $\partial K$, then we can assign that
$\partial_{\boldsymbol{n}}\tilde{v}(z)=0$, and if $z$ is a corner then the two
tangential derivatives at $z$ will suffice to define $\nabla\tilde{v}(z)$
uniquely. This implies that $Q_{K}\tilde{v}$ is well-defined. In addition,
since $Q_{K}\tilde{v}$ is uniquely determined by boundary DoFs of $\tilde{v}$
and $\tilde{v}|_{e}\in\mathbb{P}_{k}(e)$ for all $e\in\mathcal{E}_{K}$, we
have

$Q_{K}\tilde{v}=\tilde{v}\quad\text{on $\partial K$}.$

This and (6.4) show that $Q_{K}\tilde{v}\in S(K)$ and, consequently (6.3)
imply the first inequality in

$\displaystyle|\tilde{v}|_{2,K}\leq|Q_{K}\tilde{v}|_{2,K}$
$\displaystyle\lesssim h_{K}^{-2}\|Q_{K}\tilde{v}\|_{K}$
$\displaystyle\lesssim
h_{K}^{-1}\|\mathrm{Dof}^{k+1,\mathrm{c}}(Q_{K}\tilde{v})\|_{\ell^{2}}=h_{K}^{-1}\|\mathrm{Dof}_{\partial
K}^{k+1,\mathrm{c}}(Q_{K}\tilde{v})\|_{\ell^{2}}$

with the inverse estimate and the norm equivalence available for conforming VE
functions in the next two inequalities, and (6.4) in the last equality.

Let us examine each contribution to the DoFs in the expression above. Firstly,
for any $z\in\mathcal{V}_{K}$ it can be inferred that

$\displaystyle|\tilde{v}(z)|$
$\displaystyle\leq\|\tilde{v}\|_{L^{\infty}(\partial K)}\lesssim
h_{K}^{-1/2}\|\tilde{v}\|_{L^{2}(\partial K)}$ $\displaystyle\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}+|\tilde{v}|_{1,K}\lesssim(1+c(\epsilon))h_{K}^{-1}\|\tilde{v}\|_{K}+\epsilon
h_{K}|\tilde{v}|_{2,K},$

where we have used the inverse estimate for polynomials in 1d in the second
step, and the trace inequality and Lemma 6.1 in the last two steps.

Secondly, similar arguments show that

$\displaystyle h_{z}|\nabla\tilde{v}(z)|$ $\displaystyle=h_{z}|\nabla
Q_{K}\tilde{v}(z)|\leq h_{z}|Q_{K}\tilde{v}|_{1,\infty,\partial
K}\lesssim\|Q_{K}\tilde{v}\|_{\infty,\partial K}$ $\displaystyle\lesssim
h_{K}^{-1/2}\|\tilde{v}\|_{\partial K}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}+\epsilon h_{K}|\tilde{v}|_{2,K}.$

Note that the inverse inequality for polynomials is suitable here since
$\partial_{\boldsymbol{n}}Q_{K}\tilde{v}|_{e}\in\mathbb{P}_{k-1}(e)$. For the
remaining boundary moments, the Cauchy–Schwarz inequality and the inverse
estimate lead to

$\displaystyle\Big{|}\int_{e}\partial_{\boldsymbol{n}}\tilde{v}\chi_{k-2}\,\mbox{d}s\Big{|}$
$\displaystyle=\Big{|}\int_{e}\partial_{\boldsymbol{n}}(Q_{K}\tilde{v})\chi_{k-2}\,\mbox{d}s\Big{|}\leq\|\partial_{\boldsymbol{n}}(Q_{K}\tilde{v})\|_{e}h_{e}^{1/2}$
$\displaystyle\lesssim
h_{e}^{-1/2}\|Q_{K}\tilde{v}\|_{e}=h_{e}^{-1/2}\|\tilde{v}\|_{e}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}+\epsilon h_{K}|\tilde{v}|_{2,K}$

with (6.3) and Lemma 6.1 once more in the last two steps. Similarly, we can
prove that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\tilde{v}\chi_{k-3}\,\mbox{d}s\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}+\epsilon h_{K}|\tilde{v}|_{2,K}.$

These bounds allow us to write $\|\mathrm{Dof}_{\partial
K}^{k+1,\mathrm{c}}(Q_{K}\tilde{v})\|_{\ell^{2}}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}+\epsilon h_{K}|\tilde{v}|_{2,K},$ which in turn
proves that

$|\tilde{v}|_{2,K}\lesssim
h_{K}^{-2}\|\tilde{v}\|_{K}+\epsilon|\tilde{v}|_{2,K}$

and absorbing the $\epsilon$ term on the left-hand side we immediately have
the first bound in (6.1). For the second bound it suffices to combine the
first bound with Lemma 6.1. ∎

Figure 2. Sketch of a polygonal domain $K$ and three consecutive vertices
$A,B,C$. The unit vectors $\boldsymbol{t}_{1},\boldsymbol{t}_{2}$ form an
angle $\theta$ on $A$.

######  Lemma 6.4 (Poincaré-type inequality).

Let $K$ be a polygonal domain and $v\in H^{2}(K)$. If $v(A)=v(B)=v(C)$ for any
three non-collinear consecutive vertices $A,B,C$ of $K$ (see the diagram in
Figure 2), then there exists a positive constant $C_{\rm{P}}$ depending only
on the mesh regularity parameter $\rho$, such that

$|v|_{1,K}\leq C_{\rm{P}}h_{K}|v|_{2,K}.$

###### Proof.

With respect to Figure 2, let $\boldsymbol{t}_{1},\boldsymbol{t}_{2}$ be two
tangential unit vectors along the sides $AB$ and $AC$, respectively, oriented
as in the diagram (moving away from the vertex $A$), forming an angle
$\theta\in(0,\pi)$, and
$|\boldsymbol{t}_{1}\cdot\boldsymbol{t}_{2}|=|\cos\theta|$. Owing to the
transformation stability result from [14] we know that

$\min_{\boldsymbol{a}\in\mathbb{R}^{2}\setminus\\{\boldsymbol{0}\\}}\frac{(\boldsymbol{a}\cdot\boldsymbol{v})^{2}+(\boldsymbol{a}\cdot\boldsymbol{u})^{2}}{|\boldsymbol{a}|^{2}}=1-|\boldsymbol{v}\cdot\boldsymbol{u}|,$

for linearly independent unit vectors
$\boldsymbol{v},\boldsymbol{u}\in\mathbb{R}^{2}$. We use this result in our
context with $\boldsymbol{a}=\nabla v(\boldsymbol{x})$ for an interior point
$\boldsymbol{x}$ of $K$, $\boldsymbol{v}=\boldsymbol{t}_{1}$,
$\boldsymbol{u}=\boldsymbol{t}_{2}$, giving

$(1-|\cos\theta|)|\nabla v(\boldsymbol{x})|^{2}\leq(\nabla
v(\boldsymbol{x})\cdot\boldsymbol{t}_{1})^{2}+(\nabla
v(\boldsymbol{x})\cdot\boldsymbol{t}_{2})^{2}.$

Now we define $f_{j}=\nabla v\cdot\boldsymbol{t}_{j}$ for $j=1,2$. An
integration over $K$ leads to

$|v|^{2}_{1,K}\leq\frac{1}{1-|\cos\theta|}(\|f_{1}\|^{2}_{K}+\|f_{2}\|^{2}_{K}).$

On the other hand, note that since
$\int_{A}^{B}f_{1}\,\mbox{d}s=0=\int_{A}^{C}f_{2}\,\mbox{d}s$ from the
assumption $v(A)=v(B)=v(C)$, we can apply the Poincaré–Friedrichs inequality
to $f_{1}$ and $f_{2}$ (see, for example, [15]). We are then left with

$\|f_{i}\|_{K}\leq C_{\rm{PF}}h_{K}|f_{i}|_{1,K}\quad\text{for }i=1,2.$

Hence

$|v|^{2}_{1,K}\leq\frac{C^{2}_{\rm{PF}}}{1-|\cos\theta|}h_{K}^{2}(|f_{1}|^{2}_{1,K}+|f_{2}|^{2}_{1,K})\leq\frac{2C^{2}_{\rm{PF}}}{1-|\cos\theta|}h_{K}^{2}|v|^{2}_{2,K},$

which proves the sought bound with
$C_{\rm{P}}:=C_{\rm{PF}}\sqrt{\frac{2}{1-|\cos\theta|}}$. ∎

######  Lemma 6.5 (Local norm equivalence).

For $\tilde{v}\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$, there holds

$\|\tilde{v}\|_{K}\simeq
h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}.$

###### Proof.

Step 1. Proceeding as in the proof of Lemma 6.3, for the DoFs we have

$|\tilde{v}(z)|\lesssim h_{K}^{-1}\|\tilde{v}\|_{K}+|\tilde{v}|_{1,K}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}$

with the inverse inequality applied to $\tilde{v}$ from Lemma 6.3. Using the
Cauchy–Schwarz inequality, the trace inequality, and Lemma 6.3 again, we
arrive at

$\Big{|}\int_{e}\partial_{\boldsymbol{n}}\tilde{v}\chi_{k-2}\,\mbox{d}s\Big{|}\leq\|\partial_{\boldsymbol{n}}\tilde{v}\|_{e}h_{e}^{1/2}\lesssim|\tilde{v}|_{1,K}+h_{K}|\tilde{v}|_{2,K}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}.$

In addition,

$\Big{|}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\tilde{v}\chi_{k-3}\,\mbox{d}s\Big{|}\leq
h_{e}^{-1/2}\|\tilde{v}\|_{e}\lesssim h_{K}^{-1}\|\tilde{v}\|_{K}.$

The Cauchy–Schwarz inequality for the cell moments proves that

$\Big{|}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{K}\tilde{v}\chi_{k-4}\,\mbox{d}\boldsymbol{x}\Big{|}\leq|K|^{-1/2}\|\tilde{v}\|_{K}=h_{K}^{-1}\|\tilde{v}\|_{K},$

and combining these bounds together we readily obtain

(6.5) $\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}\lesssim
h_{K}^{-1}\|\tilde{v}\|_{K}.$

Step 2. On the other hand, let us consider the problem of finding
$\tilde{v}_{2}\in H_{0}^{1}(K)\cap H^{2}(K)$ such that

$\Delta^{2}\tilde{v}_{2}=\Delta^{2}\tilde{v}\quad\text{in
$K$};\qquad\partial_{\boldsymbol{n}\boldsymbol{n}}(\tilde{v}_{2})=\partial_{\boldsymbol{n}\boldsymbol{n}}\tilde{v}\quad\text{on
$\partial K$}.$

Let $\tilde{v}_{1}=\tilde{v}-\tilde{v}_{2}$. Then, it follows that

$\Delta^{2}\tilde{v}_{1}=0\quad\text{in
$K$};\qquad\partial_{\boldsymbol{n}\boldsymbol{n}}(\tilde{v}_{1})=0\
\text{and}\ \tilde{v}_{1}=\tilde{v}\quad\text{on $\partial K$}.$

Next we recall that for any $w\in S(K)$ (cf. (6.2)) we have
$a^{K}(\tilde{v}_{1},\tilde{v}_{1}-w)=0$. From the proof of Lemma 6.3 we also
recall that $Q_{K}\tilde{v}_{1}$ is well-defined and

$Q_{K}\tilde{v}_{1}=\tilde{v}_{1}=\tilde{v}\quad\text{on $\partial K$}.$

The triangle inequality and Lemma 6.4 for $\tilde{v}_{1}-Q_{K}\tilde{v}_{1}$
(applies from the definition of $Q_{K}$) result in

$\displaystyle\|\tilde{v}_{1}\|_{K}\leq\|\tilde{v}_{1}-Q_{K}\tilde{v}_{1}\|_{K}+\|Q_{K}\tilde{v}_{1}\|_{K}$
$\displaystyle\lesssim
h_{K}^{2}|\tilde{v}_{1}-Q_{K}\tilde{v}_{1}|_{2,K}+\|Q_{K}\tilde{v}_{1}\|_{K}$
$\displaystyle\lesssim
h_{K}^{2}|\tilde{v}_{1}|_{2,K}+\|Q_{K}\tilde{v}_{1}\|_{K}.$

From the proof of Lemma 6.3, we can infer that $|\tilde{v}_{1}|_{2,K}\lesssim
h_{K}^{-3/2}\|\tilde{v}_{1}\|_{\partial K}$ and
$\|Q_{K}\tilde{v}_{1}\|_{K}\lesssim\|\tilde{v}_{1}\|_{\partial K}$. This in
the previous bound results in $\|\tilde{v}_{1}\|_{K}\lesssim
h_{K}^{1/2}\|\tilde{v}_{1}\|_{\partial K}$.

Recall that
$\Delta^{2}\tilde{v}_{2}=\Delta^{2}\tilde{v}=:g_{1}\in\mathbb{P}_{k-4}(K)$ and
$\partial_{\boldsymbol{n}\boldsymbol{n}}(\tilde{v}_{2})|_{e}=\partial_{\boldsymbol{n}\boldsymbol{n}}\tilde{v}|_{e}=:g_{2}|_{e}\in\mathbb{P}_{k-2}(e)$
for all $e\in\mathcal{E}_{K}$. Therefore, after expanding
$g_{1}=\sum_{|\alpha|\leq k-4}g^{\alpha}_{1}m_{\alpha}$ and
$g_{2}|_{e}=\sum_{|\beta|\leq k-2}g^{\beta}_{2}m_{\beta}^{e}$ in terms of the
scaled monomials $m_{\alpha}\in\mathbb{M}_{k-4}(K)$ and
$m_{\beta}^{e}\in\mathbb{M}_{k-2}(e)$, an integration by parts provides

$\displaystyle|\tilde{v}_{2}|^{2}_{2,K}$
$\displaystyle=(\Delta^{2}\tilde{v}_{2},\tilde{v}_{2})_{K}+(\partial_{\boldsymbol{n}\boldsymbol{n}}(\tilde{v}_{2}),\partial_{\boldsymbol{n}}\tilde{v}_{2})_{\partial
K}$
$\displaystyle=(g_{1},\tilde{v})_{K}-(g_{1},\tilde{v}_{1})_{K}+(g_{2},\partial_{\boldsymbol{n}}\tilde{v})_{\partial
K}-(g_{2},\partial_{\boldsymbol{n}}\tilde{v}_{1})_{\partial K}$
$\displaystyle=\sum_{|\alpha|\leq
k-4}g^{\alpha}_{1}(m_{\alpha},\tilde{v})_{K}+\sum_{|\beta|\leq
k-2}g^{\beta}_{2}(m_{\beta},\partial_{\boldsymbol{n}}\tilde{v})_{\partial
K}-(g_{1},\tilde{v}_{1})_{K}-(g_{2},\partial_{\boldsymbol{n}}\tilde{v}_{1})_{\partial
K}.$

Set the notation $\vec{g}_{1}=(g_{1}^{\alpha})_{\alpha}$,
$\vec{g}_{2}=(g_{2}^{\beta})_{\beta}$ and recall from [19, Lemma 4.1] that
$h_{K}\|\vec{g}_{1}\|_{\ell^{2}}\lesssim\|g_{1}\|_{K}$ and
$h_{K}^{1/2}\|\vec{g}_{2}\|_{\ell^{2}}\lesssim\|g_{2}\|_{\partial K}$. Hence
the Cauchy–Schwarz inequality in the previous bound and the definition of DoFs
show

$\displaystyle|\tilde{v}_{2}|^{2}_{2,K}$
$\displaystyle\leq\|\vec{g}_{1}\|_{\ell^{2}}|K|\|\mathrm{Dof}_{K}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}+\|\vec{g}_{2}\|_{\ell^{2}}\|\mathrm{Dof}_{\partial
K}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}$
$\displaystyle\quad+\|g_{1}\|_{K}\|\tilde{v}_{1}\|_{K}+\|g_{2}\|_{\partial
K}\|\partial_{\boldsymbol{n}}\tilde{v}_{1}\|_{\partial K}$
$\displaystyle\lesssim
h_{K}\|g_{1}\|_{K}\|\mathrm{Dof}_{K}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}+h_{K}^{-1/2}\|g_{2}\|_{\partial
K}\|\mathrm{Dof}_{\partial K}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}$
$\displaystyle\quad+\|g_{1}\|_{K}\|\tilde{v}_{1}\|_{K}+\|g_{2}\|_{\partial
K}((1+\epsilon)h_{K}^{1/2}|\tilde{v}_{1}|_{2,K}+C(\epsilon)h_{K}^{-3/2}\|\tilde{v}_{1}\|_{K})$

with $\|\partial_{\boldsymbol{n}}\tilde{v}_{1}\|_{\partial K}\lesssim
h_{K}^{-1/2}|\tilde{v}_{1}|_{1,K}+h_{K}^{1/2}|\tilde{v}_{1}|_{2,K}\lesssim(1+\epsilon)h_{K}^{1/2}|\tilde{v}_{1}|_{2,K}+C(\epsilon)h_{K}^{-3/2}\|\tilde{v}_{1}\|_{K}$
from the trace inequality and Lemma 6.1 in the last step. Note also that,
thanks to [20], we can assert that

$\|g_{1}\|_{K}=\|\Delta^{2}\tilde{v}_{2}\|_{K}\lesssim
h_{K}^{-2}|\tilde{v}_{2}|_{2,K},$

and using the inverse inequality on $g_{2}|_{e}\in\mathbb{P}_{k-2}(e)$, the
trace inequality, and Lemma 6.4, we are left with

$\|g_{2}\|_{\partial K}\lesssim h_{K}^{-1}|\tilde{v}_{2}|_{1,\partial
K}\lesssim
h_{K}^{-3/2}|\tilde{v}_{2}|_{1,K}+h_{K}^{-1/2}|\tilde{v}_{2}|_{2,K}\lesssim
h_{K}^{-1/2}|\tilde{v}_{2}|_{2,K}.$

Hence, all the above bounds result in

$|\tilde{v}_{2}|_{2,K}\lesssim
h_{K}^{-1}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}+h_{K}^{-3/2}\|\tilde{v}\|_{\partial
K}.$

We can then invoke again Lemma 6.4 to obtain $\|\tilde{v}_{2}\|_{K}\lesssim
h_{K}^{2}|\tilde{v}_{2}|_{2,K}$, and so

$\|\tilde{v}\|_{K}\leq\|\tilde{v}_{1}\|_{K}+\|\tilde{v}_{2}\|_{K}\lesssim
h_{K}^{1/2}\|\tilde{v}\|_{\partial
K}+h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}.$

Since $\tilde{v}$ is a polynomial along each $e\in\mathcal{E}_{K}$, then
standard scaling arguments imply that $\|\tilde{v}\|_{\partial K}\simeq
h_{K}^{1/2}\|\mathrm{Dof}_{\partial
K}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}},$ and therefore

(6.6) $\|\tilde{v}\|_{K}\lesssim
h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}.$

Finally, the desired result follows from combining the estimates (6.5) (from
Step 1) and (6.6) (from Step 2). ∎

######  Lemma 6.6.

The inverse estimates and norm equivalence results hold for any $v\in
V_{h}^{k,\mathrm{nc}}(K)$.

###### Proof.

Given $v\in V_{h}^{k,\mathrm{nc}}(K)$, construct
$\tilde{v}\in\widetilde{V}_{h}^{k,\mathrm{nc}}(K)$ with
$\mathrm{Dof}^{k,\mathrm{nc}}(v)=\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})$.
Such $\tilde{v}$ can be found since the DoFs of both local VE spaces
$V_{h}^{k,\mathrm{nc}}(K)$ and $\widetilde{V}_{h}^{k,\rm{nc}}(K)$ coincide.
Starting from the triangle inequality
$|v|_{2,K}\leq|v-\tilde{v}|_{2,K}+|\tilde{v}|_{2,K},$ we apply integration by
parts, the Cauchy–Schwarz inequality, and the inverse inequality on
$\Delta^{2}(v-\tilde{v})\in\mathbb{P}_{k}(K)$ to obtain

$\displaystyle|v-\tilde{v}|^{2}_{2,K}=a^{K}(v-\tilde{v},v-\tilde{v})$
$\displaystyle=\int_{K}\Delta^{2}(v-\tilde{v})\,(v-\tilde{v})\,\mbox{d}\boldsymbol{x}$
$\displaystyle\leq\|\Delta^{2}(v-\tilde{v})\|_{K}\|v-\tilde{v}\|_{K}\lesssim
h_{K}^{-2}|v-\tilde{v}|_{2,K}\|v-\tilde{v}\|_{K}.$

This bound together with the triangle inequality show that
$|v-\tilde{v}|_{2,K}\lesssim h_{K}^{-2}(\|v\|_{K}+\|\tilde{v}\|_{K})$.
Combining this last bound with Lemma 6.3-6.5 readily implies that

$|v|_{2,K}\lesssim
h_{K}^{-2}(\|v\|_{K}+\|\tilde{v}\|_{K}),\qquad\|\tilde{v}\|_{K}\simeq
h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}=h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(v)\|_{\ell^{2}}.$

Then we can follow the arguments developed in the proof of Lemma 6.3 to get

$|v|_{2,K}\lesssim h_{K}^{-2}\|v\|_{K}+\epsilon|v|_{2,K},$

and the proof of the inverse estimate is concluded after absorbing the
$\epsilon$ term on the left-hand side. Regarding the norm equivalence result,
we note that steps 1 and 2 from the proof of Lemma 6.5 apply to any $v\in
V_{h}^{k,\mathrm{nc}}(K)$, and decompose $v=v_{1}+v_{2}$. The proof follows
analogously only differing in the presence of the term
$\Delta^{2}{v}_{2}=\Delta^{2}{v}=:g_{1}\in\mathbb{P}_{k}(K)$, now written as

$(\Delta^{2}v_{2},v)_{K}=\sum_{|\alpha|\leq
k-4}g^{\alpha}_{1}(m_{\alpha},{v})_{K}+\sum_{k-4\leq|\alpha|\leq
k}g^{\alpha}_{1}(m_{\alpha},v)_{K}.$

The first term on the right-hand side is treated just as in Lemma 6.5. For the
second term we can apply the definition of the space
$V_{h}^{k,\mathrm{nc}}(K)$ and the Cauchy–Schwarz inequality to obtain the
estimate

$\displaystyle\sum_{k-4\leq|\alpha|\leq k}g^{\alpha}_{1}(m_{\alpha},v)_{K}$
$\displaystyle=\sum_{k-4\leq|\alpha|\leq
k}g^{\alpha}_{1}(m_{\alpha},\Pi^{\nabla^{2\\!}}_{k}v)_{K}$ $\displaystyle\leq
h_{K}^{-1}\|g_{1}\|_{K}\|m_{\alpha}\|_{K}\|\Pi^{\nabla^{2\\!}}_{k}v\|_{K}\approx\|g_{1}\|_{K}\|\Pi^{\nabla^{2\\!}}_{k}v\|_{K}$

with the observation $\|m_{\alpha}\|_{K}\approx h_{K}$ in the last step. Since
$\Pi^{\nabla^{2\\!}}_{k}v$ is uniquely determined by the DoFs of $v$ and
$\mathrm{Dof}^{k,\mathrm{nc}}(v)=\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})$, we
can conclude that $\Pi^{\nabla^{2\\!}}_{k}v=\Pi^{\nabla^{2\\!}}_{k}\tilde{v}$.
This and the triangle inequality show
$\|\Pi^{\nabla^{2\\!}}_{k}v\|_{K}=\|\Pi^{\nabla^{2\\!}}_{k}\tilde{v}\|_{K}\leq\|\Pi^{\nabla^{2\\!}}_{k}\tilde{v}-\tilde{v}\|_{K}+\|\tilde{v}\|_{K}$.
The Poincaré–Friedrichs inequality and the inverse inequality provide
$\|\Pi^{\nabla^{2\\!}}_{k}\tilde{v}-\tilde{v}\|_{K}\lesssim\|\tilde{v}\|_{K}$.
These bounds together with Lemma 6.5 establish

$\displaystyle\|\Pi^{\nabla^{2\\!}}_{k}v\|_{K}$
$\displaystyle\lesssim\|\tilde{v}\|_{K}\lesssim
h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}(\tilde{v})\|_{\ell^{2}}=h_{K}\|\mathrm{Dof}^{k,\mathrm{nc}}({v})\|_{\ell^{2}},$

and this observation concludes the proof of norm equivalence. ∎

###  6.3. Modified companion map

In this subsection, we consider the lowest-order ($k=2$) nonconforming VE
space $V_{h}^{2,\mathrm{nc}}$ and the aim is to modify the companion operator
$J_{1}v_{h}$ for $v_{h}\in V_{h}^{2,\mathrm{nc}}$ from Theorem 5.1 so that the
new companion $J_{2}^{*}v_{h}$ satisfies the $H^{1}$-orthogonality
$\nabla(v_{h}-J_{2}^{*}v_{h})\perp(\mathbb{P}_{0}(\mathcal{T}_{h}))^{2}$ in
addition to the $H^{2}$\- and $L^{2}$-orthogonalities established in Theorem
5.2(b)-5.2(d).

If $e\in\mathcal{E}^{i}$ is an interior edge, we assume that it’s shared by
two triangles $T^{+}\subset K^{+}$ and $T^{-}\subset K^{-}$ inside two
neighbouring polygons $K^{+}$ and $K^{-}$, and set $\omega_{e}:=K^{+}\cup
K^{-}$, and if $e\in\mathcal{E}^{b}$ is a boundary edge, we assume that it
only belongs to a triangle $T^{+}\subset K^{+}$ and set $\omega_{e}:=K^{+}$.
Let $\psi_{e},\phi_{e}\in H^{2}_{0}(T^{+}\cup T^{-})$ be two edge bubble-
functions from [10, 21] satisfying the following properties:

  * •

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\psi_{e}\;\,\mbox{d}s=1,|\psi_{e}|_{2,T^{\pm}}\approx
h_{e}^{-1}$,

  * •

$\phi_{e}\equiv 0$ on $\partial T^{\pm}$,
$\int_{e}\partial_{\boldsymbol{n}}\phi_{e}\,\mbox{d}s=1,|\phi_{e}|_{2,T^{\pm}}\approx
h_{e}^{-1}$.

Step 1. Given $J_{1}v_{h}$ and the bubble-function $\psi_{e}$, define

(6.7) $\displaystyle
J_{1}^{*}v_{h}:=J_{1}v_{h}+\sum_{e\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}(v_{h}-J_{1}v_{h})\,\mbox{d}s\Big{)}\psi_{e}\in
V.$

Observe from the definition of $J_{1}$ and $\psi_{e}$ that
$J_{1}^{*}v_{h}(z)=v_{h}(z)$ for any $z\in\mathcal{V}$ 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}(v_{h}-J_{1}^{*}v_{h})\,\mbox{d}s=0$ for any
$e\in\mathcal{E}$. The Cauchy–Schwarz inequality and the scaling of $\psi_{e}$
from above show that

$\displaystyle\Big{|}\Big{(}\mathchoice{{\vbox{\hbox{$\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}-J_{1}v_{h})\,\mbox{d}s\Big{)}\Big{|}\|\psi_{e}\|_{\omega_{e}}\lesssim
h_{e}^{-3/2}\|v_{h}-J_{1}v_{h}\|_{e}$ $\displaystyle\qquad\lesssim
h_{e}^{-2}\|v_{h}-J_{1}v_{h}\|_{\omega_{e}}+h_{e}^{-1}|v_{h}-J_{1}v_{h}|_{1,\omega_{e}}\lesssim|v_{h}-J_{1}v_{h}|_{2,\omega_{e}}$

with the trace inequality and Theorem 5.1 in the last two steps. This proves
that $|v_{h}-J_{1}^{*}v_{h}|_{2,h}\lesssim|v_{h}-J_{1}v_{h}|_{2,h}$. Similarly
the scaling $|\psi_{e}|_{1,T^{\pm}}\approx 1$ and Theorem 5.1(d) result in
$|v_{h}-J_{1}^{*}v_{h}|_{1,h}\lesssim h|v_{h}-J_{1}v_{h}|_{2,h}$.

Step 2. Given $J_{1}^{*}v_{h}$ and the bubble-function $\phi_{e}$, define

(6.8) $\displaystyle
J_{1}^{**}v_{h}:=J_{1}^{*}v_{h}+\sum_{e\in\mathcal{E}}\Big{(}\int_{e}\partial_{\boldsymbol{n}}(v_{h}-J_{1}^{*}v_{h})\,\mbox{d}s\Big{)}\phi_{e}\in
V.$

Since $\phi_{e}|_{e}\equiv 0$, we have $J_{1}^{**}v_{h}(z)=v_{h}(z)$ for any
$z\in\mathcal{V}$ 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}(v_{h}-J_{1}^{**}v_{h})\,\mbox{d}s=0$ for any
$e\in\mathcal{E}$. Note from
$\int_{e}\partial_{\boldsymbol{n}}\phi_{e}\,\mbox{d}s=1$ that
$\int_{e}\partial_{\boldsymbol{n}}(v_{h}-J_{1}^{**}v_{h})\,\mbox{d}s=0$. Again
as in Step 1, it is easy to prove that

$\displaystyle\Big{|}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{e}\partial_{\boldsymbol{n}}(v_{h}-J_{1}^{*}v_{h})\,\mbox{d}s\Big{)}\Big{|}\|\phi_{e}\|_{\omega_{e}}\lesssim|v_{h}-J_{1}^{*}v_{h}|_{2,\omega_{e}},$

and consequently,

$|v_{h}-J_{1}^{**}v_{h}|_{2,h}\lesssim|v_{h}-J_{1}^{*}v_{h}|_{2,h}\lesssim|v_{h}-J_{1}v_{h}|_{2,h}.$

Step 3. Next we construct the operator $J_{2}^{*}$ and the design employs the
tools from the construction of $J_{2}$. Recall the element bubble-function
$b_{h}|_{K}=b_{K}\in H^{2}_{0}(K)$ and suppose $v_{2}\in\mathbb{P}_{2}(K)$ is
the Riesz representative of the linear functional
$\mathbb{P}_{2}(K)\to\mathbb{R},$ defined by,
$w_{2}\mapsto(v_{h}-J_{1}^{**}v_{h},w_{2})_{K}$, for
$w_{2}\in\mathbb{P}_{2}(K)$ in the Hilbert space $\mathbb{P}_{2}(K)$ endowed
with the weighted scalar product $(b_{K}\bullet,\bullet)_{K}$. Given $v_{h}\in
V_{h}^{2,\mathrm{nc}}$, the function
$\tilde{v}_{2}\in\mathbb{P}_{2}(\mathcal{T}_{h})$ with
$\tilde{v}_{2}|_{K}:=v_{2}$ satisfy
$(b_{h}\tilde{v}_{2},w_{2})_{\Omega}=(v_{h}-J_{1}^{**}v_{h},w_{2})_{\Omega}$
for all $w_{2}\in\mathbb{P}_{2}(\mathcal{T}_{h})$ and define

(6.9) $\displaystyle J_{2}^{*}v_{h}:=J_{1}^{**}v_{h}+b_{h}\tilde{v}_{2}\in V.$

######  Theorem 6.7.

The modified conforming companion operator $J_{2}^{*}$ satisfies the following
properties.

  1. (a)

$\mathrm{dof}_{j}^{2,\mathrm{nc}}(J_{2}^{*}v_{h})=\mathrm{dof}_{j}^{2,\mathrm{nc}}(v_{h})$
for all $j=1,\dots,N_{2}^{2,\mathrm{nc}}$,

  2. (b)

$\nabla^{2}(v_{h}-J_{2}^{*}v_{h})\perp(\mathbb{P}_{0}(\mathcal{T}_{h}))^{2\times
2}$ in $(L^{2}(\Omega))^{2\times 2}$,

  3. (c)

$\nabla(v_{h}-J_{2}^{*}v_{h})\perp(\mathbb{P}_{0}(\mathcal{T}_{h}))^{2}$ in
$(L^{2}(\Omega))^{2}$,

  4. (d)

$v_{h}-J_{2}^{*}v_{h}\perp\mathbb{P}_{2}(\mathcal{T}_{h})$ in $L^{2}(\Omega)$,

  5. (e)

$\displaystyle\sum_{j=0}^{2}h^{j-2}|v_{h}-J_{2}^{*}v_{h}|_{j,h}\lesssim\inf_{\chi\in\mathbb{P}_{2}(\mathcal{T}_{h})}|v_{h}-\chi|_{2,h}+\inf_{v\in
V}|v_{h}-v|_{2,h}.$

###### Proof.

We provide only the proof of modified property 6.7(c) and the remaining
properties follow analogously as in the proof of Theorem 5.2. Since $b_{K}=0$
on $\partial K$, it results from that definition of $J_{2}^{*}$ and
$J_{1}^{**}$ that $\mathchoice{{\vbox{\hbox{$\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_{2}^{*}v_{h}\,\mbox{d}s=\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_{1}^{**}v_{h}\,\mbox{d}s=\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}\,\mbox{d}s$. Hence, for
$\vec{\chi}\in(\mathbb{P}_{0}(K))^{2}$ and for any $K\in\mathcal{T}_{h}$, an
integration by parts shows

$(\nabla(v_{h}-J_{2}^{*}v_{h}),\vec{\chi})_{K}=-(v_{h}-J_{2}^{*}v_{h},\text{div}(\vec{\chi}))_{K}+(v_{h}-J_{2}^{*}v_{h},\vec{\chi}\cdot\boldsymbol{n}_{K})_{\partial
K}=0,$

and therefore it concludes the proof of Theorem 6.7(c). ∎

###  6.4. Reliability

Recall $(u_{h},p_{h})$ is the nonconforming VE solution to the discrete
problem and the notation
$T(\bullet)=\partial_{\boldsymbol{n}}(\Delta\bullet+\partial_{\boldsymbol{\tau}\boldsymbol{\tau}}\bullet)$.
We proceed to define the local contributions for the estimator

$\displaystyle\eta_{1,K}^{2}$
$\displaystyle:=h_{K}^{4}\|\tilde{f}-\Pi_{k}\tilde{f}\|_{K}^{2}+h_{K}^{4}\|\tilde{f}-\Delta^{2}\Pi^{\nabla^{2\\!}}_{k}u_{h}-\Pi_{k}u_{h}-\alpha\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}\|_{K}^{2},$ $\displaystyle\eta_{2,K}^{2}$
$\displaystyle:=h_{K}^{2}\|\tilde{g}-\Pi_{\ell}\tilde{g}\|_{K}^{2}+h_{K}^{2}\|\tilde{g}+\gamma\nabla\cdot\Pi_{\ell-1}\nabla
p_{h}-\beta\Pi_{\ell}p_{h}+\alpha\nabla\cdot\Pi_{k-1}\nabla u_{h}\|^{2}_{K},$
$\displaystyle\eta_{3,K}^{2}$
$\displaystyle:=\sum_{e\in\mathcal{E}^{i}_{K}\cup\mathcal{E}^{s}_{K}}h_{e}\|[\partial_{\boldsymbol{n}\boldsymbol{n}}(\Pi^{\nabla^{2\\!}}_{k}u_{h})]\|^{2}_{e},$
$\displaystyle\eta_{4,K}^{2}$
$\displaystyle:=\sum_{e\in\mathcal{E}^{i}_{K}}h_{e}^{3}\|[T(\Pi^{\nabla^{2\\!}}_{k}u_{h})+\alpha\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}]\|^{2}_{e},$ $\displaystyle\eta_{5,K}^{2}$
$\displaystyle:=\sum_{e\in\mathcal{E}^{i}_{K}\cup\mathcal{E}^{c}_{K}}h_{e}\|[\alpha\Pi_{k-1}\nabla
u_{h}\cdot\boldsymbol{n}+\gamma\Pi_{\ell-1}\nabla
p_{h}\cdot\boldsymbol{n}]\|^{2}_{e},$ $\displaystyle\eta_{6,K}^{2}$
$\displaystyle:=(1+\alpha^{1/2}h_{K}+h_{K}^{2})^{2}S_{\nabla^{2}}^{K}((1-\Pi^{\nabla^{2\\!}}_{k})u_{h},(1-\Pi^{\nabla^{2\\!}}_{k})u_{h})$
$\displaystyle\qquad+S_{\nabla}^{K}((1-\Pi^{\nabla}_{\ell})p_{h},(1-\Pi^{\nabla}_{\ell})p_{h})+S_{2,0}^{K}((1-\Pi_{\ell})p_{h},(1-\Pi_{\ell})p_{h}),$
$\displaystyle\eta_{7,K}^{2}$
$\displaystyle:=\alpha\|\text{Dof}^{k,\rm{nc}}(u_{h}-\Pi^{\nabla}_{\ell}u_{h})\|^{2}_{\ell^{2}},$
$\displaystyle\eta_{8,K}^{2}$
$\displaystyle:=\sum_{e\in\mathcal{E}_{K}}h_{e}^{-1}(\|[\nabla\Pi^{\nabla^{2\\!}}_{k}u_{h}]\|^{2}_{e}+\|[\Pi_{\ell}p_{h}]\|^{2}_{e}).$

Denote $\eta_{i}^{2}:=\sum_{K\in\mathcal{T}_{h}}\eta_{i,K}^{2}$ for
$i=1,\dots,8$ and the following theorem shows that the sum of these
contributions form an upper bound for the error in energy norm.

######  Theorem 6.8.

With the aforementioned notation, there holds, for $k=2$ and $\ell=1$,

$\|\vec{\boldsymbol{u}}-\vec{\boldsymbol{u}}_{h}\|_{\mathbf{H}_{\epsilon}^{h}}^{2}\lesssim\eta^{2}:=\sum_{i=1}^{8}\eta_{i}^{2}.$

###### Proof.

Let $J\vec{\boldsymbol{u}}_{h}:=(J_{2}^{*}u_{h},J_{4}p_{h})$ and
$\vec{\boldsymbol{e}}:=\vec{\boldsymbol{u}}-J\vec{\boldsymbol{u}}_{h}:=(e^{u},e^{p})\in\mathbf{H}_{\epsilon}$
with $\vec{\boldsymbol{e}}_{I}:=(e_{I}^{u},e_{I}^{p})$. Even though
$J_{2}^{*}$ is constructed for the lowest-order case ($k=2$), we prefer to
write the proof with the notation $k$ and $\ell$ to point out the challenges
for general values. The coercivity of the continuous bilinear form
$\mathcal{A}$ leads to

$\displaystyle\|\vec{\boldsymbol{e}}\|^{2}_{\mathbf{H}_{\epsilon}}$
$\displaystyle\lesssim\mathcal{A}(\vec{\boldsymbol{u}},\vec{\boldsymbol{e}})-\mathcal{A}(J\vec{\boldsymbol{u}}_{h},\vec{\boldsymbol{e}})=\mathcal{F}(\vec{\boldsymbol{e}})-\mathcal{F}_{h}(\vec{\boldsymbol{e}}_{I})+\mathcal{A}_{h}(\vec{\boldsymbol{u}}_{h},\vec{\boldsymbol{e}}_{I})-\mathcal{A}(J\vec{\boldsymbol{u}}_{h},\vec{\boldsymbol{e}}).$

The identities
$(\Pi_{k}\tilde{f},e_{I}^{u}-J_{2}^{*}e_{I}^{u})_{\Omega}=0=(\Pi_{\ell}\tilde{g},e_{I}^{p}-J_{4}e_{I}^{p})_{\Omega}$
from Theorems 6.7.d and 5.4.c, and the $L^{2}$-orthogonality of $\Pi_{k}$ and
$\Pi_{\ell}$ show

$\displaystyle\mathcal{F}(\vec{\boldsymbol{e}})-\mathcal{F}_{h}(\vec{\boldsymbol{e}}_{I})=(\tilde{f},e^{u})_{\Omega}-(\Pi_{k}\tilde{f},J_{2}^{*}e_{I}^{u})_{\Omega}+(\tilde{g},e^{p})_{\Omega}-(\Pi_{\ell}\tilde{g},J_{4}e_{I}^{p})_{\Omega}$
$\displaystyle=(\tilde{f},v)_{\Omega}+(\tilde{f}-\Pi_{k}\tilde{f},J_{2}^{*}e_{I}^{u}-\Pi^{\nabla^{2\\!}}_{k}e_{I}^{u})_{\Omega}+(\tilde{g},q)_{\Omega}+(\tilde{g}-\Pi_{\ell}\tilde{g},J_{4}e_{I}^{p}-\Pi^{\nabla}_{\ell}e_{I}^{p})_{\Omega}$

with $v=e^{u}-J_{2}^{*}e_{I}^{u}$ and $q=e^{p}-J_{4}e_{I}^{p}$. The
definitions of $\mathcal{A}_{h}$ and $\mathcal{A}$ provide

$\displaystyle\mathcal{A}_{h}(\vec{\boldsymbol{u}}_{h},\vec{\boldsymbol{e}}_{I})-\mathcal{A}(J\vec{\boldsymbol{u}}_{h},\vec{\boldsymbol{e}})=(a_{1}^{h}(u_{h},e_{I}^{u})-a_{1}(J_{2}^{*}u_{h},e^{u}))+(a_{2}(J_{4}p_{h},e^{u})-a_{2}^{h}(p_{h},e_{I}^{u}))$