# A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley
FEM with original and modified right-hand sides

Carsten Carstensen111Department of Mathematics, Humboldt-Universität zu
Berlin, 10099 Berlin, Germany. Distinguished Visiting Professor, Department of
Mathematics, Indian institute of Technology Bombay, Powai, Mumbai-400076,
India<EMAIL_ADDRESS>and Neela Nataraj222Department of Mathematics,
Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.
<EMAIL_ADDRESS>
###### Abstract

This article on nonconforming schemes for $m$ harmonic problems simultaneously
treats the Crouzeix-Raviart ($m=1$) and the Morley finite elements ($m=2$) for
the original and for modified right-hand side $F$ in the dual space
$V^{*}:=H^{-m}(\Omega)$ to the energy space $V:=H^{m}_{0}(\Omega)$. The
smoother $J:V_{\mathrm{nc}}\to V$ in this paper is a companion operator, that
is a linear and bounded right-inverse to the nonconforming interpolation
operator $I_{\mathrm{nc}}:V\to V_{\mathrm{nc}}$, and modifies the discrete
right-hand side $F_{h}:=F\circ J\in V_{\mathrm{nc}}^{*}$. The best-
approximation property of the modified scheme from Veeser et al. (2018) is
recovered and complemented with an analysis of the convergence rates in weaker
Sobolev norms. Examples with oscillating data show that the original method
may fail to enjoy the best-approximation property but can also be better than
the modified scheme. The a posteriori analysis of this paper concerns data
oscillations of various types in a class of right-hand sides $F\in V^{*}$. The
reliable error estimates involve explicit constants and can be recommended for
explicit error control of the piecewise energy norm. The efficiency follows
solely up to data oscillations and examples illustrate this can be
problematic.

Keywords: nonconforming schemes, Crouzeix-Raviart, Morley, biharmonic problem,
medius error analysis, best-approximation, a priori and a posteriori analysis

AMS Classification: 65N30, 65N12, 65N50

##  1 Introduction

The beginning of the nonconforming finite element methodology, its motivating
examples in fluid mechanics, and its connections with mixed formulations are
outlined in [7] together with the relevant literature that (partly) led to the
Strang lemmas (also named after Strang-Fix and Strang-Berger etc.) in finite
element textbooks. This article aims at a simple a priori and a posteriori
error analysis for $m$ harmonic problems, that involve the energy space
$V:=H^{m}_{0}(\Omega)$ endowed with the energy scalar product $a:V\times
V\to\mathbb{R}$. The emphasis is on the lowest-order nonconforming schemes
with the piecewise polynomial, nonconforming discrete space
$V_{\mathrm{nc}}\subset P_{m}(\mathcal{T})$, that simultaneously describes the
Crouzeix-Raviart (CR) for $m=1$ and Morley finite element methods (FEM) for
$m=2$. Those nonconforming FEMs allow for an interpolation operator
$I_{\mathrm{nc}}:V\to V_{\mathrm{nc}}$ with the best-approximation property
and interpolation error estimates with explicit (and small) constants (called
$\kappa_{m}$ throughout) from [17, 16] for $m=1$ and [12] for $m=2$. The word
Strang lemma is no longer used in this paper and a companion operator
$J:V_{\mathrm{nc}}\to V$ circumvents the medius analysis from [38].

$T$

$T$

Figure 1: Crouzeix-Raviart FEM (left) and Morley FEM (right)

A (conforming) companion operator $J:V_{\mathrm{nc}}\to V$ is defined in this
paper as a right-inverse of the interpolation operator $I_{\mathrm{nc}}:V\to
V_{\mathrm{nc}}$ with additional benefits such as the $L^{2}$ orthogonality
$v_{\mathrm{nc}}-Jv_{\mathrm{nc}}\perp P_{m}(\mathcal{T})$ for all
$v_{\mathrm{nc}}\in V_{\mathrm{nc}}$. The extra $L^{2}$ orthogonality is not
needed for the sole best-approximation [51, 53, 52], but it enables a duality
argument for error estimates in weaker Sobolev norms [15].

The paper departs with a revisit of the medius analysis [38], the best-
approximation results from [51, 53, 52] and recovers the same best-
approximation constant $C_{\rm qo}$ for a modified right-hand side $F_{h}$ in
Section 3. It thereafter contributes a comprehensive analysis of convergence
rates in Sobolev norms weaker and stronger than the energy norm in $V$ in
Section 4.

The precise convergence rate $\min\\{2\sigma,m+\sigma-s\\}$ is guaranteed for
quasi-uniform meshes for the error $u-Ju_{\mathrm{nc}}$ of the post-processed
approximation $Ju_{\mathrm{nc}}$ in the norm of $H^{s}(\Omega)$ for $-m\leq
s<m+{1/2}$ with the positive index $\sigma\leq 1$ in terms of elliptic
regularity. This holds under the generic assumption that the convergence rate
in the energy norm is exactly $\sigma<m$ and so excludes untypical smooth
solutions. The only former result is the seminal contribution [39] with a
proof that the convergence rate in the $L^{2}$ error in nonconforming schemes
is no better than $2\sigma$ not only for one model scenario but for many other
nonconforming schemes.

The paper will focus on shape-regular triangulations of the planar bounded
polygonal Lipschitz domain $\Omega\subset{\mathbb{R}}^{2}$ into triangles for
simplicity. The abstract analysis holds verbatim for any space dimensions once
the companion operator $J$ is designed. This started in 2D for $m=1$ with [15]
for the Crouzeix-Raviart finite element method (CRFEM) and is almost
immediately generalized to any space dimension. For $m=2$ and the Morley FEM
it started in the PhD thesis [34] and details can be found in [35, 53] in 2D,
while the companion operator $J$ for generalization of the Morley FEM from
[44] is designed in [25] in 3D. The most refined analysis in 2D with explicit
constants and the connection to a posteriori error estimates are summarized in
[18, 26]. The paper will focus on lowest-order schemes because, first, the
interpolation operators are less clear for higher-order CRFEM in 2D and 3D
[27], hence, second, conforming companions are unknown. In fact, there are two
aspects that make the nonconforming FEM attractive in comparison to conforming
FEM: The interpolation operator is local (simplex-oriented) and defined for
any function in the energy space. Those advantages are possible for simplicial
meshes in any space dimension, but not available for all higher polynomial
degrees.

The modified nonconforming FEM specifies the discrete right-hand side
$F_{h}:=F\circ J\in V_{\mathrm{nc}}^{*}$ with the companion operator
$J:V_{\mathrm{nc}}\to V$ as a smoother. Thereby the modified version allows
for general data $F\in V^{*}$. Modifications of the right-hand side $F_{h}$ of
this type had been introduced in [2, 8], before [51, 53, 52] systematically
studied the resulting best-approximation property and established the name
smoother. The modified FEM is a Petrov-Galerkin scheme discussed in Section 3
and hence quasi-optimally convergent (once stable). If the source term $F\in
V^{*}\setminus L^{2}(\Omega)$ is not a Lebesgue function, the application of
the original discrete problem is less obvious. This paper studies a class of
right-hand sides $F$ in $V^{*}$ that includes point forces for $m=2$ in the
model example to define $F_{h}$ (without any smoother) adapted to the data.
This provokes a comparison of the nonconforming FEM with the two different
discrete right-hand sides with advantages for both sides. The model example is
rich enough to allow for a counterexample for the best-approximation of the
original method. The latter holds up to the data oscillations only; cf. [24]
for $m=1$ and $F\in L^{2}(\Omega)$. Conversely, another example illustrates
that the modified nonconforming method (with the discrete right-hand side
$F\circ J$) can be worse than the original discretization.

The a posteriori error control for the (original) nonconforming FEM and $F\in
L^{2}(\Omega)$ started with [31] and continued with [10, 9, 19, 20, 50, 11,
14] for $m=1$ and [41, 40, 13, 5] for $m=2$. The initial work was based on a
Helmholtz decomposition of the piecewise energy norm of the nonconforming
error. The Helmholtz decompositions become Hodge decompositions for multiply-
connected domains and are not all known for 3D. Their discrete counterparts
from [3] for $m=1$ and [14] for $m=2$ are not known for all domains and space
dimensions. That motivates the use of other techniques and this paper suggests
a paradigm shift: Utilize the companion operator $J$ for a split in the
conforming error $u-Ju_{\mathrm{nc}}$ and in the nonconforming error
$u_{\mathrm{nc}}-Ju_{\mathrm{nc}}$. This approach leads to a posteriori error
estimates even if a Helmholtz decomposition is unknown and without a piecewise
integration by parts. As a consequence, the error estimators do not contain
weighted jumps of the normal components of derivatives of the discrete
solution $u_{\mathrm{nc}}$ across interior edges. On the other hand, the
efficiency requires some other additional benefit of the companion operator,
which actually motivated its first design in a posteriori error control. Rate-
optimal adaptive nonconforming FEM are analyzed in [25, 15, 4, 14, 22, 47] and
the references therein. Amongst all second-order schemes, the Morley FEM
appears to be the most simple and method of choice for fourth-order problems
[21]. Hence the generalization of the methodology in this article for $m\geq
3$ with nonconforming finite elements from [55] appears appealing.

Outline of the structure. The remaining parts of this paper are organised as
follows. In Section 2 nonconforming schemes, interpolation and companion
operators, and examples are presented. The best-approximation results
available in literature are reviewed and an abstract best-approximation result
is established for data $F\in H^{-m}(\Omega)$ in Section 3. Section 4 presents
lower-order error estimates and precise convergence rates under further
hypotheses. Section 5 discusses examples of data in $V^{*}$, an alternate
choice of the right-hand side functional that leads to a modified scheme and a
comparison of the original and the modified scheme. This is followed by an a
posteriori analysis in Section 6.

General notation. Standard notation of Lebesgue and Sobolev spaces, their
norms, and $L^{2}$ scalar products applies throughout the paper such as the
abbreviation $\|\bullet\|$ for $\|\bullet\|_{L^{2}(\Omega)}$. For a shape-
regular triangulation $\mathcal{T}$ of $\Omega$ into triangles, the product
space $\prod_{T\in\mathcal{T}}H^{s}(T)$ denoted as $H^{s}(\mathcal{T})$ is
defined as $\\{v_{\mathrm{pw}}\in L^{2}(\Omega):\forall
T\in\mathcal{T},\;v_{\mathrm{pw}}|_{T}\in H^{s}(T)\\}$ and is equipped with
the Euclid norm of those contributions $\|\bullet\|_{H^{s}(T)}$ for all
$T\in\mathcal{T}$. The triple norm
$|\\!|\\!|\bullet|\\!|\\!|:=|\bullet|_{H^{m}(\Omega)}$ is the energy norm and
$|\\!|\\!|\bullet|\\!|\\!|_{\text{pw}}:=|\bullet|_{H^{m}(\mathcal{T})}:=\|D^{m}_{\text{pw}}\bullet\|$
is its piecewise version with the piecewise partial derivatives
$D_{\text{pw}}^{m}$ of order $m\in{\mathbb{N}}$. For non-integer real
${\boldmath{t}}=\ell+s$ with $\ell\in{\mathbb{N}}_{0}$ and $0<s<1$, the 2D
Sobolev-Slobodeckii semi-norm [37] of $f\in H^{t}(\Omega)$ reads

$\displaystyle|f|_{H^{\boldmath{t}}(\Omega)}:=\left(\sum_{|\beta|=\ell}\int_{\Omega}\int_{\Omega}\frac{|\partial^{\beta}f(x)-\partial^{\beta}f(y)|^{2}}{|x-y|^{2+2s}}{\rm\,dx}{\rm\,dy}\right)^{1/2}.$
(1.1)

The vector space of piecewise polynomials of at most degree $k\in{\mathbb{N}}$
is denoted by $P_{k}(\mathcal{T})$,
$S^{k}(\mathcal{T}):=P_{k}(\mathcal{T})\cap C^{0}({\Omega})\subset
H^{1}(\Omega)$, and $S^{k}_{0}(\mathcal{T}):=P_{k}(\mathcal{T})\cap
C^{0}({\Omega})\subset H^{1}_{0}(\Omega)$ includes the homogeneous boundary
conditions. For a Lebesgue-measurable set $\omega\subseteq{\mathbb{R}}^{n}$
and a Lebesgue-measurable function $v:\omega\rightarrow X$ with values in a
finite-dimensional real Banach space $X$, the integral with respect to the
$d$-dimensional Lebesgue measure is denoted by
$\fint_{\omega}v{\rm\,dx}:=1/|\omega|\int_{\omega}v{\rm\,dx}$. The integral
over a $(n-1)$-dimensional hypersurface $\Gamma$ with respect to the
$(n-1)$-dimensional Hausdorff measure reads as $\int_{\Gamma}v{\rm\,ds}.$ The
notation $A\lesssim B$ abbreviates $A\leq CB$ for some positive generic
constant $C$, which depends on the solution $u$; $A\approx B$ abbreviates
$A\lesssim B\lesssim A$. The set of all $2\times 2$ real symmetric matrices is
denoted by ${\mathbb{S}}$. The symmetric part of $A\in{\mathbb{R}}^{2\times
2}$ is denoted by $\text{sym }(A)$. That is, $\text{sym }(A)=1/2(A+A^{T})$.
For any differentiable scalar function $v$ resp. vector field
$\Phi=(\phi_{1},\phi_{2})^{T}$,

${\rm Curl}\,v=\begin{pmatrix}-\frac{\partial v}{\partial x_{2}}\\\
\frac{\partial v}{\partial x_{1}}\end{pmatrix}\quad\text{resp.}\quad{\rm
Curl}\>\Phi:={\rm Curl}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=\begin{pmatrix}-\frac{\partial\phi_{1}}{\partial
x_{2}}&\frac{\partial\phi_{1}}{\partial x_{1}}\\\
-\frac{\partial\phi_{2}}{\partial x_{2}}&\frac{\partial\phi_{2}}{\partial
x_{1}}\end{pmatrix}.$

The symbol $D^{1}v:=\nabla v$ (resp. $\nabla_{\mathrm{pw}}v$) denotes the
gradient (resp. piecewise gradient) of a scalar function $v$ and
$\boldsymbol{\sigma}:=D^{2}u$ (resp. $D^{2}_{\mathrm{pw}}u$) denotes the
Hessian (piecewise Hessian) of $v$ with the four entries $\sigma_{11}$,
$\sigma_{12}$, $\sigma_{21}$, and $\sigma_{22}$ in the form of a $2\times 2$
matrix.

##  2 Nonconforming schemes

###  2.1 Continuous model problem

Suppose $u\in V:=H^{m}_{0}(\Omega)$ solves the $m$-harmonic equation
$(-1)^{m}\Delta^{m}u=F$ for a given right-hand side $F\in V^{*}\equiv
H^{-m}(\Omega)$ in a planar bounded Lipschitz domain
$\Omega\subset\mathbb{R}^{2}$ with polygonal boundary $\partial\Omega$,
$m\in{\mathbb{N}}$. The weak form of this equation reads

$\displaystyle a(u,v)=F(v)\quad\text{for all }v\in V$ (2.1)

with $a(v,w):=\int_{\Omega}D^{m}v:D^{m}w{\rm\,dx}$ for all $v,w\in V$. The
examples in this paper are merely for $m=1,2$, but the abstract framework can
be written for any $m\in{\mathbb{N}}$. It is well known that (2.1) has a
unique solution $u$ and elliptic regularity [1, 6, 36, 46] holds in the sense
that $F\in H^{-s}(\Omega)$ implies $u\in H^{2m-s}(\Omega)$ for all
$m-\sigma\leq s\leq m$ with the positive index of elliptic regularity
$\sigma$. The latter is well established for $m=1,2$ and an overall hypothesis
in this paper establishes for other $m$. The lowest-order nonconforming finite
element schemes suggest linear convergence rates in the energy norm only
$|\\!|\\!|\bullet|\\!|\\!|:=a(\bullet,\bullet)^{1/2}\approx\|\bullet\|_{V}$
and so it is sufficient to assume throughout this paper that the quantity
$\sigma\leq 1$ is the minimum of the index of elliptic regularity and one,
whence $0<\sigma\leq 1$ depends exclusively on $\Omega$ and $m$. The index of
elliptic regularity $\sigma=1$ follows for a convex polygonal domain $\Omega$
while $1/2<\sigma<1$ characterizes a nonconvex polygonal domain $\Omega$ [1,
6, 36, 46].

###  2.2 Nonconforming discretisation

Suppose that $V_{\mathrm{nc}}\subset P_{m}(\mathcal{T})$ is a nonconforming FE
space based on a shape-regular triangulation $\mathcal{T}$ of $\Omega$ into
triangles. Let the semi-scalar product $a_{\mathrm{pw}}$ be defined by the
piecewise differential operator $D^{m}_{\mathrm{pw}}$ (the gradient for $m=1$
and the Hessian for $m=2$)

$a_{\mathrm{pw}}(v_{\mathrm{pw}},w_{\mathrm{pw}}):=\sum_{T\in\mathcal{T}}\int_{T}D^{m}v_{\mathrm{pw}}:D^{m}w_{\mathrm{pw}}{\rm\,dx}\quad\text{for
all }v_{\mathrm{pw}},w_{\mathrm{pw}}\in H^{m}(\mathcal{T}),$

that induces a piecewise $H^{m}$ seminorm
$|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}=a_{\mathrm{pw}}(\bullet,\bullet)^{1/2}$
that is also a norm in $V_{\mathrm{nc}}$. Then
$(V_{\mathrm{nc}},a_{\mathrm{pw}})$ is a (finite-dimensional) Hilbert space so
that, given any $F_{h}\in V_{\mathrm{nc}}^{*}$, there exists a unique discrete
solution $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to

$\displaystyle
a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})=F_{h}(v_{\mathrm{nc}})\quad\text{for
all }v_{\mathrm{nc}}\in V_{\mathrm{nc}}.$ (2.2)

The nonconforming FEM is accompanied with an interpolation operator
$I_{\mathrm{nc}}:V\rightarrow V_{\mathrm{nc}}$ and a best-approximation
property in $P_{m}(\mathcal{T})$, i.e.,

$\displaystyle a_{\mathrm{pw}}(v-I_{\mathrm{nc}}v,w_{m})$
$\displaystyle=0\quad\text{for all }v\in V\text{ and all }w_{m}\in
P_{m}(\mathcal{T}).$ (2.3)

Suppose that there exists a universal constant $\kappa_{m}>0$ with

$\displaystyle\|h_{\mathcal{T}}^{-m}(v-I_{\mathrm{nc}}v)\|$
$\displaystyle\leq\kappa_{m}|\\!|\\!|v-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}\quad\text{for
all }v\in V$ (2.4)

for the $L^{2}$ norm $\|\bullet\|:=\|\bullet\|_{L^{2}(\Omega)}$ and the
piecewise $H^{m}$ seminorm $|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}$ for the
piecewise constant mesh-size $h_{\mathcal{T}}\in P_{0}(\mathcal{T})$ with
$h_{\mathcal{T}}|_{T}:=h_{T}=\text{ diam }(T)$.

######  Remark 2.1 (Pythagoras).

The semi-inner product $a_{\mathrm{pw}}$ in
$H^{m}(\mathcal{T}):=\\{v_{\mathrm{pw}}\in L^{2}(\Omega):\forall
T\in\mathcal{T},\;v_{\mathrm{pw}}|_{T}\in H^{m}(T)\\}$ allows for the concept
of orthogonality: Two piecewise Sobolev functions
$v_{\mathrm{pw}},w_{\mathrm{pw}}\in H^{m}(\mathcal{T})$ are orthogonal if
$a_{\mathrm{pw}}(v_{\mathrm{pw}},w_{\mathrm{pw}})=0$ holds and then the
Pythagoras theorem

$|\\!|\\!|v_{\mathrm{pw}}+w_{\mathrm{pw}}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w_{\mathrm{pw}}|\\!|\\!|_{\mathrm{pw}}^{2}$

follows. An example is the orthogonality of $v-I_{\mathrm{nc}}v$ and
$w_{\mathrm{nc}}$ from (2.3) that guarantees in particular

$\displaystyle|\\!|\\!|v-w_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|v-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w_{\mathrm{nc}}-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}^{2}$
(2.5)

for all $v\in V$ and $w_{\mathrm{nc}}\in V_{\mathrm{nc}}$.

######  Remark 2.2 ($V+V_{\mathrm{nc}}$ is a Hilbert space with inner product
$a_{\mathrm{pw}}$).

Recall the assumption that $|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}$ is a norm
in $V_{\mathrm{nc}}$ and that some $I_{\mathrm{nc}}:V\rightarrow
V_{\mathrm{nc}}$ satisfies (2.3)-(2.4). Then
$|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}$ is a norm in $V+V_{\mathrm{nc}}$.
Since $|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}$ is always a seminorm, the
definiteness has to be clarified: Suppose that
$|\\!|\\!|v+v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=0$ for some $v\in V$ and
$v_{\mathrm{nc}}\in V_{\mathrm{nc}}$. This and
$w_{\mathrm{nc}}:=-v_{\mathrm{nc}}$ imply in (2.5) that
$|\\!|\\!|v-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}=0=|\\!|\\!|v_{\mathrm{nc}}+I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}$.
Since $|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}$ is a norm in
$V_{\mathrm{nc}}$, it follows $v_{\mathrm{nc}}=-I_{\mathrm{nc}}v$. This shows
in (2.4) that $v+v_{\mathrm{nc}}=0$ a.e. as claimed.

Consequently, $V+V_{\mathrm{nc}}$ is a pre-Hilbert space with inner product
$a_{\mathrm{pw}}$ with a complete subspace $V$ and a finite-dimensional
subspace $V_{\mathrm{nc}}$. The completeness of $V+V_{\mathrm{nc}}$ follows
from this by standard arguments with the closed subspace $W:=V\cap
V_{\mathrm{nc}}$ and its orthogonal complement $W^{\perp}$ in
$V+V_{\mathrm{nc}}$: Let $Q$ and $R$ denote the orthogonal projection onto $W$
and $V\cap W^{\perp}$, respectively. Given any $v+v_{\mathrm{nc}}\in
W^{\perp}$ for $v\in V$ and $v_{\mathrm{nc}}\in V_{\mathrm{nc}}$, the
consequence $v+v_{\mathrm{nc}}=(1-Q)v+(1-Q)v_{\mathrm{nc}}$ leads to
$W^{\perp}=(V\cap W^{\perp})\oplus(V_{\mathrm{nc}}\cap W^{\perp})$. The
orthogonal split
$v+v_{\mathrm{nc}}=(1-Q)v+R(1-Q)v_{\mathrm{nc}}+(1-R)(1-Q)v_{\mathrm{nc}}$
leads to $W^{\perp}=(V\cap W^{\perp})\operp(1-R)(V_{\mathrm{nc}}\cap
W^{\perp})$. The space $V+V_{\mathrm{nc}}$ is therefore the orthogonal sum of
the complete subspaces $W$, $V\cap W^{\perp}$, and $(1-R)(V_{\mathrm{nc}}\cap
W^{\perp})$ and so complete.

######  Remark 2.3 (uniqueness and extension of
$I_{\mathrm{nc}}:V+V_{\mathrm{nc}}\rightarrow V_{\mathrm{nc}}$).

Since $(V+V_{\mathrm{nc}},a_{\mathrm{pw}})$ is a Hilbert space, (2.3) shows
that $I_{\mathrm{nc}}:V\rightarrow V_{\mathrm{nc}}$ maps any $v$ to its
(unique) best-approximation $I_{\mathrm{nc}}v$ in $V_{\mathrm{nc}}$. In other
words, the operator $I_{\mathrm{nc}}$ in (2.3)-(2.4) is unique and equal to
the restriction to $V$ of the best-approximation
$I_{\mathrm{nc}}:V+V_{\mathrm{nc}}\rightarrow V_{\mathrm{nc}}$ that is the
orthogonal projection onto $V_{\mathrm{nc}}$ in the Hilbert space
$(V+V_{\mathrm{nc}},a_{\mathrm{pw}})$. Throughout this paper, the symbol
$I_{\mathrm{nc}}$ denotes this orthogonal projection in the extended space
$I_{\mathrm{nc}}:V+V_{\mathrm{nc}}\rightarrow V_{\mathrm{nc}}$ as well as its
restriction to arguments in $V$. Notice that $I_{\mathrm{nc}}=\text{ id }$ in
$V_{\mathrm{nc}}$ and
$I_{\mathrm{nc}}(v+w_{\mathrm{nc}})=I_{\mathrm{nc}}v+w_{\mathrm{nc}}$ for all
$v\in V$ and $w_{\mathrm{nc}}\in V_{\mathrm{nc}}$.

###  2.3 Examples for $m=1$ and $m=2$

For $m=1$ (resp. $m=2$), the nonconforming discretization models the Crouzeix-
Raviart (resp. Morley) FEM. Besides the best approximation property (2.3),
these examples allow for the $L^{2}$ orthogonality

$\displaystyle D^{m}(v-I_{\mathrm{nc}}v)\perp
P_{0}(\mathcal{T};{\mathbb{R}}^{2m}).$ (2.6)

####  2.3.1 Crouzeix-Raviart FEM for the harmonic equation

The set of vertices $\mathcal{V}$ (resp. edges $\operatorname{\mathcal{E}}$)
in the shape-regular triangulation $\mathcal{T}$ of the bounded polygonal
Lipschitz domain $\Omega$ into triangles is divided into the interior vertices
$\mathcal{V}(\Omega)$ (resp. edges $\operatorname{\mathcal{E}}(\Omega)$) and
the vertices $\mathcal{V}(\partial\Omega)$ (resp. edges
$\operatorname{\mathcal{E}}(\partial\Omega)$) on the boundary
$\partial\Omega$. Let $\Pi_{k}$ denote the $L^{2}$ projection onto the space
of piecewise polynomials ${P}_{k}(\mathcal{T})$ of total degree at most
$k\in{\mathbb{N}}_{0}$. The mesh-size $h_{\mathcal{T}}\in P_{0}(\mathcal{T})$
is defined by $h_{\mathcal{T}}|_{T}:=h_{T}:=|T|^{1/2}\approx\text{\rm
diam}(T)$ in any triangle $T\in\mathcal{T}$ of area $|T|$; $|E|$ denotes the
length of an edge $E\in\operatorname{\mathcal{E}}$.

The CR finite element space $V_{\mathrm{nc}}:=\text{CR}^{1}_{0}(\mathcal{T})$
for $m=1$ reads [29]

$\displaystyle{\text{CR}}^{1}(\mathcal{T})$
$\displaystyle:=\big{\\{}v_{\mathrm{CR}}\in
P_{1}(\mathcal{T})\;{{\big{|}}}\;v_{\mathrm{CR}}\text{ is continuous at all
midpoints of interior edges of }\mathcal{T}\big{\\}},$
$\displaystyle{\text{CR}}_{0}^{1}(\mathcal{T})$
$\displaystyle:=\big{\\{}v_{\mathrm{CR}}\in\text{
CR}^{1}(\mathcal{T})\;{{\big{|}}}\;v_{\mathrm{CR}}\text{ vanishes at all
midpoints of boundary edges of }\mathcal{T}\big{\\}}.$

The CR interpolation operator
$I_{\mathrm{nc}}:=I_{\mathrm{CR}}:V\rightarrow\mathrm{CR}^{1}_{0}(\mathcal{T})$
is defined by

$(I_{\mathrm{CR}}v)(\text{mid }(E))=\fint_{E}v{\rm\,ds}\text{ for all
}E\in{\operatorname{\mathcal{E}}}\;\text{ for all }v\in V.$

It satisfies (2.3)-(2.5) and (2.6) with
$\kappa_{1}=\sqrt{{j_{1,1}}^{-2}+1/48}$ for the first positive root $j_{1,1}$
of the Bessel function of the first kind [16, 12], that does not even depend
on the shape of the triangles.

For a simply-connected domain $\Omega$, the discrete Helmholtz decomposition
[3] shows

$P_{0}(\mathcal{T};{\mathbb{R}}^{2})=\nabla_{\mathrm{pw}}\mathrm{CR}^{1}_{0}(\mathcal{T})\operp{\rm
Curl}\>(S^{1}(\mathcal{T})/{\mathbb{R}})$

and, as $h\rightarrow 0$, leads to the Helmholtz decomposition

$L^{2}(\Omega;{\mathbb{R}}^{2})=\nabla H^{1}_{0}(\Omega)\operp{\rm
Curl}\>(H^{1}(\Omega)/{\mathbb{R}}).$

####  2.3.2 Morley FEM for the biharmonic equation

The nonconforming Morley finite element space
$V_{\mathrm{nc}}=\mathrm{M}(\mathcal{T})$ for $m=2$ reads [45]

$\displaystyle{\mathrm{M}}^{\prime}(\mathcal{T})$ $\displaystyle:=$
$\displaystyle\Bigg{\\{}v_{\mathrm{M}}\in
P_{2}(\mathcal{T}){{\Bigg{|}}}\begin{aligned} &\;v_{\mathrm{M}}\text{ is
continuous at the vertices and its normal derivatives }\\\
&\nu_{E}\cdot\nabla_{\text{pw}}{v_{\mathrm{M}}}\text{ are continuous at the
midpoints of interior edges}\end{aligned}\Bigg{\\}},$
$\displaystyle{\mathrm{M}}(\mathcal{T})$ $\displaystyle:=$
$\displaystyle\Bigg{\\{}v_{\mathrm{M}}\in
M^{\prime}(\mathcal{T}){{\Bigg{|}}}\begin{aligned} &\;v_{\mathrm{M}}\text{
vanishes at the vertices of }\partial\Omega\text{ and its normal
derivatives}\\\ &\;\nu_{E}\cdot\nabla{v_{\mathrm{M}}}\text{ vanish at the
midpoints of boundary edges}\end{aligned}\Bigg{\\}}.$

For any $v\in V$, the Morley interpolation operator
$I_{\mathrm{nc}}v:=I_{\mathrm{M}}v\in\mathrm{M}(\mathcal{T})$ is defined by
the degrees of freedom

$(I_{\mathrm{M}}v)(z)=v(z)\text{ for any }z\in\mathcal{V}(\Omega)\text{ and
}\fint_{E}\frac{\partial
I_{\mathrm{M}}v}{\partial\nu_{E}}{\rm\,ds}=\fint_{E}\frac{\partial
v}{\partial\nu_{E}}{\rm\,ds}\text{ for any }E\in\mathcal{E}.$

The Morley interpolation operator
$I_{\mathrm{M}}:V\rightarrow\mathrm{M}(\mathcal{T})$ satisfies (2.3)-(2.5) and
(2.6) [41, 14] with $\kappa_{2}=0.25745784465$ [12] that does not even depend
on the shape of the triangles.

For a simply-connected domain $\Omega$, the discrete Helmholtz decomposition
[14] reads

$P_{0}(\mathcal{T};{\mathbb{S}})=D^{2}_{\mathrm{pw}}\mathrm{M}(\mathcal{T})\operp{\rm
sym\>Curl}\>(S^{1}(\mathcal{T})^{2}/{\mathbb{R}}^{2})$

and, as $h\rightarrow 0$, leads to the Helmholtz decomposition

$L^{2}(\Omega;{\mathbb{R}}^{2})=D^{2}H^{2}_{0}(\Omega)\operp{\rm
sym\>Curl}\>(H^{1}(\Omega;{\mathbb{R}}^{2})/{\mathbb{R}}^{2}).$

###  2.4 Companion operator

The interpolation operator $I_{\mathrm{nc}}:V\rightarrow V_{\mathrm{nc}}$
allows for a right-inverse $J$ with benefits [26]. Recall the conforming
finite element subspace $V_{\rm c}$ of $V$ and suppose that a linear operator
$J:V_{\mathrm{nc}}\rightarrow V$ and a constant $\Lambda_{\mathrm{J}}$ exist
with

$\displaystyle I_{\mathrm{nc}}J=\text{id in }V_{\mathrm{nc}},\text{ i.e.,
}I_{\mathrm{nc}}Jv_{\mathrm{nc}}=v_{\mathrm{nc}},$ (2.7)
$\displaystyle|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|v_{\mathrm{nc}}-v|\\!|\\!|_{\mathrm{pw}},$
(2.8) $\displaystyle v_{\mathrm{nc}}-Jv_{\mathrm{nc}}\perp
P_{m}(\mathcal{T})\quad\text{ in }L^{2}(\Omega)$ (2.9)

for all $v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ and for all $v\in V$.

The companion operator $J$ in this paper (a) allows for an elegant proof (in
Subsection 3.2), (b) provides an appropriate right-hand side
$F_{h}:=F(J\bullet)\in V^{*}_{\mathrm{nc}}$ in (2.1) for general $F\in V^{*}$
with best-approximation (in Subsection 3.3), and (c) controls the
inconsistency term $\min_{v\in
V}|\\!|\\!|u_{\mathrm{nc}}-v|\\!|\\!|_{\mathrm{pw}}$ in the a posteriori error
control (in Section 6).

######  Example 2.1 (companion operators).

For any $v_{\mathrm{CR}}\in CR^{1}_{0}(\mathcal{T})$, there exists
$Jv_{\mathrm{CR}}\in(S^{1}_{0}(\mathcal{T})+P_{4}(\mathcal{T}))\cap V$ [15]
that satisfies (2.7)-(2.9). For any
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$, there exists a
$Jv_{\mathrm{M}}\in(HCT(\mathcal{T})+P_{8}(\mathcal{T}))\cap V$ (with the HCT
finite element space $HCT(\mathcal{T})$) that satisfies (2.7)-(2.9) [26, 34,
35].

######  Remark 2.4 (extra orthogonality in (2.9)).

The $L^{2}$ orthogonality in (2.9) allows a direct proof of Theorem 3.1 that
circumvents the a posteriori error analysis of the consistency term as part of
the medius analysis [38]. It also allows control over dual norm estimates of
the form
$\|v_{\mathrm{M}}-Jv_{\mathrm{M}}\|_{H^{-s}(\Omega)}\lesssim\|h_{\mathcal{T}}^{s}(v_{\mathrm{M}}-Jv_{\mathrm{M}})\|$
for $0\leq s\leq 2$. This is critical e.g. in eigenvalue analysis or for
problems with low-order terms [26, 22]. The proof of the best-approximation of
Theorem 3.2 for the modified scheme (3.9), however, does not require the
$L^{2}$ orthogonality in (2.9).

##  3 Best-Approximation

The first subsection is a reformulation of the medius analysis in the seminal
work [38].

###  3.1 First error analysis for $F\in L^{2}(\Omega)$

Suppose $F\in L^{2}(\Omega)$ and let $u$ (resp. $u_{\mathrm{nc}}$) denote the
exact (resp. discrete) solution to (2.1) (resp. (2.2)). The analysis is first
illustrated for data $F\equiv f\in L^{2}(\Omega)$ such that
$F_{h}(v_{\mathrm{nc}})=F(v_{\mathrm{nc}}):=\int_{\Omega}fv_{\mathrm{nc}}{\rm\,dx}$
for all $v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ in (2.2).

The error $e:=u-u_{\mathrm{nc}}\in V+V_{\mathrm{nc}}$ and its interpolation
$e_{\mathrm{nc}}:=I_{\mathrm{nc}}u-u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ are
naively split into

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}_{\mathrm{pw}}$
$\displaystyle=a(u,u-Ju_{\mathrm{nc}})-a_{\mathrm{pw}}(u_{\mathrm{nc}},e_{\mathrm{nc}})+a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}}).$
(3.1)

(The identity (3.1) follows from elementary algebra in the Hilbert space
$V+V_{\mathrm{nc}}$ and
$a_{\mathrm{pw}}(u_{\mathrm{nc}},u-I_{\mathrm{nc}}u)=0=a_{\mathrm{pw}}(I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}})$
from (2.3) and (2.7).) The first and second term on the right-hand side of
(3.1) allow for (2.1) and (2.2), respectively, and their sum is equal to
$F(u-Ju_{\mathrm{nc}}-e_{\mathrm{nc}})$ (recall $F_{h}\equiv F$). This and the
last term in (3.1) allow Cauchy inequalities. This, (2.7), and (2.8) lead in
(3.1) to

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}_{\mathrm{pw}}$
$\displaystyle\leq\|h_{\mathcal{T}}^{m}f\|\|h_{\mathcal{T}}^{-m}(1-I_{\mathrm{nc}})(u-Ju_{\mathrm{nc}})\|+\Lambda_{\mathrm{J}}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}.$
(3.2)

The interpolation estimate in (2.4) is followed by a triangle inequality,
(2.8) with $v=u\in V$ and (2.5) to derive

$\|h_{\mathcal{T}}^{-m}(1-I_{\mathrm{nc}})(u-Ju_{\mathrm{nc}})\|\leq\kappa_{m}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}+\kappa_{m}\Lambda_{\mathrm{J}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}\leq\kappa_{m}(1+\Lambda_{\mathrm{J}})|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}.$

The combination with (3.2) shows best-approximation up to a data approximation
term

$\displaystyle|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\max\\{\Lambda_{\mathrm{J}},\kappa_{m}(1+\Lambda_{\mathrm{J}})\\}\left(|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}+\|h_{\mathcal{T}}^{m}f\|\right).$
(3.3)

Some local efficiency is enfolded to verify the a posteriori error estimate
$\|h_{\mathcal{T}}^{m}f\|\lesssim\mathrm{osc}_{m}(f,\mathcal{T})+\|u-I_{\mathrm{nc}}u\|_{H^{m}(\mathcal{T})}$
in the examples of [38]. This is a standard argument in residual-based a
posteriori error control and links (3.3) to the alternate version in (3.4).

###  3.2 Alternative proof for $F\in L^{2}(\Omega)$

The medius analysis in [38] does not employ (2.9) to derive the slightly
improved version

$\displaystyle|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq
C_{1}{}\left(|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}+\mathrm{osc}_{m}(f,\mathcal{T})\right)$
(3.4)

in terms of the data-oscillations

$\displaystyle\mathrm{osc}_{m}(f,\mathcal{T}):=\|h_{\mathcal{T}}^{m}(f-\Pi_{m}f)\|$
(3.5)

for $f\in L^{2}(\Omega)$ and its $L^{2}$ projection $\Pi_{m}f$ onto
$P_{m}(\mathcal{T})$. The beneficial property (2.9) allows for a direct proof
and this is detailed next.

The key observation is that the split (3.1) leads to
$F((1-I_{\mathrm{nc}})(u-Ju_{\mathrm{nc}}))$ and the contribution
$F(u-I_{\mathrm{nc}}u)$ (that stems from
$a(u,u)-a_{\mathrm{pw}}(u_{\mathrm{nc}},I_{\mathrm{nc}}u)$ by (2.1)-(2.2))
does not  allow for (2.9). The alternative split

$|\\!|\\!|e|\\!|\\!|^{2}_{\mathrm{pw}}=a(u,Je_{\mathrm{nc}})-a_{\mathrm{pw}}(u_{\mathrm{nc}},e_{\mathrm{nc}})+a_{\mathrm{pw}}(u,e-Je_{\mathrm{nc}})$
(3.6)

(this follows with the arguments in (3.1)) does allow for (2.9). Because of
(2.1)-(2.2), the sum of the first two terms on the right-hand side of (3.6) is
equal to

$F((J-1)e_{\mathrm{nc}})=\int_{\Omega}(f-\Pi_{m}f)((J-1)e_{\mathrm{nc}}){\rm\,dx}\leq\mathrm{osc}_{m}(f,\mathcal{T})\|h_{\mathcal{T}}^{-m}(1-J)e_{\mathrm{nc}}\|$

with (2.9) followed by a weighted Cauchy inequality in $L^{2}(\Omega)$ in the
end. Since $(1-J)e_{\mathrm{nc}}=(1-I_{\mathrm{nc}})(1-J)e_{\mathrm{nc}}$ from
(2.7), (2.4) implies

$\displaystyle\kappa_{m}^{-1}\|h_{\mathcal{T}}^{-m}(1-J)e_{\mathrm{nc}}\|\leq|\\!|\\!|(1-J)e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}$
(3.7)

with $v=0$ in (2.8) for the second last and the Pythagoras theorem (e.g. (2.5)
with $v=u$ and $w_{\mathrm{nc}}=u_{\mathrm{nc}}$) in the last step.

The last term of (3.6) is recast with $I_{\mathrm{nc}}(e-Je_{\mathrm{nc}})=0$
from (2.7) and $a_{\mathrm{pw}}(I_{\mathrm{nc}}u,e-Je_{\mathrm{nc}})=0$ from
(2.3) as

$\displaystyle a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,e-Je_{\mathrm{nc}})$
$\displaystyle=|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|^{2}_{\mathrm{pw}}+a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,(1-J)e_{\mathrm{nc}})$
$\displaystyle\leq(1+\Lambda_{\mathrm{J}})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}$
(3.8)

with a Cauchy inequality and
$|\\!|\\!|(1-J)e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}$
from (3.7) and (2.5) in the last step. The summary of the inequalities in
(3.6) concludes the proof of Theorem 3.1 (with suboptimal constants as
outlined in Remark 3.8 below).

######  Theorem 3.1 (best-approximation up to data approximation).

The assumptions (2.1)-(2.9) imply (3.4) for
$C_{1}:=\max\\{\kappa_{m}\Lambda_{\mathrm{J}},1+\Lambda_{\mathrm{J}}\\}.$

###  3.3 Best-approximation for the modified scheme

The discrete scheme (2.2) requires a discrete right-hand side $F_{h}$ for a
general $F\in H^{-m}(\Omega)$. The evaluation of $F_{h}:=F\circ J$ is feasible
with $F_{h}(v_{\mathrm{nc}}):=F(Jv_{\mathrm{nc}})$ for all $v_{\mathrm{nc}}\in
V_{\mathrm{nc}}$ and the (modified) nonconforming scheme seeks the solution
$u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to

$a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})=F(Jv_{\mathrm{nc}})\quad\text{for
all }v_{\mathrm{nc}}\in V_{\mathrm{nc}}.$ (3.9)

Let $\Lambda_{0}$ denote the norm of $1-J$ of the right-inverse $J\in
L(V_{\mathrm{nc}};V)$ regarded as a linear map between $V_{\mathrm{nc}}$ and
$V$,

$\Lambda_{0}:=\sup_{v_{\mathrm{nc}}\in
V_{\mathrm{nc}}\setminus\\{0\\}}|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}/|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}.$
(3.10)

######  Theorem 3.2 (best-approximation).

(a) Under the assumptions (2.3)-(2.4), (2.7)-(2.8), the exact solution $u$ to
(2.1) and the discrete solution $u_{\mathrm{nc}}$ to (3.9) satisfy

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\sqrt{1+\Lambda_{0}^{2}}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$

(b) The best-approximation constant $\sqrt{1+\Lambda_{0}^{2}}=C_{\rm qo}$ is
attained in the sense that there exist some $F=a(u,\bullet)\in V^{*}$ and
$u\in V$ with $|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=C_{\rm
qo}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}>0$ for the discrete
solution $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to (3.9).

Proof of (a). Recall the error $e:=u-u_{\mathrm{nc}}\in V+V_{\mathrm{nc}}$ and
its interpolation
$e_{\mathrm{nc}}:=I_{\mathrm{nc}}u-u_{\mathrm{nc}}=I_{\mathrm{nc}}e\in
V_{\mathrm{nc}}$. The key identity

$\displaystyle
a(u,Je_{\mathrm{nc}})=F(Je_{\mathrm{nc}})=a_{\mathrm{pw}}(u_{\mathrm{nc}},e_{\mathrm{nc}})$
(3.11)

follows directly from (2.1) (resp. (3.9)) with the test function
$Je_{\mathrm{nc}}\in V$ (resp. $e_{\mathrm{nc}}$). This proves in (3.6) that

$|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}^{2}=a_{\mathrm{pw}}(u,e-Je_{\mathrm{nc}})=a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,e-Je_{\mathrm{nc}})$
(3.12)

with $a_{\mathrm{pw}}(I_{\mathrm{nc}}u,e-Je_{\mathrm{nc}})=0$ in the last
step. The term $a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,e-Je_{\mathrm{nc}})$ is
estimated in (3.2). It is true that $u_{\mathrm{nc}}$ denotes another discrete
solution in (3.7)-(3.2), but the related arguments remain valid verbatim in
this proof as well and lead to the upper bound
$(1+\Lambda_{\mathrm{J}})|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$.
This concludes an easy proof of the best-approximation result

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq(1+\Lambda_{\mathrm{J}})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$

The proof of the optimal best-approximation constant in Theorem 3.2 revisits
(3.12) and the Pythagoras theorem $|\\!|\\!|e|\\!|\\!|^{2}_{\mathrm{pw}}\
=|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}$
(e.g. (2.5) with $v=e$ and $w_{\mathrm{nc}}=0$). The combination with (3.12)
proves

$|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,e_{\mathrm{nc}}-Je_{\mathrm{nc}}).$

The Cauchy inequality and the continuity constant $\Lambda_{0}$ show that

$|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$

(after a division of $|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$, if
positive). This and the above Pythagoras theorem result in

$|\\!|\\!|e|\\!|\\!|^{2}_{\mathrm{pw}}\
=|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}\leq(1+\Lambda_{0}^{2})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}.$

This concludes the proof under conditions weaker than (2.8) and with a better
constant.

Proof of (b). The second assertion on the attainment is immediate for
$\Lambda_{0}=0$; for $V_{\mathrm{nc}}\subset V$; see Remark 3.1 for a
discussion of this case. Hence suppose $\Lambda_{0}>0$ in the sequel.

Note that $\Lambda_{0}^{2}$ from (3.10) is an eigenvalue and there exists
$v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ with
$|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$ and

$a_{\mathrm{pw}}(v_{\mathrm{nc}}-Jv_{\mathrm{nc}},w_{\mathrm{nc}}-Jw_{\mathrm{nc}})=\Lambda_{0}^{2}a_{\mathrm{pw}}(v_{\mathrm{nc}},w_{\mathrm{nc}})\text{
for all }w_{\mathrm{nc}}\in V_{\mathrm{nc}}.$ (3.13)

For $w_{\mathrm{nc}}=v_{\mathrm{nc}}$, (3.13) shows
$|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}.$
Since $a_{\mathrm{pw}}(v_{\mathrm{nc}}-Jv_{\mathrm{nc}},w_{\mathrm{nc}})=0$,
(3.13) implies

$a_{\mathrm{pw}}(Jv_{\mathrm{nc}},Jw_{\mathrm{nc}})=(\Lambda_{0}^{2}+1)a_{\mathrm{pw}}(v_{\mathrm{nc}},w_{\mathrm{nc}})\text{
for all }w_{\mathrm{nc}}\in V_{\mathrm{nc}}.$ (3.14)

Define $F:=-a(Jv_{\mathrm{nc}},\bullet)$; so that (2.1) has the exact solution
$u=-Jv_{\mathrm{nc}}\in V$ and the discrete problem for $u_{\mathrm{nc}}\in
V_{\mathrm{nc}}$ reads

$a_{\mathrm{pw}}(u_{\mathrm{nc}},w_{\mathrm{nc}})=F(Jw_{\mathrm{nc}})=-a(Jv_{\mathrm{nc}},Jw_{\mathrm{nc}})\text{
for all }w_{\mathrm{nc}}\in V_{\mathrm{nc}}.$ (3.15)

The combination of (3.14)-(3.15) shows
$u_{\mathrm{nc}}=-(1+\Lambda_{0}^{2})v_{\mathrm{nc}}.$ Consequently,
$u-u_{\mathrm{nc}}=v_{\mathrm{nc}}-Jv_{\mathrm{nc}}+\Lambda_{0}^{2}v_{\mathrm{nc}}$
and $v_{\mathrm{nc}}=-I_{\mathrm{nc}}u$. This and the orthogonality
$a(v_{\mathrm{nc}}-Jv_{\mathrm{nc}},v_{\mathrm{nc}})=0$ lead to

$\displaystyle|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}$
$\displaystyle=|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}+\Lambda_{0}^{4}|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=\Lambda_{0}^{2}(1+\Lambda_{0}^{2})>0$

with
$|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}$
and $|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$ in the last step.
Since
$|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}=|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}$
from the design of $u=-Jv_{\mathrm{nc}}$ with
$I_{\mathrm{nc}}u=-v_{\mathrm{nc}}$, this proves
$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\sqrt{1+\Lambda^{2}_{0}}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$
∎

######  Remark 3.1 ($V_{\mathrm{nc}}\nsubset V$).

The case $\Lambda_{0}=0$ corresponds to the optimal case $\|J\|=1$ and implies
that $J$ is the identity in $V_{\mathrm{nc}}\subset V$. This is reserved for
the conforming Ritz-Galerkin scheme and excluded throughout this paper.

######  Remark 3.2 ($V_{\mathrm{nc}}\neq P_{m}(\mathcal{T})$).

It is clear that $V_{\mathrm{nc}}\subset P_{m}(\mathcal{T})$ and
$V_{\mathrm{nc}}=P_{m}(\mathcal{T})$ is excluded in the following sense. If a
linear operator $I_{\mathrm{nc}}:V\rightarrow V_{\mathrm{nc}}$ exists with
(2.3) for $V_{\mathrm{nc}}=P_{m}(\mathcal{T})$ and dimension $n=2$, $m=1$ or
$m=2$, then there exists no right-inverse $J$ of $I_{\mathrm{nc}}$. The proof
argues by contradiction, so let $J:P_{m}(\mathcal{T})\rightarrow V$ be a
right-inverse of $I_{\mathrm{nc}}$ and start with the observation that (2.3)
implies $D_{\mathrm{pw}}^{m}I_{\mathrm{nc}}v=\Pi_{0}D^{m}v$ for all
$v=Jv_{\mathrm{nc}}\in V$, $v_{\mathrm{nc}}\in P_{m}(\mathcal{T})$. Since
$I_{\mathrm{nc}}v=v_{\mathrm{nc}}$ (from
$I_{\mathrm{nc}}Jv_{\mathrm{nc}}=v_{\mathrm{nc}}$), this means

$\displaystyle
D_{\mathrm{pw}}^{m}v_{\mathrm{nc}}=\Pi_{0}{D^{m}}Jv_{\mathrm{nc}}\text{ for
all }v_{\mathrm{nc}}\in P_{m}(\mathcal{T}).$ (3.16)

For a contradiction, let $\varphi_{\rm c}\in S^{1}_{0}(\mathcal{T})\setminus
P_{0}(\Omega)$ (so that $|\\!|\\!|\varphi_{\rm c}|\\!|\\!|\neq 0$) for $m=1$
and let $\Psi_{\rm c}\in S^{1}(\mathcal{T};{\mathbb{R}}^{2})\setminus
P_{0}(\Omega;{\mathbb{R}}^{2})$ (so that $|\\!|\\!|\Psi_{\rm c}|\\!|\\!|\neq
0$) and then define $v_{m}\in P_{m}(\mathcal{T})$ for $x\in T\in\mathcal{T}$
by

$\displaystyle v_{1}(x)$ $\displaystyle:=(\text{Curl }\varphi_{\rm
c})\cdot(x-\text{ mid }(T))$ $\displaystyle\text{ resp. }v_{2}(x)$
$\displaystyle:=\frac{1}{2}(x-\text{ mid }(T))\cdot\text{ (sym}(\text{Curl
}\Psi_{\rm c}))(x-\text{ mid }(T)).$

The point is that $\nabla_{\mathrm{pw}}v_{1}=\text{Curl }\varphi_{\rm c}\in
L^{2}(\Omega;{\mathbb{R}}^{2})$ resp. $D^{2}_{\mathrm{pw}}v_{2}=\text{ sym
}(\text{Curl }\Psi_{\rm c})\in L^{2}(\Omega;{\mathbb{S}})$ is $L^{2}$
perpendicular to $\nabla v$ resp. $D^{2}v$ for $v:=Jv_{m}\in V$. Hence

$0=\int_{\Omega}D^{m}_{\mathrm{pw}}v_{m}:D^{m}v{\rm\,dx}=\int_{\Omega}D^{m}_{\mathrm{pw}}v_{m}:D^{m}Jv_{m}{\rm\,dx}=|\\!|\\!|v_{m}|\\!|\\!|_{\mathrm{pw}}^{2}$

with (3.16) for $v_{m}\in P_{m}(\mathcal{T})$ in the last step. This is a
contradiction to the nontrivial choice of $\varphi_{\rm c}$ resp. $\Psi_{\rm
c}$ with $|\\!|\\!|v_{m}|\\!|\\!|_{\mathrm{pw}}>0$.

###  3.4 Interpretation as a Petrov-Galerkin scheme

This section continues with another interpretation of the modified
nonconforming finite element method as a Petrov-Galerkin scheme in the Hilbert
space $(\widehat{V},a_{\mathrm{pw}})$ for $\widehat{V}=V+V_{\mathrm{nc}}$. The
best-approximation operator $I_{\mathrm{nc}}:\widehat{V}\to V_{\mathrm{nc}}$
in the Hilbert space $(\widehat{V},a_{\mathrm{pw}})$ is characterized by the
orthogonality $a_{\mathrm{pw}}(u_{\mathrm{nc}},v-I_{\mathrm{nc}}v)=0$ for all
$v\in V$. Since $J$ is a right-inverse, this and $v=Jv_{\mathrm{nc}}$ for
$v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ imply
$a_{\mathrm{pw}}(u_{\mathrm{nc}},Jv_{\mathrm{nc}}-v_{\mathrm{nc}})=0$. Suppose
$u\in V$ defines $F:=a(u,\bullet)\in V^{*}$ (or vice versa owing to the Riesz
isomorphism), this and
$a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})=a_{\mathrm{pw}}(u_{\mathrm{nc}},Jv_{\mathrm{nc}})$
lead in (3.9) to

$a_{\mathrm{pw}}(u-u_{\mathrm{nc}},Jv_{\mathrm{nc}})=0\quad\text{for all
}v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ (3.17)

(and this is equivalent to (3.9) for $F=a(u,\bullet)$). This defines a Petrov-
Galerkin operator $u\mapsto u_{\mathrm{nc}}$ that is naturally extended to
$\widehat{V}:=V+V_{\mathrm{nc}}$ with (3.17) as follows. Any
$\widehat{u}\in\widehat{V}$ solves (3.9) with $F(Jv_{\mathrm{nc}})$ replaced
by $a_{\mathrm{pw}}(\widehat{u},Jv_{\mathrm{nc}})$ for all $v_{\mathrm{nc}}\in
V_{\mathrm{nc}}$ with the solution $P\widehat{u}:=u_{\mathrm{nc}}\in
V_{\mathrm{nc}}$. Therefore $P:\widehat{V}\to\widehat{V}$ is characterized by

$a_{\mathrm{pw}}(\widehat{v}-P\widehat{v},Jv_{\mathrm{nc}})=0\quad\text{for
all }v_{\mathrm{nc}}\in V_{\mathrm{nc}}\text{ and for all
}\widehat{v}\in\widehat{V}.$ (3.18)

This characterizes a Petrov-Galerkin scheme in the Hilbert space
$(\widehat{V},a_{\mathrm{pw}})$ with the trial space $V_{\mathrm{nc}}$ and the
test space $V_{\rm c}:=JV_{\mathrm{nc}}$. The Petrov-Galerkin operator $P\in
L(\widehat{V})$ is regarded as a map from $\widehat{V}$ into $\widehat{V}$. It
is well defined, linear, bounded and idempotent. (In particular, any $u\in V$
is approximated by $u_{\mathrm{nc}}=Pu$.)

Based on the interpretation as a Petrov-Galerkin scheme, several conclusions
may be drawn in the subsequent subsection.

###  3.5 Comments on the best-approximation constant $C_{\rm qo}$

The subsection gives certain characterizations of the best-approximation
constant $C_{\rm qo}$ in (3.19) and shows in particular, that the
multiplicative constant $C_{\rm qo}=\sqrt{1+\Lambda_{0}^{2}}$ in Theorem 3.2
is optimal in the resulting estimate

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq C_{\rm
qo}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$ (3.19)

for the exact solution $u\in V$ to (2.1) and the discrete solution
$u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to (3.9) and an arbitrary right-hand side
$F\in V^{*}$. To be more precise, adopt the notation of a Petrov-Galerkin
operator $P:V\to V_{\mathrm{nc}}$ from (3.18) and specify

$C_{\rm qo}:=\sup_{v\in V\setminus
V_{\mathrm{nc}}}|\\!|\\!|v-Pv|\\!|\\!|_{\mathrm{pw}}/|\\!|\\!|v-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}$
(3.20)

with a supremum over all $v\in V$ with $v\neq I_{\mathrm{nc}}v$ (for this
leads to an undefined quotient $0/0$ that can be excluded) in the definition
of the best multiplicative constant (3.20) for (3.19). To avoid a pathological
situation, suppose throughout this paper that $V\setminus V_{\mathrm{nc}}$ is
non-empty and that $V_{\mathrm{nc}}\neq\\{0\\}$ is nontrivial (as well as
$V$). (The overall intention is that the nonconforming finite element space
$V_{\mathrm{nc}}\nsubset V$ is not conforming, but the case
$V_{\mathrm{nc}}\subset V$ is actually included in the analysis of this
section.) Let $\|J\|:=\|J\|_{L(V_{\mathrm{nc}};V)}$ abbreviate the operator
norm of the right-inverse $J:V_{\mathrm{nc}}\to V$.

######  Theorem 3.3 (characterization of best-approximation constant).

$C_{\rm qo}=\sqrt{1+\Lambda_{0}^{2}}=\|J\|$.

###### Proof.

Step 1 recalls $\|1-P\|_{L(\widehat{V})}=C_{\rm qo}$.  This is well known in
the mathematical foundations of Petrov-Galerkin schemes and the short proof is
given for completeness. Any $\widehat{v}\in\widehat{V}=V+V_{\mathrm{nc}}$ is
of the form $\widehat{v}=v+v_{\mathrm{nc}}=w+w_{\mathrm{nc}}$ for
$w:=(1-I_{\mathrm{nc}})v\in(1-I_{\mathrm{nc}})V=V_{\mathrm{nc}}^{\perp}$ and
$w_{\mathrm{nc}}=I_{\mathrm{nc}}v+v_{\mathrm{nc}}\in V_{\mathrm{nc}}$. Since
$P\widehat{v}=Pw+w_{\mathrm{nc}}$ and $\widehat{v}-P\widehat{v}=w-Pw$, the
orthogonal split $\widehat{v}=w+w_{\mathrm{nc}}$ shows

$\|1-P\|_{L(\widehat{V})}=\sup_{w,w_{\mathrm{nc}}}|\\!|\\!|w-Pw|\\!|\\!|_{\mathrm{pw}}(|\\!|\\!|w|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2})^{-1/2}=\sup_{w\in
V_{\mathrm{nc}}^{\perp}\setminus\\{0\\}}|\\!|\\!|w-Pw|\\!|\\!|_{\mathrm{pw}}/|\\!|\\!|w|\\!|\\!|_{\mathrm{pw}}$

with the abbreviation $\sup_{w,w_{\mathrm{nc}}}$ for the supremum over all
$(w,w_{\mathrm{nc}})\in(V_{\mathrm{nc}}^{\perp}\times
V_{\mathrm{nc}})\setminus\\{0\\}$. This and the re-substitution
$w=v-I_{\mathrm{nc}}v$ and $w-Pw=v-Pv$ for $v\in V$ in the second equality
conclude the proof,

$\|1-P\|_{L(\widehat{V})}=\|1-P\|_{L(V_{\mathrm{nc}}^{\perp};\widehat{V})}=\sup_{v\in
V\setminus
V_{\mathrm{nc}}}|\\!|\\!|v-Pv|\\!|\\!|_{\mathrm{pw}}/|\\!|\\!|v-I_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}=C_{\rm
qo}.\qed$

Step 2 establishes $\|P\|_{L(\widehat{V})}=\|P\|_{L(V_{\rm
c};V_{\mathrm{nc}})}$ for the conforming companion finite element space
$V_{\rm c}:=JV_{\mathrm{nc}}$. Since $\|P\|_{L(V_{\rm
c};V_{\mathrm{nc}})}\leq\|P\|_{L(\widehat{V})}$ is obvious (for $V_{\rm
c}\subset\widehat{V}$), the proof of the converse inequality concerns an
arbitrary $\widehat{v}\in\widehat{V}$ and its orthogonal projection $v_{\rm
c}$ onto the (finite-dimensional, whence closed) subspace $V_{\rm
c}:=JV_{\mathrm{nc}}$ in the Hilbert space $(\widehat{V},a_{\mathrm{pw}})$
with orthogonality written as $\perp$. In particular, $v_{\rm c}\in V_{\rm c}$
and $\widehat{v}-v_{\rm c}\perp V_{\rm c}$. Recall from (3.18) that
$P\widehat{v}$ is characterized by $P\widehat{v}\in V_{\mathrm{nc}}$ and
$\widehat{v}-P\widehat{v}\perp V_{\rm c}$. The two orthogonalities lead to
$v_{\rm c}-P\widehat{v}\perp V_{\rm c}$, whence $P\widehat{v}=Pv_{\rm c}$.
This and $|\\!|\\!|v_{\rm
c}|\\!|\\!|_{\mathrm{pw}}\leq|\\!|\\!|\widehat{v}|\\!|\\!|_{\mathrm{pw}}$
imply

$|\\!|\\!|P\widehat{v}|\\!|\\!|_{\mathrm{pw}}=|\\!|\\!|Pv_{\rm
c}|\\!|\\!|_{\mathrm{pw}}\leq\|P\|_{L(V_{\rm
c};V_{\mathrm{nc}})}|\\!|\\!|v_{\rm
c}|\\!|\\!|_{\mathrm{pw}}\leq\|P\|_{L(V_{\rm
c};V_{\mathrm{nc}})}|\\!|\\!|\widehat{v}|\\!|\\!|_{\mathrm{pw}}.\qed$

Step 3 computes $\|P\|_{L(V_{\rm c};V_{\mathrm{nc}})}=\|J\|$ with linear
algebra. Given any basis $(\psi_{1},\dots,\psi_{N})$ of $V_{\mathrm{nc}}$,
$(J\psi_{1},\dots,J\psi_{N})$ forms a basis of $V_{\rm c}$ (for
$v_{\mathrm{nc}}\in V_{\mathrm{nc}}$ with $Jv_{\mathrm{nc}}=0$ implies
$v_{\mathrm{nc}}=I_{\mathrm{nc}}Jv_{\mathrm{nc}}=0$). The stiffness matrices
$A,B\in\mathbb{R}^{N\times N}$ are defined by

$A_{jk}:=a_{\mathrm{pw}}(\psi_{j},\psi_{k})\quad\text{amd}\quad
B_{jk}:=a_{\mathrm{pw}}(J\psi_{j},J\psi_{k})\quad\text{for all
}j,k=1,\dots,N.$

Then $A$ and $B$ are SPD and characterize the Petrov-Galerin map $P:V_{\rm
c}\to V_{\mathrm{nc}}$ in coefficient vectors $x,y\in\mathbb{R}^{N}$ for
$v_{\mathrm{nc}}=\sum_{j=1}^{N}x_{j}\psi_{j}$ and $v_{\rm
c}=\sum_{j=1}^{N}y_{j}J\psi_{j}$ by $v_{\mathrm{nc}}=Pv_{\rm c}$ is equivalent
to $Ax=By$. The norm $\|P\|_{L(V_{\rm c};V_{\mathrm{nc}})}$ of $P:V_{\rm c}\to
V_{\mathrm{nc}}$ involves the norms with the squares $|\\!|\\!|Pv_{\rm
c}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=x\cdot
Ax$ and $|\\!|\\!|v_{\rm c}|\\!|\\!|_{\mathrm{pw}}^{2}=y\cdot By$ and lead,
for $x,y\in\mathbb{R}^{N}\setminus\\{0\\}$ with $Ax=By$, to $y\cdot By=x\cdot
AB^{-1}Ax$. This and the Rayleigh-Ritz $\min$-$\max$ principle show

$\|P\|_{L(V_{\rm c};V_{\mathrm{nc}})}^{2}=\sup_{v_{\rm c}\in V_{\rm
c}\setminus\\{0\\}}\frac{|\\!|\\!|Pv_{\rm
c}|\\!|\\!|_{\mathrm{pw}}^{2}}{|\\!|\\!|v_{\rm
c}|\\!|\\!|_{\mathrm{pw}}^{2}}=\sup_{x\in\mathbb{R}^{N}\setminus\\{0\\}}\frac{x\cdot
Ax}{x\cdot AB^{-1}Ax}=\lambda$

for the maximal eigenvalue of an eigenpair
$(\lambda,x)\in\mathbb{R}_{+}\times\mathbb{R}^{N}\setminus\\{0\\}$ of the
generalized eigenvalue problem $Ax=\lambda AB^{-1}Ax$. Since the latter is
equivalent to $Bx=\lambda Ax$, the maximal eigenvalue is

$\lambda=\sup_{x\in\mathbb{R}^{N}\setminus\\{0\\}}\frac{x\cdot Bx}{x\cdot
Ax}=\sup_{v_{\mathrm{nc}}\in
V_{\mathrm{nc}}\setminus\\{0\\}}\frac{|\\!|\\!|Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}{|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}=\|J\|^{2}.\qed$

Step 4 verifies $\|J\|^{2}=1+\Lambda_{0}^{2}$. Since $J$ is a right-inverse to
$I_{\mathrm{nc}}$, the Pythagoras theorem
$|\\!|\\!|Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}$
concludes the proof by the definition of $\|J\|$ and $\Lambda_{0}$:

$\|J\|^{2}=\sup_{v_{\mathrm{nc}}\in
V_{\mathrm{nc}}\setminus\\{0\\}}\frac{|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}{|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}=1+\sup_{v_{\mathrm{nc}}\in
V_{\mathrm{nc}}\setminus\\{0\\}}\frac{|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}{|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}}=1+\Lambda_{0}^{2}.\qed$

Step 5 concludes the proof with Kato’s lemma. Any oblique projection (that is
a bounded linear map $P\in L(\widehat{V})$ in a Hilbert space $\widehat{V}$
with $P^{2}=P$) satisfies $\|1-P\|_{L(\widehat{V})}=\|P\|_{L(\widehat{V})}$
provided $0\neq P\neq 1$. This is known as Kato’s lemma [42] with many very
different proofs in [48]. The summary of all steps concludes the proof. ∎

######  Remark 3.3.

Veeser et al. [53] provide the bound $C_{\rm qo}{=}\|J\|$, while Theorem 3.2
gives $C_{\rm qo}=\sqrt{1+\Lambda_{0}^{2}}$ and both results are optimal.

######  Remark 3.4 ($J$ vs $V_{\rm c}$).

The Petrov-Galerkin scheme and its operator $P$ require the trial space
$V_{\mathrm{nc}}$ and the test space $V_{\rm c}$, but no knowledge about $J$.
However, $V_{\rm c}:=J(V_{\mathrm{nc}})$ depends very much on the right-
inverse. In fact, given any $V_{\rm c}\subset V$ of dimension $\dim V_{\rm
c}=\dim V_{\mathrm{nc}}\in\mathbb{N}$, there is at most one right inverse: If
$I_{\mathrm{nc}}$ is not injective as a linear map from $V_{\rm c}$ into
$V_{\mathrm{nc}}$, or equivalently (for $\dim V_{\rm c}=\dim V_{\mathrm{nc}}$)
it is not surjective, then there exists no right-inverse $J:V_{\mathrm{nc}}\to
V$ with $V_{\rm c}=J(V_{\mathrm{nc}})$. But otherwise $J$ exists and is
uniquely determined by $V_{\rm c}$. (If $J:V_{\mathrm{nc}}\to V_{\rm c}$ is a
linear map and $I_{\mathrm{nc}}J$ is identity in $V_{\mathrm{nc}}$ then $J$ is
injective and hence bijective for $\dim V_{\rm c}=\dim V_{\mathrm{nc}}$. Then
$J$ is the inverse of $I_{\mathrm{nc}}$, when $I_{\mathrm{nc}}$ is regarded as
a bijection of $V_{\rm c}$ onto $V_{\mathrm{nc}}$.)

######  Remark 3.5 ($\|J\|>1$).

Theorem 3.3 asserts $\|J\|\geq 1$ and the optimal case $\|J\|=1$ for the
conforming Ritz-Galerkin scheme is discussed in Remark 3.1. For the truly
nonconforming situation $V_{\mathrm{nc}}\nsubset V$, the quality of the scheme
is controlled by $C_{\rm qo}=\|J\|>1$.

######  Remark 3.6 (another characterization of $\Lambda_{0}$).

Theorem 3.3 also reveals the identity
$\Lambda_{0}=\|P\|_{L(V_{\mathrm{nc}}^{\perp};\widehat{V})}$ the orthogonal
complement of $V_{\mathrm{nc}}$ in $\widehat{V}$. This identity is behind the
refined proof of Theorem 3.2. The proof of it considers $Pw\perp w\in
V_{\mathrm{nc}}^{\perp}$ and the Pythagoras theorem
$|\\!|\\!|w-Pw|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|Pw|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w|\\!|\\!|_{\mathrm{pw}}^{2}$
for the first equation in

$\|1-P\|_{L(V_{\mathrm{nc}}^{\perp};\widehat{V})}^{2}=\sup_{w\in
V_{\mathrm{nc}}^{\perp}\setminus\\{0\\}}(|\\!|\\!|Pw|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w|\\!|\\!|_{\mathrm{pw}}^{2})|\\!|\\!|w|\\!|\\!|_{\mathrm{pw}}^{-2}=1+\|P\|_{L(V_{\mathrm{nc}}^{\perp};\widehat{V})}^{2}$

Theorem 3.3 and the last line in Step 1 of its proof show that this is equal
to $1+\Lambda_{0}^{2}$. ∎

######  Remark 3.7 (post-processing).

The error $u-Ju_{\mathrm{nc}}$ of the post-processed discrete solution can be
measured in all Sobolev scales $H^{s}(\Omega)$ for $-m\leq s\leq m$ and we
start with $s=m$ and the affirmative estimate

$\left||\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}-|\\!|\\!|u-Ju_{\mathrm{nc}}|\\!|\\!|\right|\leq\Lambda_{\rm
J}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\rm J}C_{\rm
qo}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$

(This follows from a reverse triangle inequality and (2.8) plus Theorem 3.2.)
This estimate compares with errors in the continuous and piecewise energy
norms up to a best-approximation term on the right-hand side. In particular,
the post-processing is quasi-optimal. However, it is not quasi-optimal in
comparison with the best-approximation in $V_{\rm c}:=JV_{\mathrm{nc}}$:
Otherwise, with the Petrov-Galerkin operator $P:V\to V_{\mathrm{nc}}$ from
(3.18), any inequality
$|\\!|\\!|v-JPv|\\!|\\!|_{\mathrm{pw}}\lesssim|\\!|\\!|v-v_{\rm
c}|\\!|\\!|_{\mathrm{pw}}$ for all $v\in V$ and $v_{\rm c}\in V_{\rm c}$
implies, in particular, that $v_{\rm c}=JPv_{\rm c}$ for all $v=v_{\rm c}\in
V_{\rm c}$. Since $J$ is a right-inverse, the last identity shows that
$I_{\mathrm{nc}}v_{\rm c}=Pv_{\rm c}$ for all $v_{\rm c}\in V_{\rm c}$. But
this implies $\|P\|_{L(V_{\rm c};V_{\mathrm{nc}})}\leq 1$. Step 3 in the proof
of Theorem 3.3 leads to $\|J\|=1$. This is characterized in Remark 3.1 as the
conforming case $V_{\mathrm{nc}}=V_{\rm c}\subset V$.

######  Remark 3.8 (sharper constants in Theorem 3.1).

The constant $C_{1}$ in (3.4) is not much larger than
$\sqrt{1+\Lambda_{0}^{2}}$. Follow the proof until (3.7) and replace
$\Lambda_{\mathrm{J}}$ by $\Lambda_{0}$ and ignore the last estimate. Utilize
$\Lambda_{0}$ for
$|\\!|\\!|(1-J)e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
in the equality (3.2). This leads to an estimate for the right-hand side of
(3.6) that includes the additive term
$|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}$ (still from (3.2))
that arises also in the Pythagoras theorem
$|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}$
on the left-hand side. This proves (after a division of
$|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$ if positive)

$|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}(\kappa_{m}\mathrm{osc}_{m}(f,\mathcal{T})+|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}).$

This identity and the Pythagoras theorem again conclude the proof of

$|\\!|\\!|e_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}\leq(1+\Lambda^{2}_{0})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}^{2}+2\Lambda^{2}_{0}\kappa_{m}\mathrm{osc}_{m}(f,\mathcal{T})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}+\kappa^{2}_{m}\mathrm{osc}_{m}(f,\mathcal{T})^{2}.$

The right-hand side is equal to
$(|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}},\mathrm{osc}_{m}(f,\mathcal{T}))\cdot
A(|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}},\mathrm{osc}_{m}(f,\mathcal{T}))$
with the symmetric $2\times 2$ matrix $A$ that has entries
$A_{11}=1+\Lambda^{2}_{0}$, $A_{12}=\Lambda^{2}_{0}\kappa_{m}$, and
$A_{22}=\kappa^{2}_{m}$. The eigenvalues of $A$ can be computed and the bigger
one leads to a bound (3.4) with

$2(1+\Lambda_{0}^{2})<2C_{1}^{2}=1+\kappa_{m}^{2}+\Lambda_{0}^{2}+\sqrt{(1-\kappa_{m}^{2}+\Lambda_{0}^{2})^{2}+4\kappa_{m}^{2}\Lambda_{0}^{4}}.$

Since $\kappa_{m}$ is small, the constant $C_{1}$ is not significantly larger
than $C_{\rm qo}$ in (3.19). ∎

##  4 Weaker and piecewise Sobolev norm estimates

###  4.1 Lower-order error estimates

This section lists further conditions sufficient for a lower-order a priori
error estimate for the discrete problem (2.2) under the hypotheses (2.3)-(2.4)
and (2.7)-(2.8).

The duality argument relies on the elliptic regularity and the approximation
properties of the dual and discrete problem with constants $0<\sigma\leq 1$
and $0<C_{\rm reg}(s),C_{\rm int}(s)<\infty$ such that for any $G\in
H^{-s}(\Omega)$ with $m-\sigma\leq s\leq m$, the weak solution $z\in
V=H^{m}_{0}(\Omega)$ to the PDE $(-1)^{m}\Delta^{m}z=G$ satisfies $z\in V\cap
H^{2m-s}(\Omega)$ and

$\displaystyle\|z\|_{H^{2m-s}(\Omega)}$ $\displaystyle\leq C_{\rm
reg}(s)\|G\|_{H^{-s}(\Omega)},$ (4.1)
$\displaystyle|\\!|\\!|z-I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq C_{\rm int}(s)h_{\rm max}^{m-s}\|z\|_{H^{2m-s}(\Omega)}.$
(4.2)

Since in general $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ may not belong to
$H^{m-s}(\Omega)$, the post-processing $Ju_{\mathrm{nc}}$ is considered in the
lower-order a priori error estimate.

######  Theorem 4.1 (lower-order error estimates).

Suppose the hypotheses of Theorem 3.2, adopt the notation (4.1)-(4.2) for some
$s$ with $m-\sigma\leq s\leq m$. Given any $F\in H^{-s}(\Omega)$, the
solutions $u\in V$ to (2.1) and $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to (2.2)
satisfy $(a)\;\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}\leq
C_{2}{}(s)h_{\max}^{m-s}~{}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
and $(b)\;\|u-u_{\mathrm{nc}}\|_{{H^{s}}(\mathcal{T})}\leq
C_{3}{}(s)h_{\max}^{m-s}|\\!|\\!|{u-u_{\mathrm{nc}}}|\\!|\\!|_{\mathrm{pw}}.$

Proof of (a). The duality argument is applied for the exact solution $u\in V$
to (2.1) and the post-processing $v:=Ju_{\mathrm{nc}}\in V$ for the discrete
solution $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ to (2.2). The duality of
$H^{s}_{0}(\Omega)$ and $H^{-s}(\Omega)$ reveals

$\displaystyle\|u-v\|_{H^{s}(\Omega)}$ $\displaystyle=\sup_{0\neq G\in
H^{-s}(\Omega)}\frac{G(u-v)}{\|G\|_{H^{-s}(\Omega)}}=G(u-v)$

and the supremum is attained for some $G\in H^{-s}(\Omega)\subset V^{*}$ with
norm $\|G\|_{H^{-s}(\Omega)}=1$ (owing to a corollary of the Hahn-Banach
theorem). The functional has a unique Riesz representation $z\in V$ in the
Hilbert space $(V,a)$, $a(\bullet,z)=G\in V^{*}$; i.e., $z\in V$ the weak
solution to the PDE $(-1)^{m}\Delta^{m}z=G$. The elliptic regularity (4.1)
leads to $z\in V\cap H^{2m-s}(\Omega)$ and (4.1)-(4.2); in particular,

$\displaystyle\|u-v\|_{H^{s}(\Omega)}$
$\displaystyle=a(u-v,z)\quad\text{and}\quad\|z\|_{H^{2m-s}(\Omega)}\leq C_{\rm
reg}(s).$ (4.3)

Let $v:=Ju_{\mathrm{nc}}\in V$ and $\zeta:=JI_{\mathrm{nc}}z\in V$, where
$u_{\mathrm{nc}}$ (resp. $z$) denote the discrete (resp. dual) solution. The
test function $\zeta=JI_{\mathrm{nc}}z\in V$ in (2.1) and the test function
$I_{\mathrm{nc}}z$ in (2.2) lead to

$a_{\mathrm{pw}}(u_{\mathrm{nc}},I_{\mathrm{nc}}z)=F(\zeta)=a(u,\zeta).$

This and elementary algebra with
$a_{\mathrm{pw}}((1-J)u_{\mathrm{nc}},I_{\mathrm{nc}}z)=0=a_{\mathrm{pw}}(u_{\mathrm{nc}},(1-J)I_{\mathrm{nc}}z)$
(from (2.3) and (2.7)), result in

$\displaystyle
a(u-v,z)=a(u-v,z-\zeta)+a_{\mathrm{pw}}(u_{\mathrm{nc}}-v,\zeta-
I_{\mathrm{nc}}z).$ (4.4)

From (2.8) deduce that

$\displaystyle|\\!|\\!|v-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq\Lambda_{\mathrm{J}}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\quad\text{and}\quad|\\!|\\!|\zeta-
I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|z-I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}.$
(4.5)

The combination of (4.5) and triangle inequality leads to

$\displaystyle|\\!|\\!|u-v|\\!|\\!|$
$\displaystyle\leq|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}+|\\!|\\!|v-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq(1+\Lambda_{\mathrm{J}})|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}.$
(4.6)

Similar arguments for $(z,\zeta,I_{\mathrm{nc}}z)$ replacing
$(u,v,u_{\mathrm{nc}})$ reveal

$\displaystyle|\\!|\\!|z-\zeta|\\!|\\!|$
$\displaystyle\leq(1+\Lambda_{\mathrm{J}})|\\!|\\!|z-I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}.$
(4.7)

A Cauchy inequality for the terms on the right-hand side of (4.4) followed by
an application of (4.5)-(4.7) implies that $a(u-v,z)$ is bounded by
$((1+\Lambda_{\mathrm{J}})^{2}+\Lambda_{\mathrm{J}}^{2})|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|z-I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}$.
A substitution to (4.3) leads to

$\|u-v\|_{H^{s}(\Omega)}\leq{((1+\Lambda_{\mathrm{J}})^{2}+\Lambda_{\mathrm{J}}^{2})}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|z-I_{\mathrm{nc}}z|\\!|\\!|_{\mathrm{pw}}.$

This, (4.2), and (4.3b) lead to $C_{\ref{lower1}}(s):=C_{\rm int}(s)C_{\rm
reg}(s)((1+\Lambda_{\mathrm{J}})^{2}+\Lambda_{\mathrm{J}}^{2})$. ∎

Proof of (b). The Sobolev-Slobodeckii semi-norm over $\Omega$ involves double
integrals over $\Omega\times\Omega$ and so is larger than or equal to the sum
of the integrals over $T\times T$ for all the triangles $T\in\mathcal{T}$,
i.e.,
$\sum_{T\in\mathcal{T}}|\bullet|_{H^{s}(T)}^{2}\leq|\bullet|_{H^{s}(\Omega)}^{2}$
for any $m-1<s<m$. Hence Theorem 4.1.a implies for all $s\text{ with
}m-\sigma\leq s\leq m$ and a constant $C_{\ref{lower1}}(s)$ that

$\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\mathcal{T})}\leq
C_{\ref{lower1}}(s)\,h_{\max}^{m-s}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}.$
(4.8)

The equivalence of the Sobolev-Slobodeckii norm and the norm by interpolation
of Sobolev spaces [43, Remark 9.1], for instance for a fixed reference
triangle $T=T_{\rm ref}$ with $C_{4}{}(s)=C_{\ref{ccint3}}(s,T_{\rm ref})$,
provides for ${w}:=(u_{\mathrm{nc}}-Ju_{\mathrm{nc}})|_{T}\in H^{m}(T)$ the
estimate

$\|{w}\|_{H^{s}(T)}\leq
C_{\ref{ccint3}}(s)\,\|{w}\|_{H^{m-1}(T)}^{m-s}\|{w}\|_{H^{m}(T)}^{s-m+1}\quad\text{for
}m-1<s<m.$ (4.9)

A straightforward transformation of Sobolev norms [28, Theorem 3.1.2] shows
(4.9) for any triangle $T\in\mathcal{T}$ with
$C_{\ref{ccint3}}(s)=C_{\ref{ccint3}}(s,T)=\kappa^{1+s}C_{\ref{ccint3}}(s,T_{\rm
ref})$ for the condition number $\kappa=\sigma_{1}/\sigma_{2}$ of the affine
transformation $a+Bx$ of $T_{\rm ref}$ to $T$ with the $2\times 2$ matrix $B$
and its positive singular values $\sigma_{2}\leq\sigma_{1}$. A more detailed
analysis [23] reveals that $C_{\ref{ccint3}}(s)$ exclusively depends on $s$
(but exploits singularities as $s$ approaches the end-points $m-1$ and $m$).
The estimate (4.9) shows the first inequality in

$C_{\ref{ccint3}}(s)^{-2}\,\|{w}\|_{H^{s}(T)}^{2}\leq\|{w}\|_{H^{m-1}(T)}^{2(m-s)}\|{w}\|_{H^{m}(T)}^{2(s-m+1)}\leq\|{w}\|_{H^{m-1}(T)}^{2}+\|{w}\|_{H^{m-1}(T)}^{2(m-s)}|{w}|_{H^{m}(T)}^{2(s-m+1)}$

with the subadditivity $(a+b)^{p}\leq a^{p}+b^{p}$ for $a,b\geq 0$ and
$0<p=s-m+1<1$ (e.g. from the concavity of $x\mapsto x^{p}$ for non-negative
$x$) in the last step. An elementary estimate is followed by the Young’s
inequality $ab\leq a^{p}/p+b^{q}/q$ for $p=(m-s)^{-1}$, $q=(s-m+1)^{-1}$,
$a=\|h_{T}^{-1}{w}\|_{H^{1}(T)}^{2(m-s)}$, and $b=|{w}|_{H^{2}(T)}^{2(s-m+1)}$
to prove

$h_{\max}^{2(s-m)}\|{w}\|_{H^{m-1}(T)}^{2(m-s)}|{w}|_{H^{m}(T)}^{2(s-m+1)}\leq\|h_{\mathcal{T}}^{-1}{w}\|_{H^{m-1}(T)}^{2(m-s)}|{w}|_{H^{m}(T)}^{2(s-m+1)}\leq\|h_{\mathcal{T}}^{-1}{w}\|_{H^{m-1}(T)}^{2}+|{w}|_{H^{m}(T)}^{2}.$

This and the trivial estimate $h_{\max}^{2(s-m+1)}\leq\text{\rm
diam}(\Omega)^{2(s-m+1)}$ lead to
$C_{5}{}(s)=C_{\ref{ccint3}}(s)^{2}(1+\text{\rm diam}(\Omega)^{2(s-m+1)})$ in

$\|{w}\|_{H^{s}(T)}^{2}\leq
C_{\ref{ccint4}}(s)h_{\max}^{2(m-s)}\left(\|h_{T}^{-1}{w}\|_{H^{m-1}(T)}^{2}+|{w}|_{H^{m}(T)}^{2}\right).$

The sum over all those contributions over $T\in\mathcal{T}$ reads

$\displaystyle\|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}\|_{H^{s}(\mathcal{T})}$
$\displaystyle\leq\sqrt{C_{\ref{ccint4}}(s)}h_{\max}^{m-s}\left(\|h_{\mathcal{T}}^{-1}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}})\|_{H^{m-1}(\mathcal{T})}+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\right).$

An inverse estimate shows
$\|h_{T}^{-1}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}})\|_{H^{m-1}(T)}\lesssim
h_{T}^{-m}\|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}\|_{L^{2}(T)}$ for all
$T\in\mathcal{T}$ with a constant in $\lesssim$ that depends only on the
shape-regularity. This, (2.4), a triangle inequality, (4.8), and (2.8)
conclude the proof of $(b)$. ∎

###  4.2 Precise convergence rates for quasi-uniform meshes

This subsection comments on the sharpness of the a priori estimates of Theorem
4.1 and discusses conditions sufficient for the convergence rate
$t:=\min\\{2\sigma,m+\sigma-s\\}$ of the error $u-Ju_{\mathrm{nc}}\in
V=H^{m}_{0}(\Omega)$ in the norm of $H^{s}(\Omega)$ for $-m\leq s\leq m$ as
$h_{\rm max}\rightarrow 0$. Under the generic assumption
$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\approx h_{\max}^{\sigma}$,
the estimate

$\displaystyle\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}\approx h_{\max}^{t}\text{
for }-m\leq s<m+1/2$ (4.10)

holds for a class of quasi-uniform shape-regular triangulations $\mathcal{T}$
with sufficiently small maximal mesh-size $h_{\max}\ll 1$. Figure 2 depicts
the convergence rate $t$ in (4.10) as $h_{\max}\rightarrow 0$.

$-m$$m-\sigma$$m$$m+\frac{1}{2}$$\sigma-\frac{1}{2}$$\sigma$$2\sigma$$\frac{1}{2}$$s$${t}$
Figure 2: Rates of convergence $t$ of $\|u-Ju_{\rm nc}\|_{H^{s}(\Omega)}$ as
$h_{\rm max}\rightarrow 0$ for $-m\leq s<m+1/2$.

######  Theorem 4.2.

Given $F\in H_{0}^{\max\\{0,-s\\}}(\Omega)$ for $-m\leq s\leq m+1/2$, let
$u\in V$ solve (2.1) and let $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ solve (3.9).
Suppose $\sigma<m$, $t:=\min\\{2\sigma,m+\sigma-s\\}$, and that for a sequence
of quasi-uniform shape-regular triangulations

$h_{\max}^{\sigma}\lesssim|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
(4.11)

holds. Then $\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}\approx h_{\max}^{t}$ as
$h_{\max}\to 0$.

The proof of the theorem for $-m\leq s\leq m-\sigma$ rests on the subsequent
lemma.

######  Lemma 4.3.

For $F\in H_{0}^{\max\\{0,-s\\}}(\Omega)$ and $-m\leq s\leq m-\sigma$, the
exact solution $u\in V$ to (2.1) and the discrete solution $u_{\mathrm{nc}}\in
V_{\mathrm{nc}}$ to (3.9) satisfy

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}\leq
2\kappa_{m}h^{m}_{\max}(1+\Lambda_{\rm
J})\|F\||\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}+\|F\|_{H^{-s}(\Omega)}\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}.$

###### Proof.

The weak solution $u$ to $(-1)^{m}\Delta^{m}u=F$ satisfies $a(u,u)=F(u)$. The
discrete solution $u_{\mathrm{nc}}$ to (3.9) satisfies
$a_{\mathrm{pw}}(u_{\mathrm{nc}},2I_{\mathrm{nc}}u-u_{\mathrm{nc}})=F(2JI_{\mathrm{nc}}u-Ju_{\mathrm{nc}})$.
Consequently,

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}=F(Ju_{\mathrm{nc}}-u)+2F(u-JI_{\mathrm{nc}}u).$

The first term on the right-hand side of the above expression is
$\leq\|F\|_{H^{-s}(\Omega)}\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}$ while the
second term, for $F\in L^{2}(\Omega)$, is twice

$\displaystyle F(u-JI_{\mathrm{nc}}u)$
$\displaystyle\leq\kappa_{m}h^{m}_{\max}\|F\||\\!|\\!|u-JI_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq\kappa_{m}(1+\Lambda_{\rm
J})h^{m}_{\max}\|F\||\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$

with (2.4) in the first step and a triangle inequality plus (2.8) in the last
step. A combination of the estimates concludes the proof. ∎

###### Proof of Theorem 4.2 for $-m\leq s\leq m-\sigma$.

Suppose that $\Omega$ is a (possibly nonconvex) polygon with elliptic
regularity index $1/2<\sigma\leq 1$ and $\mathcal{T}$ is quasi-uniform with
mesh-size $h_{\max}$. The estimate
$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\lesssim h_{\max}^{\sigma}$
follows from Theorem 3.2 and the interpolation approximation estimate
$|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}\lesssim h_{\max}^{\sigma}$
for $u\in H^{m+\sigma}(\Omega)$. The additional assumption (4.11) is that the
convergence rate is not better than $\sigma$. This and Lemma 4.3 imply

$C_{6}{}h_{\max}^{2\sigma}\leq\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}+h_{\max}^{m+\sigma}$

for a positive constant $C_{\ref{con:lemsharp}}\approx 1$. In case
$m=1=\sigma$, this estimate is inconclusive. But otherwise $2\sigma<m+\sigma$
(e.g. for $m=2$ or $\sigma<1$) and
$h_{\max}^{2\sigma}(C_{\ref{con:lemsharp}}-h_{\max}^{m-\sigma})\leq\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}$
shows, at least for sufficiently small $h_{\max}\ll 1$, that
$h_{\max}^{2\sigma}\lesssim\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}$. The
Sobolev embedding for $s\leq m-\sigma$ shows
$\|u-Ju_{\mathrm{nc}}\|_{H^{s}(\Omega)}\lesssim\|u-Ju_{\mathrm{nc}}\|_{H^{m-\sigma}(\Omega)}$.
Theorem 4.1 and $|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\lesssim
h_{\max}^{\sigma}$ imply
$\|u-Ju_{\mathrm{nc}}\|_{H^{m-\sigma}(\Omega)}\lesssim h_{\max}^{2\sigma}$.
The combination of all those estimates concludes the proof for $-m\leq s\leq
m-\sigma$. ∎

Suppose $\mathcal{T}$ is quasi-uniform and $0<\varepsilon<1/2$ throughout the
remaining parts of this subsection. A quasi-uniform mesh allows inverse
estimates even in the Sobolev-Slobodeckii semi-norm. Throughout the remaining
part of this subsection, the generic factors hidden in $\lesssim$ may also
depend on $\varepsilon$.

######  Lemma 4.4 (inverse estimate).

It holds
$\varepsilon^{1/2}\|u-Ju_{\mathrm{nc}}\|_{H^{m+\varepsilon}(\Omega)}\lesssim
h_{\max}^{\sigma-\varepsilon}|u|_{H^{m+\sigma}(\Omega)}.$

###### Proof.

The best-approximation property leads to
$\|u-Ju_{\mathrm{nc}}\|_{H^{m}(\Omega)}\lesssim
h^{\sigma}_{\max}|u|_{H^{m+\sigma}(\Omega)}$ and controls the low-order
contributions. Hence the proof focusses on the Sobolev-Slobodeckii semi-norm
of each partial-dertivative $\partial^{\alpha}$ of order $|\alpha|=m$. Let
$f:=\partial^{\alpha}u$, $g:=\partial^{\alpha}Ju_{\mathrm{nc}}$. The triangle
inequality motivates the split

$|f-g|_{H^{\varepsilon}(\Omega)}\leq|f-\Pi_{0}f|_{H^{\varepsilon}(\Omega)}+|\Pi_{0}f-g|_{H^{\varepsilon}(\Omega)}.$

The first term is controlled by a Poincare inequality for the Sobolev-
Slobodeckii semi-norm

$\|f-\Pi_{0}f\|_{H^{\varepsilon}(\Omega)}\lesssim\varepsilon^{-1/2}h_{\max}^{s-\varepsilon}|f|_{H^{s}(\Omega)}.$

(This is a standard estimate for the reference triangle e.g. by interpolation
and a scaling argument completes the proof with a generic constant that may
depend on $\varepsilon$. The claimed multiplicative constant
$\approx\varepsilon^{-1/2}$ follows from a refined estimation [23].) The
second term is an inverse estimate for $Ju_{\mathrm{nc}}\in V$ that is a
piecewise polynomial (e.g. with respect to a HCT refinement of $\mathcal{T}$)
with mesh-sizes equivalent to $h_{\mathcal{T}}$. Since $0<\varepsilon<1/2$,
any piecewise polynomial $p_{r}\in P_{r}(\mathcal{T})$ of a degree at most
$r\in{\mathbb{N}}_{0}$ belongs to $H^{\varepsilon}(\Omega)$ and the inverse
estimate
$\|p_{r}\|_{H^{\varepsilon}(\Omega)}\lesssim\varepsilon^{-1/2}\|h_{\mathcal{T}}^{-\varepsilon}p_{r}\|_{L^{2}(\Omega)}$
holds [30] for a quasi-uniform mesh (the multiplicative constant
$\approx\varepsilon^{-1/2}$ follows from a refined estimation [23]). Since
this is available for the above piecewise polynomial $\Pi_{0}f-g$, it implies

$\varepsilon^{1/2}|\Pi_{0}f-g|_{H^{\varepsilon}(\Omega)}\lesssim\|h_{\mathcal{T}}^{-\varepsilon}(\Pi_{0}f-g)\|\lesssim{h_{\rm
min}^{-\varepsilon}}|\\!|\\!|Ju_{\mathrm{nc}}-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$

for the minimal mesh-size $h_{\min}:=\min_{T\in\mathcal{T}}h_{T}$ in
$\mathcal{T}$ and $h_{\min}\approx h_{\max}$ for a quasi-uniform
triangulation. A triangle inequality with (2.5), (2.8), and Theorem 3.2 lead
to

$\displaystyle|\\!|\\!|I_{\mathrm{nc}}u-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|$
$\displaystyle\leq C_{\mathrm{qo}}(1+\Lambda_{\rm
J})|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}\lesssim
h_{\max}^{\sigma}|u|_{H^{m+\sigma}(\Omega)}.$

The combination of the above estimates for each derivative
$f=\partial^{\alpha}u$ of order $m=|\alpha|$ shows
$\varepsilon^{1/2}\|u-Ju_{\mathrm{nc}}\|_{H^{m+\varepsilon}(\Omega)}\lesssim
h_{\max}^{\sigma-\varepsilon}|u|_{H^{m+\sigma}(\Omega)}$. This concludes the
proof for a quasi-uniform triangulation. ∎

######  Lemma 4.5 (interpolation).

If $0<s<m$ and $0<\varepsilon<1/2$, then
$\|u-Ju_{\mathrm{nc}}\|_{H^{m}(\Omega)}^{s+\varepsilon}\lesssim
h_{\max}^{s(\sigma-\varepsilon)}|u|_{H^{m+\sigma}(\Omega)}^{s}\|u-Ju_{\mathrm{nc}}\|^{\varepsilon}_{H^{m-s}(\Omega)}$.

###### Proof.

The interpolation [49] of $H^{m}(\Omega)$ between $H^{m-s}(\Omega)$ and
$H^{m+\varepsilon}(\Omega)$ for $e:=u-Ju_{\mathrm{nc}}\in V\cap
H^{m+\varepsilon}(\Omega)$ leads to
$\|e\|_{H^{m}(\Omega)}\lesssim\|e\|^{s/(s+\varepsilon)}_{H^{m-s}(\Omega)}\|e\|^{\varepsilon/(s+\varepsilon)}_{H^{m+\varepsilon}(\Omega)}.$
Lemma 4.4 reveals $\|e\|_{H^{m+\varepsilon}(\Omega)}\lesssim
h_{\max}^{\sigma-\varepsilon}|u|_{H^{m+\sigma}(\Omega)}$ and concludes the
proof. ∎

###### Proof of Theorem 4.2 for $m-\sigma\leq s\leq m$.

The upper bound for the error $e:=u-Ju_{\mathrm{nc}}\in V$ in the norm of
$H^{s}(\Omega)$ follows from Theorem 4.1. The assumption (4.11) and Lemma 4.5
reveal
$h_{\max}^{\sigma+m-s}\lesssim|u|_{H^{m+\sigma}(\Omega)}^{s/(\sigma+\varepsilon)}\|e\|_{H^{s}(\Omega)}$.
This concludes the proof for $m-\sigma\leq s\leq m$. ∎

###### Proof of Theorem 4.2 for $m\leq s<m+1/2$.

The upper bounds for quasi-uniform meshes follow from Lemma 4.4. The lower
bounds are derived by interpolation for $e:=u-Ju_{\mathrm{nc}}\in V\cap
H^{m+\varepsilon}(\Omega)$. For the convex coefficients
$\theta={\varepsilon}/({\sigma+\varepsilon})$ and
$1-\theta={\sigma}/({\sigma+\varepsilon})$ in
$m=\theta(m-\sigma)+(1-\theta)(m+\varepsilon)$, an interpolation [49] shows

$|\\!|\\!|e|\\!|\\!|\leq
C(\varepsilon)\|e\|_{H^{m-\sigma}(\Omega)}^{\theta}\|e\|_{H^{m+\varepsilon}(\Omega)}^{1-\theta}$

for some constant $C(\varepsilon)$ that exclusively depends on $\varepsilon$
and $\Omega$. This and Theorem 4.1 result in
$|\\!|\\!|e|\\!|\\!|^{\sigma+\varepsilon}\lesssim
h_{\max}^{2\sigma}\varepsilon\|e\|_{H^{m+\varepsilon}(\Omega)}^{\sigma}$.
Recall (4.11) and derive the asserted lower bound
$h_{\max}^{\sigma-\varepsilon}\lesssim\|e\|_{H^{m+\varepsilon}(\Omega)}$. This
concludes the proof. ∎

##  5 Comments and examples of data in $H^{-{m}}(\Omega)$

###  5.1 The space $H^{-m}(\Omega)$

The space $H^{-m}(\Omega)$ is the dual to $H^{m}_{0}(\Omega)$ and is a real
Hilbert space. It is known that for any functional $F\in H^{-m}(\Omega)$,
there exist $f_{\alpha}\in L^{2}(\Omega)$ for all multi-indices $\alpha$ with
$|\alpha|\leq m$ such that

$F(v)=\sum_{|\alpha|\leq
m}\int_{\Omega}f_{\alpha}\partial_{\alpha}v{\rm\,dx}\text{ for all }v\in
H^{m}_{0}(\Omega)$ (5.1)

(with multi-index notation
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\in{\mathbb{N}}_{0}^{n}$ and
$|\alpha|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ for the partial derivatives
$\partial_{\alpha}v:=\partial^{|\alpha|}v/(\partial^{\alpha_{1}}x_{1}\cdots\partial^{\alpha_{n}}x_{n})$
etc). The family $(f_{\alpha}:|\alpha|\leq m)$ of $L^{2}(\Omega)$ functions
with (5.1) is not unique. For instance, given any ${F}\in L^{2}(\Omega)$ one
may solve $(-1)^{m}\Delta^{m}u={F}$ for a weak solution $u\in
H^{m}_{0}(\Omega)$ and consider $G:=D^{m}u\in
L^{2}(\Omega;{\mathbb{R}}^{(n^{m})})$ with

$F(v)=a(u,v)=\int_{\Omega}G:D^{m}v{\rm\,dx}\text{ for all }v\in
H^{m}_{0}(\Omega).$

Hence $f_{\alpha}:=\begin{cases}F\text{ if }\alpha=(0,0,\cdots,0)\\\ 0\text{
if }1\leq|\alpha|\leq m\end{cases}$ and $f_{\alpha}:=\begin{cases}0\text{ if
}|\alpha|\leq m-1\\\ {G_{\alpha}}\text{ if }|\alpha|=m\end{cases}$ with
$G_{\alpha}=\partial_{\alpha}u$ define two sets of data
$(f_{\alpha}:|\alpha|\leq m)$ on the right-hand side of (5.1) for the same
functional $F$. It is also known [49, Lemma 15.9, p.78] or [32, Thm.1. in 5.9,
p.253] for $m=1$ that $\|F\|_{H^{-m}(\Omega)}$ is the infimum of
$\sqrt{\sum_{|\alpha|\leq m}\|f_{\alpha}\|^{2}_{L^{2}(\Omega)}}$ for all
$(f_{\alpha}:|\alpha|\leq m)$ with (5.1).

It cannot be overemphasized that the preprocessing of the data set
$(f_{\alpha}:|\alpha|\leq m)$ in (5.1) is a subtle issue that essentially
depends on the way the right-hand side $F$ is provided. For instance, point
forces may apply, a right-hand side $f\in L^{1}(\Omega)$ for $m\geq 2$ and
line integrals may be modeled for $m\geq 1$. The main idea will be explained
for a model example with the right-hand side $F\in H^{-m}(\Omega)$,
$\Omega\subset{\mathbb{R}}^{2}$ in (5.1) of the form

$F(v):=\int_{\Omega}G:D^{m}v{\rm\,dx}+\int_{\Omega}gv{\rm\,dx}\quad\text{for
all }v\in V=H_{0}^{m}(\Omega)$ (5.2)

for given $G\in L^{2}(\Omega;{\mathbb{R}}^{2^{m}})$ and $g\in L^{2}(\Omega)$.
Consider the vector space $H({\rm div}^{m},\Omega;{\mathbb{R}}^{2^{m}})$ of
all $L^{2}$ functions $Q\in L^{2}(\Omega;{\mathbb{R}}^{2^{m}})$ such that the
$m$-th distributional divergence ${\rm div}^{m}Q\in L^{2}(\Omega)$ is a
Lebesgue function: $q\in L^{2}(\Omega)$ is called the $m$-th divergence of
some $Q\in L^{2}(\Omega;{\mathbb{R}}^{2^{m}})$ if, for all $v\in D(\Omega)$,
$\int_{\Omega}Q:D^{m}v{\rm\,dx}=(-1)^{m}\int_{\Omega}qv{\rm\,dx}.$ This holds
for all $v\in H^{m}_{0}(\Omega)$ as well (by density of $D(\Omega)$ in
$H^{m}_{0}(\Omega)$) and one writes $q=(-1)^{m}{\rm div}^{m}\>Q$. Hence any
$Q$ and any $v\in H^{m}_{0}(\Omega)$ satisfy

$F(v)=\int_{\Omega}(G-Q):D^{m}v{\rm\,dx}+\int_{\Omega}(g+{\rm
div}^{m}\>Q)\>v{\rm\,dx}$ (5.3)

In other words, the data $G$ and $g$ of $F$ in (5.2) can be modified to
$G^{\prime}:=G-Q$ and $g^{\prime}:=g+{\rm div}^{m}\>Q$ without changing the
action of $F$ on the continuous level. This data preprocessing may reduce the
data oscillations as they arise e.g. in the a posteriori error control.

###  5.2 Point forces $(m=2)$

The mechanical model often involves point loads. The Dirac delta functional
$\delta_{z}\in H^{-2}(\Omega)$ is the evaluation of the continuous test
function $u\in V=H^{2}_{0}(\Omega)\hookrightarrow C({\overline{\Omega}})$ at a
fixed point $z\in\overline{\Omega}$, called atom of the delta functional.
Unlike for $m=1$, $\delta_{z}:V\rightarrow{\mathbb{R}}$ is linear and bounded
for $m\geq 2$. It is well known that (if $\langle\bullet,\bullet\rangle$
abbreviates the duality brackets in $H^{-2}(\Omega)\times H^{2}(\Omega)$ that
extends the $L^{2}$ scalar product)

$\langle\delta_{z},v\rangle=v(z)=\frac{1}{2\pi}\int_{\Omega}\Delta
v(x)\log|x-z|{\rm\,dx}\quad\text{for all }v\in V=H_{0}^{m}(\Omega).$

(cf. e.g. [33, Ex.1, page 82] or [32, Thm 1, page 23] for further details and
proofs.) Hence $\delta_{z}$ is represented by
$G{(x)}:=\frac{1}{2\pi}\log|x-z|1_{2\times 2}$ for $x\in\Omega$ with the
$2\times 2$ unit matrix $1_{2\times 2}$ and $g\equiv 0$. Since $G\in
L^{2}(\Omega;{\mathbb{S}})$, the results of this survey apply to this case as
well, but the practical computation may avoid integration with the weakly
singular function $\log|x-z|$ and just apply $\delta_{z}$ on the point
evaluation. The point force at $a\in A\subset\mathcal{V}(\Omega)$ does not
contribute to the upper bound below in Proposition 5.1 when located at an
interior vertex of the triangulation. The model example will suppose this for
simplicity and because it can easily arranged by appropriate triangulations in
practice at least for a small number of point forces.

Before the next subsection continues with the corresponding model example, the
effect of a general location shall at least be illuminated.

######  Example 5.1 (point force not located at a vertex).

Suppose that there is a point force at $z\in\Omega$ that is not correlated to
the triangulation $\mathcal{T}$. The original Morley FEM has to model
$\delta_{z}$ also in case $z\in\partial T_{+}\cap\partial T_{-}=E$ is shared
by two triangles $T_{+}$ and $T_{-}$. The test function
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ is (in general) discontinuous at
$z$ and

$\displaystyle F_{h}$
$\displaystyle:=\mu\delta_{T_{+},z}+(1-\mu)\delta_{T_{-},z}$ (5.4)

applies with the piecewise point evaluation
$\langle\delta_{T_{\pm},z},v_{\mathrm{pw}}\rangle_{H^{2}(\mathcal{T})}:=(v_{\mathrm{pw}}|_{T_{\pm}})(z)$
for all $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$ and a convex coefficient
$0\leq\mu\leq 1$; $\mu=1/2$ appears natural. The choice of $\mu$ does not
affect the modified Morley FEM, but it will do so for the original scheme in
general. The a priori error analysis of the best-approximation for the
modified Morley FEM remains valid and the Proposition 5.1 is easily modified.
A standard Bramble-Hilbert argument shows for $w\in H^{2}(T)$,
$T\in\mathcal{T}$, that $w:=v_{\mathrm{M}}-Jv_{\mathrm{M}}$ satisfies

$\|w\|_{L^{\infty}(T)}\leq C_{7}{}h_{T}|w|_{H^{2}(T)}$

(In fact, the $P_{1}$ FEM interpolation error analysis shows in the above
displayed inequality for a left-hand side $|w|_{H^{1}(T)}$ based on $w\in
H^{2}(T)$ with $w(z)=0$ for all vertices $z\in\mathcal{V}(T)$. The $H^{1}$
seminorm and the $L^{\infty}$ norm do not scale in 2D [28] and this leads to
the assertion.) It follows that the absolute value of

$F_{h}(v_{\mathrm{M}})-F(Jv_{\mathrm{M}})=\mu(w|_{T_{+}})(z)+(1-\mu)(w|_{T_{-}})(z)$

is smaller than or equal to
$\mu\|w\|_{L^{\infty}(T_{+})}+(1-\mu)\|w\|_{L^{\infty}(T_{-})}\leq
C_{\ref{conj}}\left(\mu
h_{T_{+}}+(1-\mu)h_{T_{-}}\right)|\\!|\\!|{w}|\\!|\\!|_{\mathrm{pw}}.$ This
leads to the additional data-oscillation term $\mu
h_{T_{+}}+(1-\mu)h_{T_{-}}$.

###  5.3 Model example

The smoother $J$ in the right-hand side of the discrete problem (3.9) leads to
an a priori best-approximation result for general data $F\in H^{-m}(\Omega)$.
The a posteriori error control requires data evaluation in explicit residual-
based a posteriori error estimates. Throughout this section the particular
format of the right-hand side $F\in H^{-m}(\Omega)$ in (2.1) is assumed of the
form

$F(v):=\int_{\Omega}G:D^{m}v{\rm\,dx}+\int_{\Omega}gv{\rm\,dx}+\sum_{a\in
A}\beta(a)v(a)\text{ for all }v\in V$ (5.5)

for given $G\in L^{2}(\Omega;{\mathbb{R}}^{(n^{m})})$, $g\in L^{2}(\Omega)$,
$A$ is a finite set of interior vertices, and $a\in
A\subset\mathcal{V}(\Omega)$ is the position and $\beta(a)$ the strength of
the point force. The natural right-hand side
$F_{h}:=\widehat{F}|_{V_{\mathrm{nc}}}\in V_{\mathrm{nc}}^{*}$ in the discrete
problem for (5.5) reads, for all
$\widehat{v}=v+v_{\mathrm{nc}}\in\widehat{V}\equiv V+V_{\mathrm{nc}}$,

$\widehat{F}(\widehat{v}):=\int_{\Omega}G:D_{\mathrm{pw}}^{m}\widehat{v}{\rm\,dx}+\int_{\Omega}g\widehat{v}{\rm\,dx}+\sum_{a\in
A}\beta(a)\widehat{v}(a).$ (5.6)

###  5.4 Comparison

The two schemes (3.9) and (2.2) for $F_{h}=\widehat{F}|_{V_{\mathrm{nc}}}$
from (5.6) are in fact different. While the modified scheme (3.9) with right-
hand side $F\circ J$ enjoys a best-approximation property, the original
version for the right-hand side $\widehat{F}|_{V_{\mathrm{nc}}}$ from (5.6)
does so only up to data-oscillations.

######  Proposition 5.1.

Given $F$ and ${\widehat{F}}$ in (5.5)-(5.6), let $u_{\mathrm{nc}}^{\rm
org}\in V_{\mathrm{nc}}$ solve (2.2) and let $u_{\mathrm{nc}}^{\rm mod}$ solve
(3.9). Then $\Lambda_{0}^{-1}|\\!|\\!|u_{\mathrm{nc}}^{\rm
org}-u_{\mathrm{nc}}^{\rm
mod}|\\!|\\!|_{\mathrm{pw}}\leq\|G-\Pi_{0}G\|+\kappa_{m}\mathrm{osc}_{m}(g,\mathcal{T}).$

###### Proof.

Since $\displaystyle|\\!|\\!|u_{\mathrm{nc}}^{\rm org}-u_{\mathrm{nc}}^{\rm
mod}|\\!|\\!|^{2}_{\mathrm{pw}}=\widehat{F}((1-J)(u_{\mathrm{nc}}^{\rm
org}-u_{\mathrm{nc}}^{\rm
mod}))\leq\|\widehat{F}\circ(1-J)\|_{V^{*}_{\mathrm{nc}}}\>|\\!|\\!|u_{\mathrm{nc}}^{\rm
org}-u_{\mathrm{nc}}^{\rm mod}|\\!|\\!|_{\mathrm{pw}}$, it remains to analyse
$\|\widehat{F}\circ(1-J)\|_{V^{*}_{\mathrm{nc}}}$. For any $v_{\mathrm{nc}}\in
V_{\mathrm{nc}}$ with $|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$, the
definition (5.6) shows

$\displaystyle\widehat{F}\circ(1-J)(v_{\mathrm{nc}})$
$\displaystyle=\int_{\Omega}G:D^{2}_{\mathrm{pw}}(v_{\mathrm{nc}}-Jv_{\mathrm{nc}}){\rm\,dx}+\int_{\Omega}g(v_{\mathrm{nc}}-Jv_{\mathrm{nc}}){\rm\,dx}$
$\displaystyle=\int_{\Omega}(G-\Pi_{0}G):D^{2}_{\mathrm{pw}}(v_{\mathrm{nc}}-Jv_{\mathrm{nc}}){\rm\,dx}+\int_{\Omega}(g-\Pi_{m}g)(v_{\mathrm{nc}}-Jv_{\mathrm{nc}}){\rm\,dx}$

with the assumptions that
$\Pi_{0}D^{2}Jv_{\mathrm{nc}}=D^{2}_{\mathrm{pw}}v_{\mathrm{nc}}$ and
$\Pi_{m}Jv_{\mathrm{nc}}=v_{\mathrm{nc}}$ for $m=1$ and $m=2$ in the last step
that holds for CR and Morley FEM (see Sections 2.3.1 and 2.3.2). This and
$\|h_{\mathcal{T}}^{-m}(v_{\mathrm{nc}}-Jv_{\mathrm{nc}})\|\leq\kappa_{m}|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
from (2.4) and (2.7) prove

$\widehat{F}\circ(1-J)(v_{\mathrm{nc}})\leq(\|G-\Pi_{0}G\|+\kappa_{m}\>\mathrm{osc}_{m}(g,\mathcal{T}))|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}.$

The definition of $\Lambda_{0}$ in (3.10) shows
$|\\!|\\!|v_{\mathrm{nc}}-Jv_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}|\\!|\\!|v_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}.$
The combination of the preceding estimates proves
$\widehat{F}\circ(1-J)(v_{\mathrm{nc}})\leq\Lambda_{0}(\|G-\Pi_{0}G\|+\kappa_{m}\>\mathrm{osc}_{m}(g,\mathcal{T})).$
This concludes the proof. ∎

The data-oscillation term $\mathrm{osc}_{m}(g,\mathcal{T})=o(h_{\rm max})$ can
be of higher-order if $g\in H^{1}(\mathcal{T})$ is piecewise smooth for $m\geq
2$, while this is less clear for the data-oscillation $\|G-\Pi_{0}G\|$.

The subsequent example shows that $(a)$ the original scheme can be optimal
while the modified one is not, $(b)$ the bound in the Proposition 5.1 is sharp
in general (at least if $g\equiv 0$). The methodology is similar to the
analysis in Theorem 3.3; in fact $(c)$ is another proof of the optimality of
$C_{\mathrm{qo}}=1+\Lambda_{0}^{2}$ (recall $\Lambda_{0}>0$ throughout the
paper).

######  Example 5.2.

There exist $G:=D^{m}Jz_{\mathrm{nc}}\in D^{m}V_{\rm c}$ and $g:=0$ in (5.5)
for some $z_{\mathrm{nc}}\in V_{\mathrm{nc}}$ such that the exact solution
$u\in V$ to (2.1), the discrete solution $u_{\mathrm{nc}}^{\rm org}$ to the
original scheme (2.2) and the discrete solution $u_{\mathrm{nc}}^{\rm mod}\in
V_{\mathrm{nc}}$ to (3.9) with $F$ and $F_{h}=\widehat{F}|_{V_{h}}$ from
(5.5)-(5.6), satisfy $(a)\;u_{\mathrm{nc}}^{\rm org}=I_{\mathrm{nc}}u,$
$(b)\;|\\!|\\!|u_{\mathrm{nc}}^{\rm org}-u_{\mathrm{nc}}^{\rm
mod}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}^{2},$ and
$(c)\;|\\!|\\!|u-u_{\mathrm{nc}}^{\rm
mod}|\\!|\\!|^{2}_{\mathrm{pw}}=|\\!|\\!|u-u_{\mathrm{nc}}^{\rm
org}|\\!|\\!|^{2}_{\mathrm{pw}}+\Lambda_{0}^{4}={\Lambda_{0}^{2}(1+\Lambda_{0}^{2}).}$

Proof of $(a)$. Given $V_{\mathrm{nc}}$ and $J$ from Subsections 2.2 and 2.4,
there exists $z_{\mathrm{nc}}\in V_{\mathrm{nc}}$ with
$|\\!|\\!|z_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$ and
$|\\!|\\!|z_{\mathrm{nc}}-Jz_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}|\\!|\\!|z_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}$.
Let $G:=D^{m}Jz_{\mathrm{nc}}$ and recall
$\Pi_{0}G=D^{m}_{\mathrm{pw}}I_{\mathrm{nc}}Jz_{\mathrm{nc}}=D^{m}_{\mathrm{pw}}z_{\mathrm{nc}}$
with $\|\Pi_{0}G\|=|\\!|\\!|z_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$ and
$\|G-\Pi_{0}G\|=|\\!|\\!|z_{\mathrm{nc}}-Jz_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=\Lambda_{0}$.
For all $v_{\mathrm{nc}}\in V_{\mathrm{nc}}$, (2.3) shows

$a_{\mathrm{pw}}(z_{\mathrm{nc}},v_{\mathrm{nc}})=a_{\mathrm{pw}}(Jz_{\mathrm{nc}},v_{\mathrm{nc}})=\int_{\Omega}G:D^{m}_{\mathrm{pw}}v_{\mathrm{nc}}{\rm\,dx}$

with $G:=D^{m}Jz_{\mathrm{nc}}$ in the last step. Hence $u_{\mathrm{nc}}^{\rm
org}=z_{\mathrm{nc}}$. Since $g=0$, for $u=Jz_{\mathrm{nc}}$, (5.5) shows

$a(u,v)=a(Jz_{\mathrm{nc}},v)=\int_{\Omega}G:D^{m}v{\rm\,dx}\text{ for all
}v\in V.$

This and
$I_{\mathrm{nc}}u=I_{\mathrm{nc}}Jz_{\mathrm{nc}}=z_{\mathrm{nc}}=u_{\mathrm{nc}}^{\rm
org}$ proves the equality in $(a)$.

Proof of $(b)$. Recall

$\displaystyle\Lambda_{0}^{2}$
$\displaystyle=|\\!|\\!|z_{\mathrm{nc}}-Jz_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}=a_{\mathrm{pw}}(Jz_{\mathrm{nc}}-z_{\mathrm{nc}},Jz_{\mathrm{nc}})$
$\displaystyle=a(u,u)-a_{\mathrm{pw}}(z_{\mathrm{nc}},z_{\mathrm{nc}})=\int_{\Omega}G:D^{m}Jz_{\mathrm{nc}}{\rm\,dx}-\int_{\Omega}G:D^{m}_{\mathrm{pw}}z_{\mathrm{nc}}{\rm\,dx}$
$\displaystyle=a_{\mathrm{pw}}(u_{\mathrm{nc}}^{\rm mod}-u_{\mathrm{nc}}^{\rm
org},z_{\mathrm{nc}}).$

A Cauchy inequality in the semi-scalar inner product $a_{\mathrm{pw}}$ with
$|\\!|\\!|z_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}=1$ leads to

$\Lambda_{0}^{2}=a_{\mathrm{pw}}(u_{\mathrm{nc}}^{\rm
mod}-u_{\mathrm{nc}}^{\rm
org},z_{\mathrm{nc}})\leq|\\!|\\!|u_{\mathrm{nc}}^{\rm
mod}-u_{\mathrm{nc}}^{\rm org}|\\!|\\!|_{\mathrm{pw}}.$

For $g=0$ and $\|G-\Pi_{0}G\|=\Lambda_{0}$ from above, the Proposition 5.1
shows $|\\!|\\!|u_{\mathrm{nc}}^{\rm mod}-u_{\mathrm{nc}}^{\rm
org}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}^{2}$. This concludes the proof of
$(b)$. ∎

Proof of $(c)$. The assertion $(c)$ follows from the above and the Pythagoras
theorem

$|\\!|\\!|u-u_{\mathrm{nc}}^{\rm
mod}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|Jz_{\mathrm{nc}}-z_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|u_{\mathrm{nc}}^{\rm
mod}-u_{\mathrm{nc}}^{\rm org}|\\!|\\!|_{\mathrm{pw}}^{2}.\qed$

###  5.5 Counterexamples for best-approximation

The best-approximation property for the original nonconforming FEM with the
natural right-hand side fails in general: The data oscillation terms cannot be
removed. Suppose $\Omega$ a simply-connected domain throughout this
subsection.

####  5.5.1 Counterexample to best-approximation of CRFEM for $m=1$

Suppose $|\mathcal{T}|\geq 2$ so that there exists some
$\beta_{\mathrm{CR}}\in\mathrm{CR}^{1}(\mathcal{T})\setminus
S^{1}(\mathcal{T})$. Since the quotient space
$S^{1}(\mathcal{T})/{\mathbb{R}}$ is a Hilbert space with scalar product
$a_{\mathrm{pw}}(\bullet,\bullet)$, the linear functional $v_{\rm c}\mapsto
a_{\mathrm{pw}}(\beta_{\mathrm{CR}},v_{\rm
c})\in(S^{1}(\mathcal{T})\setminus{\mathbb{R}})^{*}$ has a Riesz
representation $\beta_{\rm c}$ so that
$b_{\mathrm{CR}}:=\beta_{\mathrm{CR}}-\beta_{\rm
c}\in\mathrm{CR}^{1}(\mathcal{T})\setminus S^{1}(\mathcal{T})$ satisfies
$a_{\mathrm{pw}}(b_{\mathrm{CR}},v_{\rm c})=0$ for all $v_{\rm c}\in
S^{1}(\mathcal{T})$. Without loss of generality, suppose
$|\\!|\\!|b_{\mathrm{CR}}|\\!|\\!|_{\mathrm{pw}}=1$. Since ${\rm
Curl}_{\mathrm{pw}}b_{\mathrm{CR}}$ is a rotated gradient
$\nabla_{\mathrm{pw}}b_{\mathrm{CR}}$ etc. in $2$D, the $L^{2}$ orthogonality
${\rm Curl}_{\mathrm{pw}}b_{\mathrm{CR}}\perp{\rm Curl}\>S^{1}(\mathcal{T})$
in $L^{2}(\Omega;{\mathbb{R}}^{2})$ follows. The companion operators in this
paper obey homogeneous boundary conditions but are naturally extended to
functions without any restriction. This leads to some $b:=Jb_{\mathrm{CR}}\in
H^{1}(\Omega)$ with $b_{\mathrm{CR}}:=I_{\mathrm{CR}}b$. Given those
functions, define the data $G:={\rm Curl}\>b\perp\nabla H^{1}_{0}(\Omega)$ and
$g\equiv 0$ in (5.5)-(5.6) for $F$ and ${\widehat{F}}$. Then the exact
solution $u\equiv 0$ and hence the best approximation provides
$u_{\mathrm{CR}}^{\rm mod}\equiv 0$ as well. The investigation of the original
CRFEM solution $u_{\mathrm{nc}}^{\rm org}\in\mathrm{CR}_{0}^{1}(\mathcal{T})$
to (2.2) recalls the discrete Helmholtz decomposition from Subsection 2.3.1:
$P_{0}(\mathcal{T};{\mathbb{R}}^{2})=\nabla_{\mathrm{pw}}\mathrm{CR}^{1}_{0}(\mathcal{T})\operp{\rm
Curl}\>(S^{1}(\mathcal{T})\setminus{\mathbb{R}})$. The $L^{2}$ orthogonality
of $\Pi_{0}G=\Pi_{0}\>{\rm Curl}\>b={\rm
Curl}_{\mathrm{pw}}\>I_{\mathrm{CR}}b={\rm Curl}_{\mathrm{pw}}$
$b_{\mathrm{CR}}\perp{\rm Curl}\>S^{1}(\mathcal{T})$ in
$L^{2}(\Omega;{\mathbb{R}}^{2})$ therefore proves
$\Pi_{0}G=\nabla_{\mathrm{pw}}u_{\mathrm{CR}}$ for some
$u_{\mathrm{CR}}\in\mathrm{CR}^{1}_{0}(\mathcal{T})$. In fact,
$u_{\mathrm{CR}}$ solves the original CRFEM because of

$a_{\mathrm{pw}}(u_{\mathrm{CR}},v_{\mathrm{CR}})=\int_{\Omega}\Pi_{0}G\cdot\nabla_{\mathrm{pw}}v_{\mathrm{CR}}{\rm\,dx}=\int_{\Omega}G\cdot\nabla_{\mathrm{pw}}v_{\mathrm{CR}}{\rm\,dx}$

for all $v_{\mathrm{CR}}\in\mathrm{CR}^{1}_{0}(\mathcal{T})$. That is,
$|\\!|\\!|u_{\mathrm{CR}}^{\rm
org}|\\!|\\!|_{\mathrm{pw}}=|\\!|\\!|u_{\mathrm{CR}}|\\!|\\!|_{\mathrm{pw}}=\|\Pi_{0}G\|=|\\!|\\!|I_{\mathrm{CR}}b|\\!|\\!|_{\mathrm{pw}}=|\\!|\\!|b_{\mathrm{CR}}|\\!|\\!|_{\mathrm{pw}}=1$.
This contradicts the best-approximation property.

####  5.5.2 Counterexample to best-approximation of Morley FEM for $m=2$

Suppose $|\mathcal{T}|\geq 2$ and recall
$S^{1}(\mathcal{T})\subsetneq\mathrm{CR}^{1}_{0}(\mathcal{T})$ for the
existence of
$\beta_{\mathrm{CR}}\in\mathrm{CR}^{1}(\mathcal{T};{\mathbb{R}}^{2})\setminus
S^{1}(\mathcal{T};{\mathbb{R}}^{2})$. The linear Green strain
$\varepsilon:={\rm sym}\>D$ from linear elasticity allows for a Korn
inequality with the rigid body motions

${\rm RH}(\Omega):=\\{\gamma\in
P_{1}(\Omega;{\mathbb{R}}^{2}):D\gamma+D\gamma^{T}=0\\}$

(a linear subspace of dimension $3$ of the form
$\gamma(x)-(a,b)+c(x_{2},-x_{1})$ for all $x\in\Omega$). The semi-scalar
product $(\varepsilon(\bullet),\varepsilon(\bullet))_{L^{2}(\Omega)}$ in the
vector space $S^{1}(\mathcal{T};{\mathbb{R}}^{2})/{\rm RH}(\Omega)$ is a
scalar product (from the Korn inequality). The Riesz representation
$\beta_{\rm c}\in S^{1}(\mathcal{T};{\mathbb{R}}^{2})/{\rm RH}(\Omega)$ of the
functional $v_{\rm
c}\mapsto(\varepsilon_{\mathrm{pw}}(\beta_{\mathrm{CR}}),\varepsilon(v_{\rm
c}))$ in $(S^{1}(\mathcal{T};{\mathbb{R}}^{2})/{\rm RH}(\Omega))^{*}$ provides
$b_{\mathrm{CR}}:=\beta_{\mathrm{CR}}-\beta_{\rm
c}\in\mathrm{CR}^{1}(\mathcal{T};{\mathbb{R}}^{2})\setminus
S^{1}(\mathcal{T};{\mathbb{R}}^{2})$ with
$\varepsilon_{\mathrm{pw}}(b_{\mathrm{CR}})\perp\varepsilon(S^{1}(\mathcal{T};{\mathbb{R}}^{2}))$
in $L^{2}(\Omega;{\mathbb{S}})$. Without loss of generality suppose
$\|\varepsilon(b_{\mathrm{CR}})\|=1$. Recall the companion operator
$J:\mathrm{CR}^{1}(\mathcal{T})\rightarrow H^{1}(\Omega)$ from the previous
example and apply
$Jb_{\mathrm{CR}}=(Jb_{\mathrm{CR}}(1),Jb_{\mathrm{CR}}(2))\in
H^{1}(\Omega;{\mathbb{R}}^{2})$ componentwise to
$b_{\mathrm{CR}}\in\mathrm{CR}^{1}(\mathcal{T};{\mathbb{R}}^{2})$. Then let
$G:={\rm sym}\>{\rm Curl}\>(Jb_{\mathrm{CR}}(2),-Jb_{\mathrm{CR}}(1))$ and
$g\equiv 0$. Since $G\perp D^{2}H^{2}_{0}(\Omega)$, the solutions
$u=0=u_{\mathrm{M}}^{\rm mod}$ coincide. On the other hand, for all
$\phi,\psi\in H^{1}(T;{\mathbb{R}}^{2})$,

$({\rm sym}\>{\rm Curl}\>(\phi_{2},-\phi_{1})):({\rm sym}\>{\rm
Curl}\>(\psi_{2},-\psi_{1}))=\varepsilon(\phi):\varepsilon(\psi)\text{ a.e. in
}T.$

This relation shows

$\displaystyle\Pi_{0}G$ $\displaystyle={\rm sym}\>{\rm
Curl}_{\mathrm{pw}}\>I_{\mathrm{CR}}(Jb_{\mathrm{CR}}(2),-Jb_{\mathrm{CR}}(1))={\rm
sym}\>{\rm Curl}_{\mathrm{pw}}\>(b_{\mathrm{CR}}(2),-b_{\mathrm{CR}}(1))$
$\displaystyle\perp{\rm sym}\>{\rm
Curl}\;S^{1}(\mathcal{T};{\mathbb{R}}^{2})\text{ in
}L^{2}(\Omega;{\mathbb{S}}).$

The discrete Helmholtz decomposition
$P_{0}(\mathcal{T};{\mathbb{S}})=D^{2}_{\mathrm{pw}}\mathrm{M}(\mathcal{T})\operp{\rm
sym}\>{\rm Curl}\;S^{1}(\mathcal{T};{\mathbb{R}}^{2}/{\mathbb{R}}^{2})$ from
Subsection 2.3.2 implies $\Pi_{0}G=D^{2}_{\mathrm{pw}}u_{\mathrm{M}}$ for some
$u_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$. Then
$u_{\mathrm{M}}=u_{\mathrm{M}}^{\rm org}$ and $|\\!|\\!|u_{\mathrm{M}}^{\rm
org}|\\!|\\!|_{\mathrm{pw}}=\|\Pi_{0}G\|=\|b_{\mathrm{CR}}\|_{L^{2}(T)}=1$.
This contradicts the best-approximation property.

##  6 A posteriori analysis

Unlike conforming schemes where the guaranteed error bound involves constants
from approximation in the error estimates of a quasi-interpolation operator,
the constants in the nonconforming schemes appear small and are related to the
simplicial element domain (and not the shape of nodal patches). Hence the
resulting a posteriori error estimators can indeed be employed directly in the
a posteriori error control.

###  6.1 Reliable error estimates

The natural right-hand side in the discrete problem is
$F_{h}:=\widehat{F}|_{V_{\mathrm{nc}}}\in V_{\mathrm{nc}}^{*}$ with
$\widehat{F}$ from (5.6).

######  Theorem 6.1 (a posteriori estimate for
$F_{h}=\widehat{F}|_{V_{\mathrm{nc}}}$).

Suppose the $L^{2}$ orthogonality in (2.6). Let $u\in V$ solve (2.1) and let
$u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ solve (2.2) with right-hand side
$F_{h}:=\widehat{F}|_{V_{\mathrm{nc}}}$ from (5.6). Then

$\displaystyle(a)\quad|\\!|\\!|u-Ju_{\mathrm{nc}}|\\!|\\!|^{2}$
$\displaystyle+|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}\leq(\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}})^{2},$
$\displaystyle(b)\quad|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}$
$\displaystyle+|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}\leq(\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+2\widehat{F}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}}).$

The former analysis in [41, 13, 5] is based on Helmholtz decompositions and
worked out for simply-connected domains. This is problematic and the three-
dimensional application in [41] requires some 3D Helmholtz decompositions for
a domain that is simply-connected and has a boundary that is connected and a
discrete counterpart is unknown. The present analysis is rather simple and can
immediately be extended to any bounded polyhedral Lipschitz domain with the
help of the companion operator $J$ in 3D from [25].

Proof of (a). For $e:=u-Ju_{\mathrm{nc}}$, elementary algebra and (2.1)-(2.3)
lead to

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}$
$\displaystyle=a(u,e)+a_{\mathrm{pw}}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}},e)-a_{\mathrm{pw}}(u_{\mathrm{nc}},I_{\mathrm{nc}}e)$
$\displaystyle=\widehat{F}(e-I_{\mathrm{nc}}e)+a_{\mathrm{pw}}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}},e-I_{\mathrm{nc}}e).$
(6.1)

This, the definition of $\widehat{F}$ in (5.6), (2.6), and a Cauchy inequality
show

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}$
$\displaystyle\leq\int_{\Omega}(G-\Pi_{0}G):D^{m}(e-I_{\mathrm{nc}}e){\rm\,dx}+\int_{\Omega}g(e-I_{\mathrm{nc}}e){\rm\,dx}$
$\displaystyle\quad+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}.$

This, (weighted) Cauchy inequalities and
$\|h_{\mathcal{T}}^{-m}(e-I_{\mathrm{nc}}e)\|\leq\kappa_{m}|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}$
from (2.4) imply

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}$
$\displaystyle\leq(\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}})|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}.$

The Pythogoras theorem (2.5) shows
$|\\!|\\!|e|\\!|\\!|^{2}=|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}^{2}$.
This and Young’s inequality conclude the proof. ∎

Proof of $(b)$. Elementary algebra and
$a_{\mathrm{pw}}(u_{\mathrm{nc}},u-I_{\mathrm{nc}}u)=0=a_{\mathrm{pw}}(I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}})$
from (2.3) result in

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}=a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}})+a_{\mathrm{pw}}(u,u-Ju_{\mathrm{nc}})-a_{\mathrm{pw}}(u_{\mathrm{nc}},I_{\mathrm{nc}}u-u_{\mathrm{nc}}).$

This, the Pythagoras theorem (2.5), a Cauchy inequality, the continuous
problem (2.1) with the test function $u-Ju_{\mathrm{nc}}$, and (2.2) with the
test function $I_{\mathrm{nc}}u-u_{\mathrm{nc}}$ lead to

$\displaystyle|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|^{2}_{\mathrm{pw}}+|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}$
$\displaystyle\leq|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\quad+\widehat{F}(u-I_{\mathrm{nc}}u)+\widehat{F}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}}).$
(6.2)

The definition of $\widehat{F}$ in (5.6), (2.6), (weighted) Cauchy
inequalities, and (2.4) show

$\displaystyle\widehat{F}(u-I_{\mathrm{nc}}u)$
$\displaystyle=\int_{\Omega}(G-\Pi_{0}G):D^{m}(u-I_{\mathrm{nc}}u){\rm\,dx}+\int_{\Omega}g(u-I_{\mathrm{nc}}u){\rm\,dx}$
$\displaystyle\leq(\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|)|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$

The last two displayed estimates and Young’s inequality conclude the proof. ∎

######  Remark 6.1.

The term $\widehat{F}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}})$ can be computed and
so $(b)$ is an a posteriori error estimate. Moreover, (2.9) and (weighted)
Cauchy inequalities show

$\widehat{F}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}})\leq(\|G-\Pi_{0}G\|+\kappa_{m}\mathrm{osc}_{m}(g,\mathcal{T}))|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}.$

The combination with (6.1) leads to (a) up to the extra factor 2.

The modified choice of the right-hand side reads $F_{h}=F\circ J$ for the
right-inverse $J$ of $I_{\mathrm{nc}}$ from Section 2.4. The discrete problem
for this choice of $F_{h}$ from (3.9) leads to a (new) a posteriori estimate
in Theorem 6.2.

######  Theorem 6.2 (a posteriori estimate for $F_{h}=F\circ J$).

Suppose (5.5) and the $L^{2}$ orthogonality in (2.6). Let $u\in V$ solve (2.1)
and let $u_{\mathrm{nc}}\in V_{\mathrm{nc}}$ solve (3.9) with right-hand side
$F_{h}:=F\circ J$. Then

$\displaystyle(a)\;|\\!|\\!|u-Ju_{\mathrm{nc}}|\\!|\\!|$
$\displaystyle\leq\sqrt{1+\Lambda_{0}^{2}}\|G-\Pi_{0}G\|+\sqrt{(\kappa_{m}\|h_{\mathcal{T}}^{m}g\|+|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}})^{2}+\kappa_{m}^{2}\Lambda_{0}^{2}\mathrm{osc}^{2}_{m}(g,\mathcal{T})},$
$\displaystyle(b)\;|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq\sqrt{2(|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|^{2}+{\rm
apx}(F)^{2})}$

with ${\rm apx}(F):=(1+\Lambda_{\rm
J})\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|+\kappa_{m}\Lambda_{\rm
J}\mathrm{osc}_{m}(g,\mathcal{T})$ and $\Lambda_{0}$ (resp. $\Lambda_{\rm J}$)
from (3.10) (resp. (2.8)).

Proof of (a). Elementary algebra with $e:=u-Ju_{\mathrm{nc}}\in V$, (2.1),
$a_{\mathrm{pw}}(u_{\mathrm{nc}},e-I_{\mathrm{nc}}e)~{}=0$ from (2.3),
$a_{\mathrm{pw}}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}},~{}I_{\mathrm{nc}}e)~{}=~{}0$
from (2.7), and (3.9) lead to

$\displaystyle|\\!|\\!|e|\\!|\\!|^{2}$
$\displaystyle=a(u,e)+a_{\mathrm{pw}}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}},e)-a_{\mathrm{pw}}(u_{\mathrm{nc}},e)$
$\displaystyle=F(e-JI_{\mathrm{nc}}e)+a_{\mathrm{pw}}(u_{\mathrm{nc}}-Ju_{\mathrm{nc}},e-I_{\mathrm{nc}}e).$
(6.3)

For any $v\in V$, (2.9) and (2.6) imply
$\int_{\Omega}\Pi_{0}G:D^{m}(v-JI_{\mathrm{nc}}v){\rm\,dx}=0$. This, the
definition (5.5), and (2.9) result in

$\displaystyle F(v-JI_{\mathrm{nc}}v)$
$\displaystyle=\int_{\Omega}(G-\Pi_{0}G):D^{m}(v-JI_{\mathrm{nc}}v){\rm\,dx}+\int_{\Omega}g(v-I_{\mathrm{nc}}v){\rm\,dx}$
$\displaystyle\quad+\int_{\Omega}(g-\Pi_{0}g)((1-J)I_{\mathrm{nc}}v){\rm\,dx}$
$\displaystyle\leq\|G-\Pi_{0}G\||\\!|\\!|v-JI_{\mathrm{nc}}v|\\!|\\!|_{\mathrm{pw}}+\|h_{\mathcal{T}}^{m}g\|\|h_{\mathcal{T}}^{-m}(v-I_{\mathrm{nc}}v)\|$
$\displaystyle\quad+\mathrm{osc}_{m}(g,\mathcal{T})\|h_{\mathcal{T}}^{-m}(1-J)I_{\mathrm{nc}}v\|$
(6.4)

with (weighted) Cauchy inequalities in the last step. For
$\Lambda_{0}=\|1-J\|_{L(V_{\mathrm{nc}},\widehat{V})}$ from (3.10), a triangle
inequality and
$|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}^{2}$
from (2.5) prove

$|\\!|\\!|e-JI_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}\leq|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}+|\\!|\\!|(1-J)I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}\leq\sqrt{1+\Lambda_{0}^{2}}|\\!|\\!|e|\\!|\\!|_{\mathrm{pw}}.$

Recall from (2.4) and (2.5) that

$\|h_{\mathcal{T}}^{-m}(e-I_{\mathrm{nc}}e)\|\leq\kappa_{m}|\\!|\\!|e-I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}\text{
and
}\|h_{\mathcal{T}}^{-m}(1-J)I_{\mathrm{nc}}e\|\leq\kappa_{m}\Lambda_{0}|\\!|\\!|I_{\mathrm{nc}}e|\\!|\\!|_{\mathrm{pw}}.$

Substitute the last three displayed inequalities in (6.1) with $v:=e$ and then
substitute the resulting estimate in (6.1). Apply a Cauchy inequality for the
second term in the right-hand side of (6.1) and use (2.5) to conclude the
proof of $(a)$. ∎

Proof of (b). Elementary algebra with
$a_{\mathrm{pw}}(u_{\mathrm{nc}},u-I_{\mathrm{nc}}u)=0=a_{\mathrm{pw}}(I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}})$
from (2.3) and the continuous problem (2.1) with test function
$u-Ju_{\mathrm{nc}}$ show

$|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}=F(u-Ju_{\mathrm{nc}})+a_{\mathrm{pw}}(u-I_{\mathrm{nc}}u,Ju_{\mathrm{nc}}-u_{\mathrm{nc}})-a_{\mathrm{pw}}(u_{\mathrm{nc}},I_{\mathrm{nc}}u-u_{\mathrm{nc}}).$

The Pythagoras theorem (2.5), a Cauchy inequality, and (3.9) reveal in the
last identity that

$|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|^{2}_{\mathrm{pw}}+|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}\leq|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}+F(u-JI_{\mathrm{nc}}u).$
(6.5)

The choice of $v:=u$ in (6.1), a triangle inequality, (2.4), and twice (2.8)
for
$|\\!|\\!|I_{\mathrm{nc}}u-JI_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\rm
J}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}$ lead to

$\displaystyle
F(u-JI_{\mathrm{nc}}u)\leq|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}((1+\Lambda_{\rm
J})\|G-\Pi_{0}G\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g\|$
$\displaystyle\qquad\quad\qquad\quad+\kappa_{m}\Lambda_{\rm
J}\mathrm{osc}_{m}(g,\mathcal{T}))={\rm
apx}(F)|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}.$ (6.6)

A substitution of (6.1) in (6.5) and Young’s inequality show

$\frac{1}{2}|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|^{2}_{\mathrm{pw}}+|\\!|\\!|I_{\mathrm{nc}}u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}\leq|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}+{\rm
apx}(F)^{2}.$

This concludes the proof. ∎

######  Remark 6.2 (generalization).

The model scenario of the right-hand side in (5.5) solely coincides with the
lowest and highest order terms in (5.1) because oscillations occur in those
terms because of (2.3) and (2.9). The generalization of Theorems 6.1-6.2 to
the general situation (5.1) is straightforward and the extra terms read
$\displaystyle\sum_{1\leq|\alpha|\leq
m-1}\|h_{\mathcal{T}}^{m-|\alpha|}f_{\alpha}\|_{L^{2}(\Omega)}.$

######  Remark 6.3 (a posteriori estimates for (5.3)).

For the right-hand side in (5.3), the discretisation will change because
$\widehat{F}$ changes to

$\widehat{F}(\widehat{v}):=\int_{\Omega}(G-Q):D_{\mathrm{pw}}^{m}\widehat{v}{\rm\,dx}+\int_{\Omega}(g+{\rm
div}^{m}\>Q)\widehat{v}{\rm\,dx}\text{ for all
}\widehat{v}=v+v_{\mathrm{nc}}\in\widehat{V}\equiv V+V_{\mathrm{nc}}.$

It may be seen as an advantage of the modified scheme (3.9) that
$u_{\mathrm{nc}}$ is not affected by a change of $(G,g)$ to $(G{-}Q,g+{\rm
div}^{m}\>Q)$ for any $Q$. The a posteriori error control involves the terms
$\|G-\Pi_{0}G\|+\|h^{m}_{\mathcal{T}}g\|$ and this motivates the minimisation

$\displaystyle\min_{Q\in H({\rm
div}^{m},Q)}\left(\|(1-\Pi_{0})(G-Q)\|^{m}+\kappa_{m}\|h_{\mathcal{T}}^{m}(g+{\rm
div}^{m}\>Q)\|\right).$ (6.7)

The modified data $(G^{\prime},g^{\prime})=(G{-}Q,g+{\rm div}^{m}\>Q)$ for a
minimizer $Q$ leads to optimal a posteriori error control in Theorem 6.2 when
the upper bound is simplified to

$\displaystyle\frac{1}{2}|\\!|\\!|u-u_{\mathrm{nc}}|\\!|\\!|^{2}_{\mathrm{pw}}\leq|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}^{2}+2(1+\Lambda_{\rm
J})^{2}\left(\|G^{\prime}-\Pi_{0}G^{\prime}\|+\kappa_{m}\|h_{\mathcal{T}}^{m}g^{\prime}\|\right)^{2}.$
(6.8)

The situation is less clear in Theorem 6.1 because $u_{\mathrm{nc}}$ may be
affected by a change of $\widehat{F}$ and this may affect
$|\\!|\\!|u_{\mathrm{nc}}-Ju_{\mathrm{nc}}|\\!|\\!|_{\mathrm{pw}}$. However,
it is unrealistic to solve the minimisation problem (6.7) exactly for general
$G$ and $g$. Nevertheless, a preprocessing of the data $G$ and $g$ can reduce
the data approximation terms in (guaranteed) upper error bounds. In return,
the data selected can be poor as well and ruin the efficiency of the reliable
error estimates in Theorems 6.1 and 6.2.

###  6.2 Efficiency up to data-oscillations

The efficiency of the term $\|h_{\mathcal{T}}^{m}g\|_{L^{2}(\Omega)}$ up to
data-oscillations is established next.

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

Let $u$ solve (2.1) with right-hand side (5.5) for $G,g\in L^{2}(\Omega)$.
Then
$\displaystyle\|h_{\mathcal{T}}^{m}g\|_{L^{2}(\Omega)}\lesssim|\\!|\\!|u-I_{\mathrm{nc}}u|\\!|\\!|_{\mathrm{pw}}+\mathrm{osc}_{m}(g,\mathcal{T})+\|G-\Pi_{0}G\|.$

###### Proof.

The efficiency holds in a local form for each triangle $T\in\mathcal{T}$. Let
$\varphi_{z}\in S^{1}(\mathcal{T})$ be the $P_{1}$ nodal basis function
associated to a vertex $z\in\mathcal{V}(T)=\\{a,b,c\\}$ of the triangle $T$,
i.e., $\varphi_{z}|_{T}$ is a barycentric coordinate of $z$ in $T$ and let
$b_{T}:=27\varphi_{a}\varphi_{b}\varphi_{c}\in P_{3}(T)\cap H^{1}_{0}(T)$ be
the cubic-bubble function. Then $b_{T}^{m}\in H^{m}_{0}(T)\subset V$ satisfies
$0\leq b_{T}^{m}\leq 1$ and inverse estimates
$|\\!|\\!|b_{T}^{m}|\\!|\\!|\lesssim h^{-m}_{T}\|b_{T}\|$. Abbreviate
$g_{m}:=\Pi_{m}g|_{T}\in P_{m}(T)$ and observe

$\displaystyle\|h_{T}^{m}g\|_{L^{2}(T)}\leq\|h_{T}^{m}g_{m}\|_{L^{2}(T)}+\mathrm{osc}_{m}(g,T)$
(6.9)

from the definition of $\mathrm{osc}_{m}(g,T)$ and a triangle inequality. An
inverse estimate $\|g_{m}\|_{L^{2}(T)}\leq
C_{8}{}$$\|b_{T}^{m/2}g_{m}\|_{L^{2}(T)}$ follows from the equivalence of the
norms $\|b_{T}\bullet\|_{L^{2}(T)}\approx\|\bullet\|_{L^{2}(T)}$ in
$P_{m}(T)$. (A scaling argument reveals that the equivalence constant does not
depend on the size or shape of the triangles and exclusively depends on $m$.)
The inverse estimate shows

$\displaystyle C_{\ref{gg}}^{-2}h_{T}^{2m}\|g_{m}\|_{L^{2}(T)}^{2}$
$\displaystyle\leq
h_{T}^{2m}\|b_{T}^{m/2}g_{m}\|^{2}_{L^{2}(T)}=h_{T}^{2m}\int_{T}b_{T}^{m}g_{m}(g_{m}-g){\rm\,dx}+h_{T}^{2m}\int_{T}b_{T}^{m}g_{m}g{\rm\,dx}$
$\displaystyle\leq\mathrm{osc}_{m}(g,\mathcal{T})h_{T}^{m}\|b_{T}^{m}g_{m}\|_{L^{2}(T)}+h_{T}^{m}\int_{T}gv_{T}{\rm\,dx}$
(6.10)

with a Cauchy inequality and the abbreviation
$v_{T}:=h_{T}^{m}b_{T}^{m}g_{m}\in P_{4m}(T)\cap H^{2}_{0}(T)\subset V$ in the
last step. Since
$a(u,v_{T})=\int_{T}gv_{T}{\rm\,dx}+\int_{T}G:D^{m}v_{T}{\rm\,dx}$ (from (2.1)
for the test function $v_{T}\in V$), the crucial term in (6.2) is equal to

$h_{T}^{m}\int_{T}gv_{T}{\rm\,dx}=h_{T}^{m}\int_{T}(D^{2}u-G):D^{m}v_{T}{\rm\,dx}.$
(6.11)

The bubble-function methodology due to [54] observes at this point that
$\Pi_{0}D^{m}v_{T}=0$ from an integration by parts with $v_{T}\in
H^{m}_{0}(T)$, whence $D^{m}v_{T}=0$ on $\partial T$, and so
$\int_{T}D^{m}v_{T}{\rm\,dx}=\int_{\partial
T}(D^{m-1}v_{T})\nu_{T}{\rm\,ds}=0.$ Consequently,

$\displaystyle\int_{T}(D^{m}u-G):D^{m}v_{T}{\rm\,dx}$
$\displaystyle=\int_{T}D^{m}v_{T}:(1-\Pi_{0})(D^{m}u-G){\rm\,dx}$
$\displaystyle\leq({|u-I_{\mathrm{nc}}u|_{H^{2}(T)}}+\|G-\Pi_{0}G\|_{L^{2}(T)})\|D^{m}v_{T}\|_{L^{2}(T)}.$

A standard inverse estimate $\|D^{m}v_{T}\|_{L^{2}(T)}\lesssim
h_{T}^{-m}\|v_{T}\|_{L^{2}(T)}$ for the polynomial $v_{T}\in P_{4m}(T)$
depends on $m$ and on the shape-regularity of the triangle $T$. Note that
$h_{T}^{-m}\|v_{T}\|_{L^{2}(T)}=\|b_{T}^{m}g_{m}\|_{L^{2}(T)}\leq\|g_{m}\|_{L^{2}(T)}$
follows from the definition of $v_{T}=h_{T}^{m}b_{T}^{m}g_{m}$ and $0\leq
b_{T}\leq 1$. The aforementioned estimates lead in (6.11) to

$h_{T}^{m}\int_{T}gv_{T}{\rm\,dx}\leq
C_{9}{}(|u-I_{\mathrm{nc}}u|_{H^{2}(T)}+\|G-\Pi_{0}G\|_{L^{2}(T)})h_{T}^{m}\|g_{m}\|_{L^{2}(T)}$

for some generic constant $C_{\ref{ss}}\approx 1$ that depends on the shape of
$T$ and $m$. The combination of the previous estimate with (6.2) proves

$h_{T}^{2m}\|g_{m}\|^{2}_{L^{2}(T)}\leq
C_{\ref{gg}}h_{T}^{m}\|g_{m}\|_{L^{2}(T)}(\mathrm{osc}_{m}(g,T)+C_{\ref{ss}}(|u-I_{\mathrm{nc}}u|_{H^{2}(T)}+\|G-\Pi_{0}G\|_{L^{2}(T)})),$

whence
$h_{T}^{m}\|g_{m}\|_{L^{2}(T)}\lesssim|u-I_{\mathrm{nc}}u|_{H^{2}(T)}+\|G-\Pi_{0}G\|_{L^{2}(T)}+\mathrm{osc}_{m}(g,T).$
This and (6.9) lead to the asserted local efficiency
$\|h_{T}^{m}g\|^{2}_{L^{2}(T)}\lesssim|u-I_{\mathrm{nc}}u|^{2}_{H^{2}(T)}+\|G-\Pi_{0}G\|^{2}_{L^{2}(T)}+\mathrm{osc}_{m}(g,T)^{2}$
for each $T\in\mathcal{T}$. The sum over all $T\in\mathcal{T}$ concludes the
proof. ∎

###  6.3 Example with dominating data oscillations

The data-oscillation term $\|G-\Pi_{0}G\|$ dominates the error zero and makes
the error estimates useless in the following alarming example. Suppose $m=2$
(the situation is simpler for $m=1$) and choose some function $z_{\rm c}\in
S^{1}(\mathcal{T};{\mathbb{R}}^{2})$. Given $z_{\rm c}$, consider any $z\in
H^{1}(\Omega;{\mathbb{R}}^{2})$ with $\int_{E}z{\rm\,ds}=\int_{E}z_{\rm
c}{\rm\,ds}$ for all edges $E\in\operatorname{\mathcal{E}}$. (This can be
done, e.g, by volume bubble functions added to $z_{\rm c}$ or other functions
that do not change the above edge integrals.) Let $g\equiv 0$ and $G:={\rm
sym\;Curl}\>z\in L^{2}(\Omega;{\mathbb{S}})$ in (5.5). Since $\Pi_{0}\>{\rm
Curl}\>z={\rm Curl}_{\mathrm{pw}}\>I_{\mathrm{CR}}z={\rm Curl}\>z_{\rm c},$ it
follows that
$\int_{\Omega}G:D^{2}v{\rm\,dx}=0=\int_{\Omega}G:D_{\mathrm{pw}}^{2}v_{\mathrm{M}}{\rm\,dx}$
for all $v\in V$ and $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$. The first
identity follows immediately from an integration by parts and the second from
a piecewise integration by parts; for instance, any
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ satisfies

$\displaystyle\int_{\Omega}G:D_{\mathrm{pw}}^{2}v_{\mathrm{M}}{\rm\,dx}$
$\displaystyle=\int_{\Omega}{\rm
sym\;Curl}\>z:D^{2}_{\mathrm{pw}}v_{\mathrm{M}}{\rm\,dx}=\int_{\Omega}{\rm
Curl}\>z:D^{2}_{\mathrm{pw}}v_{\mathrm{M}}{\rm\,dx}$
$\displaystyle=\int_{\Omega}{\rm Curl}\>z_{\rm
c}:D^{2}_{\mathrm{pw}}v_{\mathrm{M}}{\rm\,dx}=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\nu_{E}\cdot\langle{\rm
Curl}\>z_{\rm c}\rangle_{E}[\nabla_{\mathrm{pw}}v_{\mathrm{M}}]_{E}{\rm\,ds}$
$\displaystyle=\sum_{E\in\operatorname{\mathcal{E}}}\left(\int_{E}[\nabla_{\mathrm{pw}}v_{\mathrm{M}}]_{E}{\rm\,ds}\right)\cdot\langle{\rm
Curl}\>z_{\rm c}\rangle_{E}\nu_{E}{\rm\,ds}=0$

(because $[{\rm Curl}\>z_{\rm c}]_{E}\times\nu_{E}=0$ and $\langle{\rm
Curl}\>z_{\rm c}\rangle_{E}\nu_{E}=0$ on $E$ while
$\int_{E}[\nabla_{\mathrm{pw}}v_{\mathrm{M}}]_{E}{\rm\,ds}=0$). This latter
orthogonality property implies $u=0=u_{\mathrm{M}}$ for the scheme (2.2) as
well as for (3.9). On the other hand, $\|G-\Pi_{0}G\|=\|{\rm
sym\;Curl}\;(z-z_{\rm c})\|$. If $\|G-\Pi_{0}G\|_{L^{2}(T)}=0$ for
$T\in\mathcal{T}$, then $w=(w_{1},w_{2}):=z-z_{\rm c}\in
H^{1}(T;{\mathbb{R}}^{2})$ satisfies $\displaystyle{\rm sym\;Curl}\>w=0$ a.e.
in $T$. This is equivalent to $w_{1,2}=0=w_{2,1}$ and $w_{1,1}=w_{2,2}$. Hence
$(w_{2},w_{1})$ is divergence-free and there exists some $\beta\in H^{2}(T)$
with $(w_{2},w_{1})={\rm\nabla}\beta=(\partial\beta/\partial
x_{1},\partial\beta/\partial x_{2}).$ Consequently, $\beta$ satisfies
$\partial^{2}\beta/\partial x_{j}^{2}=0$ for $j=1,2$. Since $\beta$ is
harmonic, it is smooth in $T$. Without loss of generality, assume $0\in{\rm
int}\>(T)$. For $x=(x_{1},x_{2})\in T$, an integration along the line ${\rm
conv}\\{0,{x_{1}}\\}\times\\{x_{2}\\}$ shows
$\beta(x)=\beta(0,x_{2})+\int_{0}^{x_{1}}\beta_{,1}(\xi,x_{2})\>{\rm d}\xi$
(with the abbreviation $\beta_{,j}:=\partial\beta/\partial x_{j}$,
$\beta_{,ij}:=\partial^{2}\beta/\partial x_{i}\partial x_{j}$, $j=1,2$ for the
partial derivatives). Since $\beta_{,11}=0$,
$\beta_{,1}(\xi,x_{2})=\beta_{,1}(0,x_{2})$ for a.e. $\xi\in{\rm
conv}\\{0,x_{1}\\}$. Hence $\beta(x)=\beta(0,x_{2})+x_{1}\beta_{,1}(0,x_{2})$.
The same argument applies for an integration along $\\{x_{1}\\}\times{\rm
conv}\\{0,{x_{2}}\\}$ and leads to
$\beta(x)=\beta(x_{1},0)+x_{2}\beta_{,2}(x_{1},0)$. Recall that $\beta$ is
smooth in ${\rm int}\>(T)$ and consider derivatives of the two identities to
verify $\beta_{,12}(x)=\beta_{,12}(0,x_{2})=\beta_{,12}(x_{1},0)$ for all
$x\in{\rm int}\>(T)$. The latter identity shows first for $(0,x_{2})\in T$
that $\beta_{,12}(0,x_{2})=\beta_{,12}(0,0)$ and second for all $x\in{\rm
int}\>(T)$ that $\beta_{,12}(x)=\beta_{,12}(0,0)$. Consequently, the Hessian
$D^{2}\beta$ of $\beta$ is constant as $\beta\in P_{2}(T)$. Recall that
$\int_{E}w{\rm\,ds}=0=\int_{E}\nabla\beta{\rm\,ds}$ for all edges
$E\in\operatorname{\mathcal{E}}(T)$ of $T$. Therefore the affine vector field
$\nabla\beta\in P_{1}(T;{\mathbb{R}}^{2})$ vanishes at all the edges,
midpoints and so everywhere in $T$. Consequently, $\beta$ is a constant and
$w:=(w_{1},w_{2})=(\beta_{,2},\beta_{,1})=0$ vanishes a.e. in $T$. This shows
that any choice of $z\in H^{1}(\Omega;{\mathbb{R}}^{2})$ with $\|z-z_{\rm
c}\|_{L^{2}(\Omega)}>0$ leads to $\|G-\Pi_{0}G\|>0$. In other words, the error
vanishes while the data approximation does not.

## Acknowledgements

The research of the first author has been supported by the Deutsche
Forschungsgemeinschaft in the Priority Program 1748 under the project
"foundation and application of generalized mixed FEM towards nonlinear
problems in solid mechanics" (CA 151/22-2). The finalization of this paper has
been supported by SPARC project (id 235) entitled the mathematics and
computation of plates.

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