# Lowest-order equivalent nonstandard finite element methods for biharmonic
plates

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

The popular (piecewise) quadratic schemes for the biharmonic equation based on
triangles are the nonconforming Morley finite element, the discontinuous
Galerkin, the $C^{0}$ interior penalty, and the WOPSIP schemes. Those methods
are modified in their right-hand side $F\in H^{-2}(\Omega)$ replaced by
$F\circ(JI_{\mathrm{M}})$ and then are quasi-optimal in their respective
discrete norms. The smoother $JI_{\mathrm{M}}$ is defined for a piecewise
smooth input function by a (generalized) Morley interpolation $I_{\mathrm{M}}$
followed by a companion operator $J$. An abstract framework for the error
analysis in the energy, weaker and piecewise Sobolev norms for the schemes is
outlined and applied to the biharmonic equation. Three errors are also
equivalent in some particular discrete norm from [Carstensen, Gallistl,
Nataraj: Comparison results of nonstandard $P_{2}$ finite element methods for
the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data
oscillations. This paper extends the work [Veeser, Zanotti: Quasi-optimal
nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56
(2018)] to the discontinuous Galerkin scheme and adds error estimates in
weaker and piecewise Sobolev norms.

Keywords: biharmonic problem, best-approximation, a priori error estimates,
companion operator, $C^{0}$ interior penalty, discontinuous Galerkin method,
WOPSIP, Morley, comparison

AMS Classification: 65N30, 65N12, 65N50

##  1 Introduction

The paper contributes to lower-order nonstandard finite element methods for a
biharmonic plate problem in a real Hilbert space $(V,a)$. Given $F\in
L^{2}(\Omega)$, [13] compares the errors for nonstandard finite element
methods (FEM) of the clamped biharmonic plate problem based on piecewise
quadratic polynomials, namely the nonconforming Morley FEM [26], the symmetric
interior penalty discontinuous Galerkin FEM (dGFEM) [28], and the $C^{0}$
interior penalty method ($C^{0}$IP) [6], with respective solutions
$u_{\mathrm{M}}$, $u_{h}$, and $u_{\mathrm{IP}}$; Table LABEL:table:summary
displays details of the respective schemes. For $F\in L^{2}(\Omega)$, dGFEM
and hp-dGFEM for biharmonic and fourth-order problems, are extensively studied
in [27, 28, 36, 39, 37, 30, 31].

For a general right-hand side $F\in H^{-2}(\Omega)$, the standard right-hand
side $F(v_{h})$ remains undefined for nonstandard finite element methods. A
postprocessing procedure in [6] enables to introduce a new $C^{0}$IP method
for right-hand sides in $H^{-2}(\Omega)$. In [41, 43, 42], the discrete test
functions are transformed into conforming functions ($J$ is called smoother in
those works) before applying the load functional and quasi-optimal energy norm
estimates

$\displaystyle\|{u-u_{\mathrm{M}}}\|_{\text{pw}}\approx\min_{v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})}\|u-v_{\mathrm{M}}\|_{\text{pw}},\;\|{u-u_{\mathrm{IP}}}\|_{\rm
IP}\approx\min_{v_{\rm IP}\in{\rm IP}(\mathcal{T})}\|u-v_{\rm IP}\|_{\rm IP}$

are derived for the Morley FEM and $C^{0}$IP method.

The papers [41, 43, 42] discuss minimal conditions on a smoother for each
problem, while this paper presents one smoother $JI_{\mathrm{M}}$ for all
schemes; the best-approximation for the dGFEM is a new result. The smoother
$JI_{\mathrm{M}}$ also allows a post-processing with a priori error estimates
in weaker and piecewise Sobolev norms.

Table LABEL:table:summary summarizes the notation of spaces, bilinear forms,
and an operator for the four second-order methods for the biharmonic problem
detailed in Subsection 3.2 and in Sections 7, 8, and 10.

Table 1:  Overview of notation for four discrete schemes with discrete bilinear form (5.12) Notation Schemes |  Morley FEM |  dGFEM |  $C^{0}$IP |  WOPSIP
---|---|---|---|---
Section reference |  Subsec 3.2 |  Sec 7 |  Sec 8 |  Sec 10
$V_{h}$ |  $\mathrm{M}(\mathcal{T})$ |  $P_{2}(\mathcal{T})$ |  $S^{2}_{0}(\mathcal{T})$ |  $P_{2}(\mathcal{T})$
${A}_{h}$ |  $a_{\mathrm{pw}}$ in (3.3) |  $A_{\rm dG}$ in (7.1) |  $A_{\mathrm{IP}}$ in (8.1) |  $A_{\mathrm{P}}$ in (10.2)
$b_{h}$ |  0 |  $-\Theta~{}\mathcal{J}~{}-~{}\mathcal{J}^{*}$ in (6.2) |  $-\Theta~{}\mathcal{J}~{}-~{}\mathcal{J}^{*}$ in (6.2) |  0
$c_{h}$ |  0 |  $c_{\rm dG}$ in (4.3) |  $c_{\mathrm{IP}}$ in (8.2) |  $c_{\mathrm{P}}$ in (10)
$I_{h}:\mathrm{M}(\mathcal{T})\rightarrow V_{h}$ |  id |  id |  $I_{\mathrm{C}}$ in (8.4) |  id

### Contributions

The main contributions of this paper are

  1. (a)

the design and analysis of a generalized Morley interpolation operator
$I_{\mathrm{M}}$ for piecewise smooth functions in $H^{2}(\mathcal{T})$,

  2. (b)

the design of modified schemes for Morley FEM, dGFEM, $C^{0}$IP method, and a
weakly over-penalized symmetric interior penalty (WOPSIP) method for the
biharmonic problem for data in $F\in H^{-2}(\Omega)$,

  3. (c)

an abstract framework for the best-approximation property and weaker
(piecewise) Sobolev norm estimates,

  4. (d)

a priori error estimates in (piecewise) Sobolev norms for the lowest-order
nonstandard finite element methods for biharmonic plates,

  5. (e)

an extension of the results of [13] to an equivalence

$\displaystyle\lVert u-u_{\mathrm{M}}\rVert_{h}\approx\lVert
u-u_{\mathrm{IP}}\rVert_{h}\approx\lVert u-u_{\rm
dG}\rVert_{h}\approx\left\|(1-\Pi_{0})D^{2}u\right\|_{L^{2}(\Omega)}$ (1.1)

without data oscillations for the modified schemes,

  6. (f)

the proof of the best approximation for the modified dGFEM that extends [42]
and [13, Thm 4.3].

######  Remark 1.1 (medius analysis).

The quasi-optimality of nonconforming and discontinuous Galerkin methods was
established in the seminal paper [34] for the original method up to data
oscillations for $F\in L^{2}(\Omega)$. Arguments from a posteriori error
analysis [44] give new insight in the consistency term from the Strang-Fix
lemmas. The techniques in this paper circumvent any a posteriori error
analysis and take advantage of the extra benefits of the companion operator
$J$.

######  Remark 1.2 (smoother).

The fundamental series of contributions [41, 43, 42] on the quasi-optimality
concerns best-approximation for a modified scheme with $F_{h}:=F\circ J$ for a
smoother $J$. Elementary algebra indicates a key identity (of (5.5) below)
that is already mentioned in [6, (5.15)] and makes the source term in the
consistency disappear.

######  Remark 1.3 (extension to higher order).

A (general) Morley interpolation allows for a simultaneous analysis of four
lowest-order schemes, but appears to be restricted to piecewise quadratics at
first glance. But the combination of $JI_{\mathrm{M}}$ as an averaging
smoother with higher-order bubble-smoothers shall enable applications to
higher-order schemes as indicated in [42].

######  Remark 1.4 (extension to 3D).

Although the plate problem is intrinsic two-dimensional, there are three-
dimensional Morley finite elements with a recent companion operator [24] that
guarantees the fundamental properties in 3D such that the abstract framework
applies.

### Organization

The remaining parts of this paper are organised as follows. Section 2 provides
an abstract characterization of the best-approximation property that applies
to the various applications considered in this and future papers [22]. Section
3 presents preliminaries, a nonconforming discretisation, introduces a novel
generalized Morley interpolation operator for discontinuous functions, and
states a best-approximation [21] result for nonconforming discretisations with
data $F\in H^{-2}(\Omega)$. Section 4 proves a crucial equivalence result of
two discrete norms for a piecewise $H^{2}(\mathcal{T})$ function, and proves
approximation properties for the generalized Morley interpolation operator.
Section 5 provides a framework for dG methods and the proof of a best-
approximation property under a set of general assumptions. Section 6 develops
the abstract result for a priori error estimates in weaker and piecewise
Sobolev norms. Sections 7 and 8 recall the dG and $C^{0}$IP schemes and verify
the assumptions of Sections 5 and 6 for the best-approximation result in the
energy norm as well as weaker and piecewise Sobolev norms without data
oscillations. The paper concludes with the equivalence (1.1) of errors in
Section 9 and a proof of quasi-optimality up to penalty for the WOPSIP scheme
in Section 10.

### 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 real $s$, $H^{s}(\Omega)$
denotes the Sobolev space associated with the Sobolev-Slobodeckii semi-norm
$|\bullet|_{H^{\boldmath{s}}(\Omega)}$ [33]; $H^{s}(T):=H^{s}({\rm int}(T))$
abbreviates the Sobolev space with respect to the interior ${\rm
int}(T)\neq\emptyset$ of a (compact) triangle $T$. The closure of $D(\Omega)$
in $H^{s}(\Omega)$ is denoted by $H^{s}_{0}(\Omega)$ and $H^{-s}(\Omega)$ is
the dual of $H^{s}_{0}(\Omega)$. The triple norm
$|\\!|\\!|\bullet|\\!|\\!|:=|\bullet|_{H^{2}(\Omega)}$ is the energy norm and
$|\\!|\\!|\bullet|\\!|\\!|_{\text{pw}}:=|\bullet|_{H^{2}(\mathcal{T})}:=\|D^{2}_{\text{pw}}\bullet\|$
is its piecewise version with the piecewise Hessian $D_{\text{pw}}^{2}$. Given
any function $v\in L^{2}(\omega)$, define the integral mean
$\fint_{\omega}v{\rm\,dx}:={1/|\omega|}\int_{\omega}v{\rm\,dx}$; where
$|\omega|$ denotes the area of $\omega$. The notation $A\lesssim B$
abbreviates $A\leq CB$ for some positive generic constant $C$, which depends
only on $\Omega$ and the shape regulatity of $\mathcal{T}$; $A\approx B$
abbreviates $A\lesssim B\lesssim A$.

##  2 Prologue

This section characterizes the best-approximation property of a class of non-
conforming finite element methods. The biharmonic problem is put in an
abstract framework in real Hilbert spaces $X$ and $Y$ and a bounded bilinear
form $a:X\times Y\rightarrow{\mathbb{R}}$ satisfying an inf-sup condition.
Given a right-hand side $F\in Y^{*}$, the exact problem seeks $x\in X$ with

$a(x,\bullet)=F\quad\text{in }Y.$

The discrete problem is put in an analog framework with finite-dimensional
real Hilbert spaces $X_{h}$ and $Y_{h}$ and a bilinear form $a_{h}:X_{h}\times
Y_{h}\rightarrow{\mathbb{R}}$. The discrete space $X_{h}$ (resp. $Y_{h}$) is
not a subspace of $X$ (resp. $Y$) in general, but $X$ and $X_{h}$ (resp. $Y$
and $Y_{h}$) belong to one common bigger vector space that gives rise to the
sum $\widehat{X}=X+X_{h}$ (resp. $\widehat{Y}=Y+Y_{h}$). It is not supposed
that this is a direct sum, so the intersection $X\cap X_{h}$ (resp. $Y\cap
Y_{h}$) may be non-trivial. We suppose that $\widehat{X}$ and $\widehat{Y}$
are real Hilbert spaces with (complete) subspaces $X$, $X_{h}$ and $Y$,
$Y_{h}$. The linear and bounded map $Q\in L(Y_{h};Y)$ links the right-hand
side $F\in Y^{*}$ of the exact problem to the right-hand side $F_{h}:=F\circ
Q\in Y_{h}^{*}$ of the discrete problem. The map $Q$ is called smoother in
[41, 42, 43] because it maps a (possibly) discontinuous function $y_{h}\in
Y_{h}$ to a smooth function $Qy_{h}$ in applications. The resulting discrete
problem  seeks $x_{h}\in X_{h}$ with

$\displaystyle a_{h}(x_{h},\bullet)=F(Q\bullet)\text{ in }Y_{h}.$ (2.1)

We also suppose that the exact and discrete problems are well-posed and this
means in particular that $\dim X_{h}=\dim Y_{h}<\infty$ and that the bounded
bilinear forms $a$ and $a_{h}$ satisfy inf-sup conditions with positive
constants $\alpha$ and $\alpha_{h}$ and are non-degenerate such that the
associated linear and bounded operators $A\in L(X;Y^{*})$ and $A_{h}\in
L(X_{h};Y_{h}^{*})$ are bijective; the associated linear operators are defined
by $Ax:=a(x,\bullet)\in Y^{*}$ for all $x\in X$ and by
$A_{h}x_{h}:=a_{h}(x_{h},\bullet)\in Y_{h}^{*}$ for all $x_{h}\in X_{h}$.

The general discussion in [41, 42, 43], leads to an optimal smoothing
$Q=\Pi_{Y}|_{Y_{h}}$ for the orthogonal projection $\Pi_{Y}\in L(\widehat{Y})$
onto $Y$. This is a global operation in general and hence infeasible for
practical computations. Notice carefully that all examples in [41, 42, 43]
discuss $Q\in L(Y_{h};Y)$ with $Qz=z$ for all $z\in Y_{h}\cap Y,$ abbreviated
by

$\displaystyle Q={\rm id}\text{ in }Y_{h}\cap Y.$ (2.2)

This paper introduces a smoother $JI_{\mathrm{M}}$ for the discontinuous
Galerkin schemes that satisfies (2.2) and is quasi-optimal in the following
sense with a constant $\Lambda_{\rm Q}\geq 0$ that is exclusively bounded in
terms of the shape regularity of the underlying triangulations.

######  Definition 2.1 (quasi-optimal smoother).

A linear bounded operator $Q\in L(Y_{h};Y)$ is called a quasi-optimal smoother
if there exists some $\Lambda_{\rm Q}\geq 0$ such that

$\displaystyle\|y_{h}-Qy_{h}\|_{\widehat{Y}}\leq\Lambda_{\rm
Q}\|y_{h}-y\|_{\widehat{Y}}\text{ for all }y_{h}\in Y_{h}\text{ and all }y\in
Y.$ (2.3)

The proofs of Lemma 2.1 and 2.2 below rely on compactness arguments whence in
the appendix, the constants $\Lambda_{\rm P}$ and $\Lambda_{\rm Q}$ depend on
the discrete space. The point is that this paper designs a smoother in Section
4.3 with a constant that does not depend on the mesh-size.

######  Lemma 2.1.

The operator $Q\in L(Y_{h};Y)$ is a quasi-optimal smoother if and only if
(2.2) holds.

The (nonconforming) finite element method is characterized by the operator
$M\in L(X;X_{h})$ that maps any $x\in X$ to a right-hand side
$F:=a(x,\bullet)\in Y^{*}$ and then to the solution
$Mx:=x_{h}=A_{h}^{-1}(Q^{*}F)$ to (2.1), i.e.,
$Mx:=A_{h}^{-1}(a(x,Q\bullet))\in X_{h}$ for all $x\in X$, or, in operator
form,

$M:=A_{h}^{-1}Q^{*}A\in L(X;X_{h}).$

In other words, the subsequent diagram commutes.

$X$${X_{h}}$$Y^{*}$$Y_{h}^{*}$$P$$M$${A}_{h}^{-1}$${A}$$Q^{*}$

This diagram also depicts some (linear and bounded) operator $P:X_{h}\to X$
that will become a quasi-optimal smoother in the context of the best-
approximation property of $M$ below. A synonym to the best-approximation
property of $M$ is to say $M$ is quasi-opimal in the following sense.

######  Definition 2.2 (quasi-optimal).

The above operator $M$ is said to be quasi-optimal if

${\text{\bf(QO)}}\quad\exists\;C_{\rm qo}>0\;\;\forall x\in
X\quad\|x-Mx\|_{\widehat{X}}\leq C_{\rm qo}\min_{x_{h}\in
X_{h}}\|x-x_{h}\|_{\widehat{X}}.$

A first characterisation of (QO) has been given in [41] in terms of

$\displaystyle M={\rm id}\text{ in }X_{h}\cap X.$ (2.4)

######  Lemma 2.2.

[41] Under the present notation, (QO) is equivalent to (2.4).

The above lemmas characterize $C_{\rm qo}$ and $\Lambda_{\rm Q}$ by a
compactness argument and it remains to control $C_{\rm qo}$ and $\Lambda_{\rm
Q}$ in terms of mesh-size independent bounds in applications. This paper
designs in Section 4.3 a smoother in the spirit of [41, 43, 42] based on
earlier work in the context of a posteriori error control [9, 16, 12] and
adaptive mesh-refinement [25, 19, 23, 24]. The outcome is a quasi-optimal
smoother $P\in L(X_{h};X)$ with a constant $\Lambda_{\rm P}$ that depends only
on the shape regularity of the underlying finite element mesh and

$\displaystyle\|x_{h}-Px_{h}\|_{\widehat{X}}\leq\Lambda_{\rm
P}\|x_{h}-x\|_{\widehat{X}}\text{ for all }x_{h}\in X_{h}\text{ and all }x\in
X.$ (2.5)

The proof of the following characterization of best-approximation shall be
given in the appendix.

######  Theorem 2.3.

Suppose $P\in L(X_{h};X)$ and $\Lambda_{\rm P}$ satisfy (2.3). Then (QO) is
equivalent to the existence of $\Lambda_{\rm H}>0$ with

(H) $\displaystyle\quad{a}_{h}(x_{h},y_{h})-{a}(Px_{h},Qy_{h})\leq\Lambda_{\rm
H}\|x_{h}-Px_{h}\|_{\widehat{X}}\|y_{h}\|_{Y_{h}}\text{ for all }x_{h}\in
X_{h}\text{ and }y_{h}\in Y_{h}.$

In particular, if (H) holds, then (QO) follows with a constant $C_{\rm qo}$
that depends solely on $\alpha_{h},$ $\Lambda_{\rm H},$ $\Lambda_{\rm P},$
$\|Q\|$, and $\|A\|$.

The next theorem presents a key estimate that is crucial for goal-oriented
error control and duality arguments for weaker norm estimates. The proof and
the dependence of contants are presented in the appendix.The motivation for
${\widehat{\text{\bf(QO)}}}$ is exemplified in Theorem 2.5 below.

######  Theorem 2.4.

Suppose $P$ and $Q$ are quasi-optimal smoothers with (2.3)-(2.3) and suppose
(QO). Then the existence of $\widehat{C_{\text{\rm qo}}}>0$ with

$\displaystyle{\widehat{\text{\bf(QO)}}}\;\;$ $\displaystyle
a(x-PMx,y)\leq\widehat{C_{\text{\rm
qo}}}\|x-Mx\|_{\widehat{X}}\|y-y_{h}\|_{\widehat{Y}}\;\text{ for all }x\in
X,\;y\in Y,\text{ and }y_{h}\in Y_{h}\quad$

is equivalent to the existence of $\widehat{\Lambda_{\rm H}}>0$ with

$\displaystyle{\widehat{\text{\bf(H)}}}\;$
$\displaystyle{a}_{h}(x_{h}^{\prime},y_{h})-{a}(Px_{h}^{\prime},Qy_{h})\leq\widehat{\Lambda_{\rm
H}}\|x_{h}^{\prime}-Px_{h}^{\prime}\|_{\widehat{X}}\|y_{h}-Qy_{h}\|_{\widehat{Y}}\text{
for all }x_{h}^{\prime}\in X_{h}^{\prime},\text{ and }y_{h}\in Y_{h}.$

In particular, if ${\widehat{\text{\bf(H)}}}$ holds,
${\widehat{\text{\bf(QO)}}}$ follows with a constant $\widehat{C_{\text{\rm
qo}}}$ that depends solely on $\|a\|$, $\Lambda_{2}^{\prime},$ $\Lambda_{\rm
P},$ and $\Lambda_{\rm Q}$.

The a priori error estimates in weaker Sololev norms (weaker than the energy
norm) are a corollary of Theorem 2.4 and the elliptic regularity, the latter
is written in an abstract form by the assumption that $X_{s}$ and $Y_{s}$ are
two Hilbert spaces with $X\subset X_{s}$ and $Y_{s}\subset Y$ such that

${\text{\bf(R)}}\;\;\exists C_{\rm reg}>0\;\forall F\in
X_{s}^{*}\;\;\|A^{-*}F\|_{Y_{s}}\leq C_{\rm reg}\|F\|_{X_{s}^{*}}$

for the solution $y:=A^{-*}F\in Y_{s}\subset Y$ to $a(\bullet,y)=F\in
X_{s}^{*}\subset X^{*}$.

######  Theorem 2.5 (weak a priori).

Under the assumptions of Theorem 2.4, ${\widehat{\text{\bf(QO)}}}$ and (R)
imply

$\|x-PMx\|_{X_{s}}\leq\widehat{C_{\text{\rm
qo}}}\|x-Mx\|_{\widehat{X}}\sup_{\stackrel{{\scriptstyle y\in
Y_{s}}}{{\|y\|_{Y_{s}}\leq C_{\rm reg}}}}\inf_{y_{h}\in
Y_{h}}\|y-y_{h}\|_{\widehat{Y}}\text{ \rm for all }x\in X.$

###### Proof.

Given $x-PMx\in X\subset X_{s}$, a corollary of the Hahn-Banach extension
theorem leads to some $F\in X_{s}^{*}\subset X^{*}$ with norm
$\|F\|_{X_{s}^{*}}\leq 1$ in $X_{s}^{*}$ and $\|x-PMx\|_{X_{s}}=F(x-PMx)$. The
dual solution $y\in Y$ to $F=a(\bullet,y)\in X^{*}$ satisfies (R) and
${\widehat{\text{\bf(QO)}}}$ leads to

$\|x-PMx\|_{X_{s}}=a(x-PMx,y)\leq\widehat{C_{\text{\rm
qo}}}\|x-Mx\|_{\widehat{X}}\|y-y_{h}\|_{\widehat{Y}}$

for any $y_{h}\in Y_{h}$. This and $\|y\|_{Y_{s}}\leq C_{\rm
reg}\|F\|_{X_{s}^{*}}\leq C_{\rm reg}$ conclude the proof. ∎

######  Example 2.1 (standard).

For the $m$-harmonic operator $A=(-1)^{m}\Delta^{m}$ and
$X=H^{m}_{0}(\Omega)=Y$, (R) holds for
$X_{s}=H_{0}^{m-s}(\Omega),\>Y_{s}=H^{m+s}(\Omega)$ and $1/2\leq s\leq 1$,
$m=1$ or 2. Typical first-order approximation properties of the discrete
finite element spaces result in

$\displaystyle\sup_{\stackrel{{\scriptstyle y\in Y_{s}}}{{\|y\|_{Y_{s}}\leq
C_{\rm reg}}}}\inf_{y_{h}\in Y_{h}}\|y-y_{h}\|_{\widehat{Y}}=O(h_{\rm
max}^{s})$

in terms of the maximal mesh-size $h_{\rm max}$ of the underlying finite
element mesh. ∎

######  Remark 2.1 (best-approximation constant).

The paper [41] gives a formula for the best-approximation constant $C_{\rm
qo}$ for some slightly simpler problem in one Hilbert space.

######  Remark 2.2 (injective smoother).

Under the above notation $Q\in L(Y_{h};Y)$ is injective if and only if $M\in
L(X;X_{h})$ is surjective [41]. Then there exists a right-inverse $P\in
L(X_{h};X)$ to $M$ and (H) holds with $\Lambda_{\rm H}=0$ (this follows with
the arguments of the proof of Theorem 2.5 for $P^{\prime}$ that is in fact a
quasi-optimal smoother owing to (10.10).) Consequently, the discrete scheme is
equivalent to a conforming Petrov-Galerkin scheme.

######  Remark 2.3 (non injective smoother).

In case $Q\in L(Y_{h};Y)$ is not injective, the discrete problem may reduced
to the range $X_{h}^{\prime}:=\mathcal{R}(M)$ of $M$ and the orthogonal
complement $Y_{h}^{\prime}$ of the kernel of $Q$ in $Y_{h}$. However, the
explicit computation of the reduced discrete spaces $X_{h}^{\prime}$ and
$Y_{h}^{\prime}$ may be costly and hence this paper outlines a general
analysis that allows non-injective quasi-optimal smoothers.

######  Example 2.2 (smoother for Morley).

For the standard Morley interpolation operator
$I_{\mathrm{M}}:H^{2}_{0}(\Omega)\rightarrow\mathrm{M}(\mathcal{T})$ and a
companion operator $J:\mathrm{M}(\mathcal{T})\rightarrow H^{2}_{0}(\Omega)$
(cf. Lemma 3.2 below for details) the smoother $Q=JI_{\mathrm{M}}$ is
injective because $J$ is a right-inverse of $I_{\mathrm{M}}$.

######  Example 2.3 (smoother for dG).

This paper advertises a smoother $Q:=JI_{\mathrm{M}}$ for a (generalized)
Morley interpolation
$I_{\mathrm{M}}:(P_{2}(\mathcal{T})+H^{2}_{0}(\Omega))\rightarrow\mathrm{M}(\mathcal{T})$
(cf. (3.2) below for details) followed by a companion operator $J$ from the
previous example for the dG FEM. Then $\text{ dim
}P_{2}(\mathcal{T})=6|\mathcal{T}|$ is strictly larger than $\text{ dim
}\mathrm{M}(\mathcal{T})=|{\mathcal{V}}(\Omega)|+|{\mathcal{E}}(\Omega)|$;
whence $Q$ cannot be injective.

The situation for the $C^{0}$ IP with the discrete space
$S^{2}_{0}(\mathcal{T})$ (of the same dimension as $\mathrm{M}(\mathcal{T})$)
is more involved and is discussed in more details in Section 8 below.

##  3 Preliminaries

###  3.1 Continuous model problem

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

$\displaystyle a(u,v)=F(v)\quad\text{for all }v\in V$ (3.1)

with the scalar product $a(v,w):=\int_{\Omega}D^{2}v:D^{2}w{\rm\,dx}$ for all
$v,w\in V$. It is well known that (3.1) has a unique solution $u$ and elliptic
regularity [1, 3, 32, 38] holds in the sense that $F\in H^{-s}(\Omega)$
implies $u\in V\cap H^{4-s}(\Omega)$ for all $s$ with $2-\sigma_{\rm reg}\leq
s\leq 2$ with the index of elliptic regularity $\sigma_{\rm reg}>0$. The
lowest-order nonconforming finite element schemes suggest a linear convergence
rate in the energy norm for a solution $u\in H^{t}(\Omega)$ at most for all
$t\geq 3$. Therefore $\sigma:=\min\\{1,\sigma_{\rm reg}\\}$ is fixed
throughout this paper and exclusively depends on $\Omega$. The regularity is
frequently employed in the following formulation.

######  Example 3.1 (regularity).

There exists a constant $0<\sigma\leq 1$ such that $F\in H^{-s}(\Omega)$ with
$2-\sigma\leq s\leq 2$ satisfies $u\in V\cap H^{4-s}(\Omega)$ and

$\displaystyle\|u\|_{H^{4-s}(\Omega)}$ $\displaystyle\leq C_{\rm
reg}(s)\|F\|_{H^{-s}(\Omega)}$ (3.2)

for some constant $C_{\rm reg}(s)<\infty$, which depends on $\Omega$ and $s$.
(The dependence on $s$ results from the equivalence of Sobolev norms that may
depend on the index $s$ in general.)

It is true that pure Dirichlet boundary conditions in the model example lead
to $\sigma>1/2$ and then allow for a control of the traces $D^{2}u$ in the
jump terms. This paper circumvents this argument and all the results hold for
$\sigma\geq 0$. The new discrete analysis is therefore much more flexible and
allows for generalizations of the model problem e.g. for mixed and boundary
conditions of less smoothness.

$T$

${\rm mid}(T)$

Figure 1: (a) Morley (left) and (b) HCT (right) finite element

###  3.2 Nonconforming discretisation

Throughout the rest of this article, the following notations are adopted. Let
$\mathcal{T}$ denote a shape regular triangulation of the polygonal Lipschitz
domain into compact triangles. Associate its piecewise constant mesh-size
$h_{\mathcal{T}}\in P_{0}(\mathcal{T})$ with
$h_{T}:=h_{\mathcal{T}}|_{T}:={\rm diam}(T)\approx|T|^{1/2}$ in any triangle
$T\in\mathcal{T}$ of area $|T|$ and its maximal mesh-size $h_{\rm max}:={\rm
max}\;h_{\mathcal{T}}$. Let ${\mathcal{V}}$ (resp. ${\mathcal{V}}(\Omega)$ or
${\mathcal{V}}(\partial\Omega)$) denote the set of all (resp. interior or
boundary) vertices in $\mathcal{T}$. Let ${\mathcal{E}}$ (resp.
${\mathcal{E}}(\Omega)$ or ${\mathcal{E}}(\partial\Omega)$) denote the set of
all (resp. interior or boundary) edges. The length of an edge $E$ is denoted
by $h_{E}$. Let $\Pi_{k}$ denote the $L^{2}(\Omega)$ orthogonal projection
onto the piecewise polynomials $P_{k}(\mathcal{T}):=\\{v\in
L^{2}(\Omega):\forall T\in\mathcal{T},\>v|_{T}\in P_{k}(T)\\}$ of degree at
most $k\in{\mathbb{N}}_{0}$. Let the Hilbert space
$H^{1}(\mathcal{T})\equiv\prod_{T\in\mathcal{T}}H^{1}(T)$. Define the jump
$[\varphi]_{E}:=\varphi|_{T_{+}}-\varphi|_{T_{-}}$ and the average
$\langle\varphi\rangle_{E}:=\frac{1}{2}\left(\varphi|_{T_{+}}+\varphi|_{T_{-}}\right)$
across the interior edge $E=\partial T_{+}\cap\partial
T_{-}\in\operatorname{\mathcal{E}}(\Omega)$ of $\varphi\in H^{1}(\mathcal{T})$
of the adjacent triangles $T_{+}$ and $T_{-}\in\mathcal{T}$ in an order such
that the unit normal vector $\nu_{T_{+}}|_{E}=\nu_{E}=-\nu_{T_{-}}|_{E}$ along
the edge $E$ has a fixed orientation and points outside $T_{+}$ and inside
$T_{-}$; $\nu_{T}$ is the outward unit normal of $T$ along $\partial T$. The
edge-patch $\omega(E):=\text{\rm int}(T_{+}\cup T_{-})$ of the interior edge
$E=\partial T_{+}\cap\partial T_{-}\in\operatorname{\mathcal{E}}(\Omega)$ is
the interior of union $T_{+}\cup T_{-}$ of the neighboring triangles $T_{+}$
and $T_{-}$. Extend the definition of the jump and the average to an edge
$E\in\operatorname{\mathcal{E}}(\partial\Omega)$ on the boundary by
$[\varphi]_{E}:=\varphi|_{E}$ and $\langle\varphi\rangle_{E}:=\varphi|_{E}$
owing to the homogeneous boundary conditions. Jump and average are understood
componentwise for any vector function. The edge-patch $\omega(E):=\text{\rm
int}(T_{+})$ of an edge $E\in\operatorname{\mathcal{E}}(\partial\Omega)$ on
the boundary is simply the interior of the one triangle $T_{+}$ with the edge
$E$ in the triangulation $\mathcal{T}$.

The nonconforming Morley finite element space [26] reads

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

Figure 1.a depicts the degrees of freedom of the Morley finite element

$\left(T,P_{2}(T),(\delta_{z}:z\in\mathcal{V}(T))\cup(\fint_{E}\partial_{\nu_{E}}\bullet{\rm\,ds}:E\in\operatorname{\mathcal{E}}(T))\right)$

(in the sense of Ciarlet) in the triangle $T$ with set of vertices
$\mathcal{V}(T)$ and set of edges $\operatorname{\mathcal{E}}(T)$.

The semi-scalar product $a_{\mathrm{pw}}$ is defined by the piecewise
differential operator $D^{2}_{\mathrm{pw}}$ and

$\displaystyle a_{\mathrm{pw}}(v_{\mathrm{pw}},w_{\mathrm{pw}})$
$\displaystyle:=\sum_{T\in\mathcal{T}}\int_{T}D^{2}v_{\mathrm{pw}}:D^{2}w_{\mathrm{pw}}{\rm\,dx}\quad\text{for
all }v_{\mathrm{pw}},w_{\mathrm{pw}}\in H^{2}(\mathcal{T}).$ (3.3)

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

$\displaystyle
a_{\mathrm{pw}}(u_{\mathrm{M}},v_{\mathrm{M}})=F_{h}(v_{\mathrm{M}})\quad\text{for
all }v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T}).$ (3.4)

###  3.3 Interpolation of discontinuous functions

######  Lemma 3.1 (interpolation estimates I [11, 7]).

The Morley interpolation operator
$I_{\mathrm{M}}:V\rightarrow\mathrm{M}(\mathcal{T})$ is defined by
$(I_{\mathrm{M}}v)(z)=v(z)$ and $\fint_{E}\frac{\partial
I_{\mathrm{M}}v}{\partial\nu_{E}}{\rm\,ds}=\fint_{E}\frac{\partial
v}{\partial\nu_{E}}{\rm\,ds}$ for any $z\in{\mathcal{V}}(\Omega)$ and
$E\in{\mathcal{E}(\Omega)}$. It satisfies (a) the integral mean property of
the Hessian, $D^{2}_{\text{\rm pw}}I_{\mathrm{M}}=\Pi_{0}D^{2}$,
(b) $\displaystyle\sum_{m=0}^{2}h_{T}^{m-2}|v-I_{\mathrm{M}}v|_{H^{m}(T)}\leq
2\|(1-\Pi_{0})D^{2}v\|_{L^{2}(T)}$ for all $v\in H^{2}(T)$ and any
$T\in\mathcal{T}$,
(c) $|\\!|\\!|v-I_{\mathrm{M}}v|\\!|\\!|_{\mathrm{pw}}\lesssim
h^{s}_{\max}\|v\|_{H^{2+s}(\Omega)}$ for all $v\in H^{2+s}(\Omega)$ and all
$0\leq s\leq 1$. ∎

A reformulation of Lemma 3.1.a is the best-approximation property

$\displaystyle a_{\mathrm{pw}}(v-I_{\mathrm{M}}v,w_{2})$
$\displaystyle=0\quad\text{for all }v\in V\text{ and all }w_{2}\in
P_{2}(\mathcal{T}).$ (3.5)

A reformulation of Lemma 3.1.b is the existence of a universal constant
$\kappa>0$ with

$\displaystyle\|h_{\mathcal{T}}^{-2}(v-I_{\mathrm{M}}v)\|$
$\displaystyle\leq\kappa|\\!|\\!|v-I_{\mathrm{M}}v|\\!|\\!|_{\mathrm{pw}}\quad\text{for
all }v\in V.$ (3.6)

(In fact $\kappa=0.25745784465$ from [11] is independent of the shape of the
triangle $T$.)

######  Remark 3.1 (Pythagoras).

The functions $v_{\mathrm{pw}},w_{\mathrm{pw}}\in
H^{2}(\mathcal{T}):=\\{v_{\mathrm{pw}}\in L^{2}(\Omega):\forall
T\in\mathcal{T},\;v_{\mathrm{pw}}|_{T}\in H^{2}(T)\\}$ are orthogonal iff
$a_{\mathrm{pw}}(v_{\mathrm{pw}},w_{\mathrm{pw}})=0$ holds and then the
Pythagoras theorem leads to

$\displaystyle|\\!|\\!|v-w_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}^{2}=|\\!|\\!|v-I_{\mathrm{M}}v|\\!|\\!|_{\mathrm{pw}}^{2}+|\\!|\\!|w_{\mathrm{M}}-I_{\mathrm{M}}v|\\!|\\!|_{\mathrm{pw}}^{2}$
(3.7)

for all $v\in V$ and $w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$. In
particular,
$\displaystyle|\\!|\\!|v-I_{\mathrm{M}}v|\\!|\\!|_{\mathrm{pw}}=\min_{v_{2}\in
P_{2}(T)}|\\!|\\!|v-v_{2}|\\!|\\!|_{\mathrm{pw}}.$

######  Definition 3.1 ((local) Morley interpolation).

Given any $T\in\mathcal{T}$ and $v_{\mathrm{pw}}\in H^{2}(T)$, the (local)
Morley interpolation $I_{\mathrm{M}}^{\rm loc}v_{\mathrm{pw}}|_{T}\in
P_{2}(T)$ is defined by the degrees of freedom of the Morley finite element
such that, for all $z\in\mathcal{V}(T)$ and for all
$E\in\operatorname{\mathcal{E}}(T)$,

$(I_{\mathrm{M}}^{\rm
loc}v_{\mathrm{pw}}-v_{\mathrm{pw}})|_{T}(z)=0\quad\text{and}\quad\fint_{E}(\partial(I_{\mathrm{M}}^{\rm
loc}v_{\mathrm{pw}}-v_{\mathrm{pw}})|_{T}/\partial\nu_{E}){\rm\,ds}=0.$

The Morley interpolation allows for an extension (still denoted by
$I_{\mathrm{M}}$) to piecewise $H^{2}$ functions in
$H^{2}(\mathcal{T})\equiv\prod_{T\in\mathcal{T}}H^{2}(T)$ by averaging the
degrees of freedom.

######  Definition 3.2 (Morley interpolation).

Given any $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$, define
$I_{\mathrm{M}}v_{\mathrm{pw}}:=v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ by
the degrees of freedom as follows. For any interior vertex
$z\in{\mathcal{V}}(\Omega)$ with set of attached triangles $\mathcal{T}(z)$
that has cardinality $|\mathcal{T}(z)|\in\mathbb{N}$ and any interior edge
$E=\partial T_{+}\cap\partial T_{-}\in{\operatorname{\mathcal{E}}(\Omega)}$
and its mean value operator $\left<\bullet\right>_{E}$ (the arithmetic mean of
the two traces from the triangles $T_{+}$ and $T_{-}\in\mathcal{T}$ along
their common edge $E=\partial T_{+}\cap\partial T_{-}$), set

$v_{\mathrm{M}}(z):=|\mathcal{T}(z)|^{-1}\sum_{T\in\mathcal{T}(z)}(v_{\mathrm{pw}}|_{T})(z)\quad\text{and}\quad\fint_{E}\frac{\partial
v_{\mathrm{M}}}{\partial\nu_{E}}{\rm\,ds}:=\fint_{E}\left<\frac{\partial
v_{\mathrm{pw}}}{\partial\nu_{E}}\right>_{E}{\rm\,ds}.$

(The remaining degrees of freedom at vertices and edges on the boundary are
zero for homogeneous boundary conditions.)

######  Remark 3.2 ( standard Morley interpolation vs Definition 3.2).

The interpolation operator $I_{\mathrm{M}}$ of Definition 3.2 extends that of
standard Morley interpolation operator in the sense that the two definitions
coincide for functions in $H^{2}_{0}(\Omega)$. This justifies the use of the
same symbol $I_{\mathrm{M}}$.

###  3.4 Companion operator and best-approximation for the Morley FEM

A conforming finite-dimensional subspace of $H^{2}_{0}(\Omega)$ is provided by
the Hsieh-Clough-Tocher $(HCT)$ FEM [26, Chap. 6]. For any $T\in\mathcal{T}$,
let $\mathcal{K}(T):=\\{T_{E}:\ E\in\mathcal{E}(T)\\}$ denote the
triangulation of $T$ into three sub-triangles
$T_{E}:=\textup{conv}\\{E,\textup{mid}(T)\\}$ with edges $E\in\mathcal{E}(T)$
and common vertex $\textup{mid}(T)$ depicted in Figure 1.b. Then,

$\displaystyle{HCT}(\mathcal{T})$ $\displaystyle:=\\{v\in H^{2}_{0}(\Omega):\
v|_{T}\in P_{3}(\mathcal{K}(T))\text{ for all }T\in\mathcal{T}\\}.$ (3.8)

The degrees of freedom in a triangle $T\in\mathcal{T}$ are the nodal values
$\psi(z)$ and its derivative $\nabla\psi(z)$ of the function
$\psi\in{HCT}(\mathcal{T})$ at any vertex $z\in\mathcal{V}(T)$ and the values
$\partial\psi/\partial\nu_{E}(\textup{mid}(E))$ of the normal derivatives at
the midpoint $\textup{mid}(E)$ of any edge $E\in\mathcal{E}(T)$.

######  Lemma 3.2 (right-inverse [29, 23, 41]).

There exists a linear map $J:{\rm
M}(\mathcal{T})\to(HCT(\mathcal{T})+P_{8}(\mathcal{T}))\cap H^{2}_{0}(\Omega)$
and a constant $\Lambda_{\mathrm{J}}$ (that exclusively depends on the shape
regularity of $\mathcal{T}$) such that any $v_{\rm M}\in{\rm M}(\mathcal{T})$
satisfies (a)–(e).

  1. (a)

$Jv_{\rm M}(z)=v_{\rm M}(z)$ for any $z\in\mathcal{V}$;

  2. (b)

$\nabla({J}v_{\rm M})(z)=|\mathcal{T}(z)|^{-1}\sum_{T\in\mathcal{T}(z)}(\nabla
v_{\rm M}|_{T})(z)\quad\text{ for }z\in\mathcal{V}(\Omega)$;

  3. (c)

$\fint_{E}\partial Jv_{\rm M}/\partial\nu_{E}\,\textup{d}s=\fint_{E}\partial
v_{\rm M}/\partial\nu_{E}\,\textup{d}s$ for any $E\in\mathcal{E}$;

  4. (d)

$|\\!|\\!|v_{\mathrm{M}}-Jv_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}\min_{v\in
V}|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}$;

  5. (e)

$v_{\mathrm{M}}-Jv_{\mathrm{M}}\perp P_{2}(\mathcal{T})$ in $L^{2}(\Omega)$. ∎

The operator $J$ of Lemma 3.2 with $(a)$-$(c)$ is a right-inverse for
$I_{\mathrm{M}}:V\rightarrow\mathrm{M}(\mathcal{T})$, i.e.,

$\displaystyle I_{\mathrm{M}}J=\text{id in }\mathrm{M}(\mathcal{T}).$ (3.9)

Examples are provided in [29, 23, 41]. For earlier references in the
literature, see [8], [10], [18]. A right-inverse with benefits like (d)-(e) is
called companion operator and [23] defines $J:{\rm M}(\mathcal{T})\to V$ so
that (a)-(e) of Lemma 3.2 hold (cf. in particular [23, Lemma 5.1] for the
analysis of (d)-(e)).

Given $F\in L^{2}(\Omega)$, we may choose $F_{h}\equiv F$ in the discrete
scheme (3.4); but otherwise $F_{h}=F\circ J$ is the option throughout this
paper; other choices are proposed in [6, 42]. Given any Lebesgue function
$F\equiv F_{h}\in L^{2}(\Omega)$ with its $L^{2}$ projection $\Pi_{2}F$ onto
$P_{2}(\mathcal{T})$, define its oscillations
$\mathrm{osc}_{2}(F,\mathcal{T}):=\|h_{\mathcal{T}}^{2}(F-\Pi_{2}F)\|$.

######  Theorem 3.3 (best-approximation up to data approximation [34, 13]).

The constant
$C_{1}:=\max\\{\kappa\Lambda_{\mathrm{J}},1+\Lambda_{\mathrm{J}}\\}$, the
solution $u\in V$ to (3.1) with $F\in L^{2}(\Omega)$, and the solution
$u_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ to (3.4) with $F_{h}\equiv F$
satisfy
$C_{1}{}^{-1}|\\!|\\!|u-u_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}+\mathrm{osc}_{2}(F,\mathcal{T})$.
∎

The discrete scheme (3.4) requires a discrete right-hand side $F_{h}$ for a
general $F\in H^{-2}(\Omega)$. The evaluation of $F_{h}:=F\circ J$ is feasible
with $F_{h}(v_{\mathrm{M}}):=F(Jv_{\mathrm{M}})$ for all
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ and the (modified) nonconforming
scheme seeks the solution $u_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ to

$a_{\mathrm{pw}}(u_{\mathrm{M}},v_{\mathrm{M}})=F(Jv_{\mathrm{M}})\quad\text{for
all }v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T}).$ (3.10)

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

$\Lambda_{0}:=\sup_{v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})\setminus\\{0\\}}|\\!|\\!|v_{\mathrm{M}}-Jv_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}/|\\!|\\!|v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}.$
(3.11)

######  Theorem 3.4 (best-approximation [21, 43]).

The solution $u\in V$ to (3.1) with $F\in V^{*}$ and the solution
$u_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ to (3.10) satisfy
$|\\!|\\!|u-u_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}$$~{}\leq~{}\sqrt{1+\Lambda_{0}^{2}}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}.$
The constant $\sqrt{1+\Lambda_{0}^{2}}$ is optimal. ∎

######  Remark 3.3 (extra orthogonality in Lemma 3.2.e.).

The $L^{2}$ orthogonality in Lemma 3.2.e 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 in eigenvalue analysis or problems with
low-order terms; for e.g. in [23, 19]. The $L^{2}$ orthogonality in Lemma
3.2.e also allows a direct proof of Theorem 3.3 that circumvents the a
posteriori error analysis of the consistency term as part of the medius
analysis [34]. Notice that the proof of the best-approximation of Theorem 3.4
for the modified scheme does not require the $L^{2}$ orthogonality in Lemma
3.2.e.

######  Remark 3.4 (minimal assumptions on the smoother).

The series of papers [41]-[43] addresses the question on the minimal
assumptions on the smoother (partly as a right inverse only). This paper
utilizes a smoother $J$ with the properties of Lemma 3.2.a-d.

The point in the subsequent example is that the smoother $JI_{\mathrm{M}}$ may
be more costly than averaging in other examples but it is at almost no extra
costs for the case of point forces, which are of practical importance in civil
engineering.

######  Example 3.2 (point forces).

Let $m$ denote the point forces in the right-hand side, i.e. , let

$\displaystyle F$ $\displaystyle=\sum_{j=1}^{m}\alpha_{j}\delta_{a_{j}},$
(3.12)

the triangulation can be adopted such that the concentration point $a_{j}$
becomes a vertex in the triangulation. The right-inverse property of
$I_{\mathrm{M}}$ displays that
$(JI_{\mathrm{M}}v_{\mathrm{M}})(z)=v_{\mathrm{M}}(z)$ holds at any vertex
$z\in\mathcal{V}$ and for any Morley function
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$. Hence the evaluation of the
modified right-hand side $F_{h}:=F\circ J$ leads for (3.12) to
$F_{h}v_{\mathrm{M}}=\sum_{j=1}^{m}\alpha_{j}v_{\mathrm{M}}({a_{j}})$. The
averaging of $I_{\mathrm{M}}$ in Definition 3.2 shows in the more general case
$V_{h}\subset P_{2}(\Omega)$ that $F_{h}:=F\circ JI_{\mathrm{M}}$ leads to
$F_{h}v_{\mathrm{M}}=\sum_{j=1}^{m}\alpha_{j}/|\mathcal{T}(a_{j})|\sum_{T\in\mathcal{T}(a_{j})}(v_{h}|_{T})({a_{j}})$.
The same formula applies to other smoothers like the enrichment in [6] and
[41]-[43].

##  4 Interpolation of piecewise $H^{2}$ functions

###  4.1 Equivalent norms

The Hilbert space $H^{2}(\mathcal{T})\equiv\prod_{T\in\mathcal{T}}H^{2}(T)$ is
endowed with a norm $\|\bullet\|_{h}$ from [13] defined by

$\displaystyle\|v_{\mathrm{pw}}\|^{2}_{h}:=|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|^{2}_{\mathrm{pw}}+j_{h}(v_{\mathrm{pw}})^{2}\quad\text{for
all }v_{\mathrm{pw}}\in H^{2}(\mathcal{T}).$ (4.1)

The homogeneous boundary conditions in $H^{2}_{0}(\Omega)$ are included in the
the jump contributions

$\displaystyle
j_{h}(v_{\mathrm{pw}})^{2}:=\sum_{E\in\operatorname{\mathcal{E}}}\sum_{z\in{\mathcal{V}}(E)}h_{E}^{-2}|[v_{\mathrm{pw}}]_{E}(z)|^{2}+\sum_{E\in\operatorname{\mathcal{E}}}\left|\fint_{E}\left[{\partial
v_{\mathrm{pw}}}/{\partial\nu_{E}}\right]_{E}{\rm\,ds}\right|^{2}$ (4.2)

by $[v_{\mathrm{pw}}]_{E}(z)=v_{\mathrm{pw}}|_{\omega(E)}(z)$ for
$z\in\mathcal{V}(\partial\Omega)$ and $\left[\frac{\partial
v_{\mathrm{pw}}}{\partial\nu_{E}}\right]_{E}=\frac{\partial
v_{\mathrm{pw}}}{\partial\nu_{E}}|_{E}$ for
$E\in\operatorname{\mathcal{E}}(\partial\Omega)$ at the boundary with jump
partner zero owing to the homogeneous boundary conditions in (3.1).

The discontinuous Galerkin schemes of [2, 28] are associated with a another
family of norms $\lVert\bullet\rVert_{\mathrm{dG}}$ depending on the two
positive parameters $\sigma_{1},\sigma_{2}>0$ in the semi-norm scalar product

$c_{\rm
dG}(v_{\mathrm{pw}},w_{\mathrm{pw}}):=\sum_{E\in\operatorname{\mathcal{E}}}\frac{\sigma_{1}}{h_{E}^{3}}\int_{E}\left[v_{\mathrm{pw}}\right]_{E}\left[w_{\mathrm{pw}}\right]_{E}{\rm\,ds}+\frac{\sigma_{2}}{h_{E}}\int_{E}\left[\frac{\partial
v_{\mathrm{pw}}}{\partial\nu_{E}}\right]_{E}\left[\frac{\partial
w_{\mathrm{pw}}}{\partial\nu_{E}}\right]_{E}{\rm\,ds}$ (4.3)

for all $v_{\mathrm{pw}},w_{\mathrm{pw}}\in H^{2}(\mathcal{T})$. The DG norm
$\lVert\bullet\rVert_{\mathrm{dG}}$ is the square root of

$\lVert
v_{\mathrm{pw}}\rVert_{\mathrm{dG}}^{2}:=|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|_{\mathrm{pw}}^{2}+c_{\rm
dG}(v_{\mathrm{pw}},v_{\mathrm{pw}})$ (4.4)

for all $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$. It depends on the parameters
$\sigma_{1},\sigma_{2}>0$ and so do all constants in the sequel; in particular
those suppressed in the abbreviations $\lesssim$ and $\approx$. The conditions
on the ellipticity of the scheme in Lemma 7.1 below will assert that
$\sigma_{1}$ and $\sigma_{2}$ are sufficiently large. The analysis of this
paper assumes this and simplifies the notation $\sigma_{1}\approx
1\approx\sigma_{2}$.

One result in [13, Theorem 4.1] shows that
$\lVert\bullet\rVert_{h}\approx\lVert\bullet\rVert_{\mathrm{dG}}$ in
$H^{2}_{0}(\Omega)+P_{2}(\mathcal{T})$; but the two norms are equivalent in
the larger vector space $H^{2}(\mathcal{T})$.

######  Theorem 4.1
($\|\bullet\|_{h}\approx\lVert\bullet\rVert_{\mathrm{dG}}$).

The function $\|\bullet\|_{h}$ from (4.1) and
$\lVert\bullet\rVert_{\mathrm{dG}}$ from (4.4) define norms in
$H^{2}(\mathcal{T})$ with

$\|v_{\mathrm{pw}}\|_{h}\approx\lVert
v_{\mathrm{pw}}\rVert_{\mathrm{dG}}\lesssim\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}v_{\rm
pw}|_{H^{m}(\mathcal{T})}\quad\text{for all }v_{\mathrm{pw}}\in
H^{2}(\mathcal{T}).$

######  Remark 4.1 ($\\{j_{h}=0\\}\cap
P_{2}(\mathcal{T})=\mathrm{M}(\mathcal{T})$).

For any $v_{2}\in P_{2}(\mathcal{T})$, the condition $j_{h}(v_{2})=0$ is
equivalent to $v_{2}\in\mathrm{M}(\mathcal{T})$. (This follows from the
definitions of $\mathrm{M}(\mathcal{T})$ and $j_{h}$.)

###### Proof of $\|\bullet\|_{h}\lesssim\lVert\bullet\rVert_{\mathrm{dG}}$.

The (possibly discontinuous) piecewise affine interpolation $v_{1}\in
P_{1}(\mathcal{T})$ of $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$ is defined by
nodal interpolation $v_{1}|_{T}(z)=v_{\mathrm{pw}}|_{T}(z)$ at the three
vertices $z\in\mathcal{V}(T)$ in each triangle $T\in\mathcal{T}$. It is well
known from standard finite element interpolation [4, 6, 26] that the error
$w:=v_{\mathrm{pw}}-v_{1}\in H^{2}(T)$ satisfies

$\sum_{m=0}^{2}h_{T}^{m-2}|w|_{H^{m}(T)}\lesssim|v_{\mathrm{pw}}|_{H^{2}(T)}$
(4.5)

for each triangle $T\in\mathcal{T}$ with explicit constants [14] that
exclusively depend on the maximal angle in the triangulation. The nodal
interpolation implies $[v_{\mathrm{pw}}]_{E}(z)=[v_{1}]_{E}(z)$ at each vertex
$z\in\mathcal{V}(E)$ of an edge $E\in\operatorname{\mathcal{E}}$. Since
${[v_{1}]_{E}}$ is an affine function along the edge
$E\in\operatorname{\mathcal{E}}$, an inverse estimate shows

$h_{E}/6\sum_{z\in{\mathcal{V}}(E)}|[v_{1}]_{E}(z)|^{2}\leq\|[v_{1}]_{E}\|_{L^{2}(E)}^{2}\leq
2\|[v_{\mathrm{pw}}]_{E}\|_{L^{2}(E)}^{2}+2\|[w]_{E}\|_{L^{2}(E)}^{2}$ (4.6)

with a triangle inequality in the last step for $w=v_{\mathrm{pw}}-v_{1}$.
(The constant $h_{E}/6$ in the first inequality of (4.6) stems from the
eigenvalues $h_{E}/2$ and $h_{E}/6$ of the $2\times 2$ mass matrix of
piecewise linear functions in 1D.) This implies an estimate for the first term
of the definition of $j_{h}(v_{\mathrm{pw}})$:

$\sum_{E\in\operatorname{\mathcal{E}}}\sum_{z\in{\mathcal{V}}(E)}h_{E}^{-2}|[v_{\mathrm{pw}}]_{E}(z)|^{2}\leq
12\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-3}(\|[v_{\mathrm{pw}}]_{E}\|_{L^{2}(E)}^{2}+\|[w]_{E}\|_{L^{2}(E)}^{2}).$

A typical contribution $(\fint_{E}\left[{\partial
v_{\mathrm{pw}}}/{\partial\nu_{E}}\right]_{E}{\rm\,ds})^{2}$ for the second
term (in the definition of $j_{h}$) is controlled with a Cauchy inequality by
$h_{E}^{-1}\|\left[{\partial
v_{\mathrm{pw}}}/{\partial\nu_{E}}\right]_{E}\|_{L^{2}(E)}^{2}$. This results
in

$j_{h}(v_{\mathrm{pw}})^{2}\leq\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-1}12\left(h_{E}^{-2}(\|[v_{\mathrm{pw}}]_{E}\|_{L^{2}(E)}^{2}+\|[w]_{E}\|_{L^{2}(E)}^{2})+\|\left[{\partial
v_{\mathrm{pw}}}/{\partial\nu_{E}}\right]_{E}\|_{L^{2}(E)}^{2}\right).$

A triangle inequality
$\|[w]_{E}\|_{L^{2}(E)}\leq\|w|_{T_{+}}\|_{L^{2}(E)}+\|w|_{T_{-}}\|_{L^{2}(E)}$
for an interior edge $E=\partial T_{+}\cap\partial
T_{-}\in\operatorname{\mathcal{E}}(\Omega)$ shared by the two triangles
$T_{\pm}\in\mathcal{T}$ plus trace inequalities on $T_{\pm}$ show

$h_{E}^{1/2}\|[w]_{E}\|_{L^{2}(E)}\lesssim\|w\|_{L^{2}(\omega(E))}+h_{E}\|\nabla_{\mathrm{pw}}w\|_{L^{2}(\omega(E))}\lesssim
h^{2}_{E}\|D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\omega(E))}$ (4.7)

with (4.5) in the end. The omission of $T_{-}$ in the above arguments for an
edge $E\in\operatorname{\mathcal{E}}(\partial\Omega)$ on the boundary provide
(4.7) with $\overline{\omega(E)}=T_{+}$. This and the finite overlap show
$\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-3}\|[w]_{E}\|_{L^{2}(E)}^{2}\lesssim|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|^{2}_{\mathrm{pw}}$.
In conclusion,

$j_{h}(v_{\mathrm{pw}})^{2}\lesssim|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|^{2}_{\mathrm{pw}}+\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-3}\|[v_{\mathrm{pw}}]_{E}\|_{L^{2}(E)}^{2}+\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-1}\|\left[{\partial
v_{\mathrm{pw}}}/{\partial\nu_{E}}\right]_{E}\|_{L^{2}(E)}^{2}.$

The upper bound in the latter estimate is $\lVert
v_{\mathrm{pw}}\rVert_{\mathrm{dG}}^{2}$ up to the weights $\sigma_{1}\approx
1\approx\sigma_{2}$. ∎

###### Proof of $\lVert\bullet\rVert_{\mathrm{dG}}\lesssim\|\bullet\|_{h}$.

Recall the piecewise affine interpolation $v_{1}\in P_{1}(\mathcal{T})$ of
$v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$ and $w:=v_{\mathrm{pw}}-v_{1}\in
H^{2}(\mathcal{T})$ with (4.5) from the previous part of the proof. Standard
trace inequalities as in (4.7) for the first term (and an analog for the
second term $h_{E}^{-1/2}\|[\partial w/\partial\nu_{E}]_{E}\|_{L^{2}(E)}$) for
$E\in\operatorname{\mathcal{E}}$ provide

$h_{E}^{-3/2}\|[w]_{E}\|_{L^{2}(E)}+h_{E}^{-1/2}\|[\partial
w/\partial\nu_{E}]_{E}\|_{L^{2}(E)}\lesssim\|D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\omega(E))}.$
(4.8)

This and triangle inequalities result in

$\displaystyle
h_{E}^{-3/2}\|[v_{\mathrm{pw}}]_{E}\|_{L^{2}(E)}+h_{E}^{-1/2}\|[\partial
v_{\mathrm{pw}}/\partial\nu_{E}]_{E}\|_{L^{2}(E)}$ $\displaystyle\quad\lesssim
h_{E}^{-3/2}\|[v_{1}]_{E}\|_{L^{2}(E)}+h_{E}^{-1/2}\|[\partial
v_{1}/\partial\nu_{E}]_{E}\|_{L^{2}(E)}+\|D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\omega(E))}.$

The constant factor $1/2$ in the upper bound of the first subsequent
inequality (displayed as $2$ in the lower bound) stems from the eigenvalues
$h_{E}/2$ and $h_{E}/6$ of the $2\times 2$ mass matrix of piecewise linear
functions in 1D,

$2h_{E}^{-3}\|[v_{1}]_{E}\|_{L^{2}(E)}^{2}\leq
h_{E}^{-2}\sum_{z\in{\mathcal{V}}(E)}|[v_{1}]_{E}(z)|^{2}=h_{E}^{-2}\sum_{z\in{\mathcal{V}}(E)}|[v_{\mathrm{pw}}]_{E}(z)|^{2}$

with the nodal interpolation property $v_{1}|_{T}(z)=v_{\mathrm{pw}}|_{T}(z)$
for $z\in\mathcal{V}(T)$, $T\in\mathcal{T}$, in the last step. The jump
$[\partial v_{1}/\partial\nu_{E}]_{E}$ is constant along the edge $E$ and so

$\displaystyle h_{E}^{-1/2}\|[\partial
v_{1}/\partial\nu_{E}]_{E}\|_{L^{2}(E)}=|\fint_{E}[\partial
v_{1}/\partial\nu_{E}]_{E}{\rm\,ds}|$
$\displaystyle\quad\leq|\fint_{E}[\partial
v_{\mathrm{pw}}/\partial\nu_{E}]_{E}{\rm\,ds}|+|\fint_{E}[\partial
w/\partial\nu_{E}]_{E}{\rm\,ds}|$

with a triangle inequality in the last step. A Cauchy inequality $\|\nabla
w_{\mathrm{pw}}\|_{L^{1}(E)}\leq h_{E}^{1/2}\|\nabla
w_{\mathrm{pw}}\|_{L^{2}(E)}$ and a trace inequality show (as above in (4.8))
that

$|\fint_{E}[\partial w/\partial\nu_{E}]_{E}{\rm\,ds}|\leq
h_{E}^{-1/2}\|[\partial
w/\partial\nu_{E}]_{E}\|_{L^{2}(E)}\lesssim\|D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\omega(E))}.$

The combination of all aforementioned estimates reads

$\displaystyle
h_{E}^{-3}\|[v_{\mathrm{pw}}]_{E}\|^{2}_{L^{2}(E)}+h_{E}^{-1}\|[\partial
v_{\mathrm{pw}}/\partial\nu_{E}]_{E}\|_{L^{2}(E)}^{2}\lesssim\|D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\omega(E))}^{2}$
$\displaystyle\quad+h_{E}^{-2}\sum_{z\in{\mathcal{V}}(E)}|[v_{\mathrm{pw}}]_{E}(z)|^{2}+|\fint_{E}[\partial
v_{\mathrm{pw}}/\partial\nu_{E}]_{E}{\rm\,ds}|^{2}.$

The sum of all those estimates over $E\in\operatorname{\mathcal{E}}$ plus
$|\\!|\\!|v_{\mathrm{pw}}|\\!|\\!|^{2}_{\mathrm{pw}}$ leads to an estimate
with the lower bound $\lVert v_{\mathrm{pw}}\rVert_{\mathrm{dG}}^{2}$ up to
the weights $\sigma_{1}\approx 1\approx\sigma_{2}$. The finite overlap of the
edge-patches $(\omega(E):E\in\operatorname{\mathcal{E}})$ shows that the
resulting upper bound is $\lesssim\|v_{\mathrm{pw}}\|_{h}$. ∎

###### Proof of the upper bound.

The proof of the asserted inequality starts with triangle inequalities for the
jumps of $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$ and the shape regularity for
$h_{E}\approx h_{T}$ for $E\in\operatorname{\mathcal{E}}(T)$. This and a
Cauchy inequality $\|\nabla v_{\mathrm{pw}}\|_{L^{1}(E)}\leq
h_{E}^{1/2}\|\nabla v_{\mathrm{pw}}\|_{L^{2}(E)}$ lead to

$j_{h}(v_{\mathrm{pw}})^{2}\lesssim\sum_{T\in\mathcal{T}}\left(h_{T}^{-2}\sum_{z\in{\mathcal{V}}(T)}|(v_{\mathrm{pw}}|_{T})(z)|^{2}+h_{T}^{-1}\sum_{E\in\operatorname{\mathcal{E}}(T)}\|\nabla
v_{\mathrm{pw}}\|_{L^{2}(E)}^{2}\right).$

A one-dimensional trace inequality (with a factor $1$ that follows from 1D
integration)

$|(v_{\mathrm{pw}}|_{T})(z)|\leq
h_{E}^{-1/2}\|v_{\mathrm{pw}}\|_{L^{2}(E)}+h_{E}^{1/2}\|\nabla
v_{\mathrm{pw}}\|_{L^{2}(E)}$

along the edge $E\in\operatorname{\mathcal{E}}(T)$ of the triangle
$T\in\mathcal{T}$ with vertex $z\in\mathcal{V}(E)$ results in

$j_{h}(v_{\mathrm{pw}})^{2}\lesssim\sum_{T\in\mathcal{T}}\left(h_{T}^{-3}\|v_{\mathrm{pw}}\|_{L^{2}(\partial
T)}^{2}+h_{T}^{-1}\|\nabla v_{\mathrm{pw}}\|_{L^{2}(\partial
T)}^{2}\right)\lesssim\sum_{T\in\mathcal{T}}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}v_{\rm
pw}|_{H^{m}(T)}^{2}$

with standard trace inequalities on $\partial T$ for $v_{\mathrm{pw}}$ and
$\nabla v_{\mathrm{pw}}$ in the last step. The right-hand side is
$\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}v_{\rm pw}|_{H^{m}(\mathcal{T})}^{2}$ as
asserted. The remaining details are omitted for brevity. ∎

###  4.2 Interpolation errors

The interpolation error estimates are summarised in one theorem.

######  Theorem 4.2 (interpolation).

Any $v_{\rm pw}\in H^{2}(\mathcal{T})$ and its Morley interpolation $I_{\rm
M}v_{\rm pw}\in\mathrm{M}(\mathcal{T})$ from Definition 3.2 satisfy
(a) $\displaystyle\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\rm pw}-I_{\rm
M}v_{\rm pw})|_{H^{m}(\mathcal{T})}\lesssim\|(1-\Pi_{0})D^{2}_{\rm pw}v_{\rm
pw}\|+j_{h}(v_{\rm pw})\leq\|v_{\rm pw}\|_{h}$;
(b) $\displaystyle\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\rm pw}-I_{\rm
M}v_{\rm
pw})|_{H^{m}(\mathcal{T})}\approx\min_{w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})}\|v_{\rm
pw}-w_{\mathrm{M}}\|_{h}\approx\min_{w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\rm
pw}-w_{\mathrm{M}})|_{H^{m}(\mathcal{T})}$.

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

The first step  reduces the analysis to piecewise quadratic functions by the
piecewise Morley interpolation $I_{\mathrm{M}}^{\rm loc}$. Definition 3.1
shows $\int_{E}\nabla(I_{\mathrm{M}}^{\rm
loc}v_{\mathrm{pw}}-v_{\mathrm{pw}})|_{T}\,{\rm\,ds}=0$ for an edge
$E\in\operatorname{\mathcal{E}}(T)$ of the triangle $T$ and therefore
$D^{2}(I_{\mathrm{M}}^{\rm
loc}v_{\mathrm{pw}})|_{T}=\Pi_{0}D^{2}v_{\mathrm{pw}}|_{T}$ a.e. in
$T\in\mathcal{T}$. Notice that the piecewise defined Morley interpolation
$v_{2}:=I_{\mathrm{M}}^{\rm loc}v_{\mathrm{pw}}\in P_{2}(\mathcal{T})$ is
discontinuous (and shares none of the compatibility or boundary conditions) in
general. The interpolation error estimates of Lemma 3.1.b read

$\sum_{m=0}^{2}h_{T}^{m-2}|v_{2}-v_{\mathrm{pw}}|_{H^{m}(T)}\leq
2\|(1-\Pi_{0})D^{2}v_{\mathrm{pw}}\|_{L^{2}(T)}.$

This and a triangle inequality show that it remains to prove that
$v_{\mathrm{M}}:=I_{\mathrm{M}}v_{\mathrm{pw}}$ satisfies

$\sum_{m=0}^{2}h_{T}^{m-2}|v_{2}-v_{\mathrm{M}}|_{H^{m}(T)}\lesssim
j_{h}(v_{\mathrm{pw}},T)$ (4.9)

for the jump terms localised to a neighbourhood $\Omega(T)$ of
$T\in\mathcal{T}$ as follows. The neighbourhood $\Omega(T)$ is the interior of
the union $\cup\\{K\in\mathcal{T}:\operatorname{dist}(T,K)=0\\}$ of
$T\in\mathcal{T}$ plus one layer of triangles in $\mathcal{T}$ around. Then

$j_{h}(v_{\mathrm{pw}},T)^{2}:=\sum_{z\in\mathcal{V}(T)}\sum_{F\in\operatorname{\mathcal{E}}(z)}h_{F}^{-2}|[v_{\mathrm{pw}}]_{F}(z)|^{2}+\sum_{E\in\operatorname{\mathcal{E}}(T)}\left|\fint_{E}\left[\frac{\partial
v_{\mathrm{pw}}}{\partial\nu_{E}}\right]_{E}{\rm\,ds}\right|^{2}$

is the contribution from $T$ and its neighbourhood $\Omega(T)$ to the full
jump term $j_{h}(v_{\mathrm{pw}})^{2}$ with the spider
$\operatorname{\mathcal{E}}(z):=\\{F\in\operatorname{\mathcal{E}}:z\in\mathcal{V}(F)\\}$
of edges with one end-point $z\in\mathcal{V}(T)$.

The second step  reduces the analysis to piecewise quadratic functions. The
first obervation is that the averaging of the degrees of freedom in the
definition of $I_{\mathrm{M}}$ merely employs the data of
$v_{2}=I_{\mathrm{M}}^{\rm loc}v_{\mathrm{pw}}$ in the sense that
$v_{\mathrm{M}}=I_{\mathrm{M}}v_{\mathrm{pw}}=I_{\mathrm{M}}v_{2}$. This
explains why $j_{h}(v_{\mathrm{pw}},T)=j_{h}(v_{2},T)$ in the asserted
estimate (4.9). The second observation is that the left-hand side of (4.9)
involves the polyonomial $(v_{2}-v_{\mathrm{M}})|_{T}\in P_{2}(T)$ that allows
for inverse estimates

$\sum_{m=0}^{2}h_{T}^{m-2}|v_{2}-I_{\mathrm{M}}v_{2}|_{H^{m}(T)}\lesssim
h_{T}^{-2}\|v_{2}-I_{\mathrm{M}}v_{2}\|_{L^{2}(T)}.$

The overall conclusion is that it suffices to prove, for all $v_{2}\in
P_{2}(\mathcal{T})$, that

$h_{T}^{-4}\|v_{2}-I_{\mathrm{M}}v_{2}\|^{2}_{L^{2}(T)}\lesssim
j_{h}(v_{2},T)^{2}.$ (4.10)

In fact, (4.10) and the aforementioned arguments lead to a localised form of
the assertion. The sum over all $T\in\mathcal{T}$ and the bounded overlap of
$(\Omega(T):T\in\mathcal{T})$ then conclude the proof of the theorem.

The third step  reduces the proof of (4.10) to six coefficients. The six
degrees of freedom on a triangle $T\in\mathcal{T}$ are the three point
evaluations $\delta_{z}$ at the three vertices $z\in\mathcal{V}(T)$ and the
three integral means of the normal derivatives
$\fint_{E}\partial_{\nu_{E}}\bullet{\rm\,ds}$ along the three edges
$E\in\operatorname{\mathcal{E}}(T)$. The six dual basis functions $\psi_{z}$
for $z\in\mathcal{V}(T)$ and $\psi_{E}$ for
$E\in\operatorname{\mathcal{E}}(T)$ in $P_{2}(T)$ are defined by the duality
relations $\psi_{E}(z)=0=\fint_{E}\partial_{\nu_{E}}\psi_{z}{\rm\,ds}$ and
$\psi_{z}(z)=1=\fint_{E}\partial_{\nu_{E}}\psi_{E}{\rm\,ds}$ for all
$z\in\mathcal{V}(T)$ and $E\in\operatorname{\mathcal{E}}(T)$, while
$\psi_{y}(z)=0=\fint_{E}\partial_{\nu_{E}}\psi_{F}{\rm\,ds}$ for all vertices
$z\neq y\in\mathcal{V}(T)$ and edges $E\neq
F\in\operatorname{\mathcal{E}}(T)$. Those functions are known and given
explicitly (e.g., in [12] in the context of a short implementation of the
Morley FEM in 30 lines of Matlab) with a scaling (which is generally
understood and follows from the explicit formulas)

$\|\psi_{z}\|_{L^{2}(T)}\approx|T|^{1/2}\approx
h_{T}\quad\text{and}\quad\|\psi_{E}\|_{L^{2}(T)}\approx h_{T}|T|^{1/2}\approx
h_{T}^{2}$

for all $z\in\mathcal{V}(T)$ and $E\in\operatorname{\mathcal{E}}(T)$. On the
other hand, given the dual basis of $P_{2}(T)$, any function
$w_{2}:=v_{2}-v_{\mathrm{M}}\in P_{2}(T)$ for
$v_{\mathrm{M}}:=I_{\mathrm{M}}v_{2}$ allows for a representation

$w_{2}=v_{2}-v_{\mathrm{M}}=\sum_{z\in\mathcal{V}(T)}w(z)\,\psi_{z}+\sum_{E\in\operatorname{\mathcal{E}}(T)}w(E)\,\psi_{E}\quad\text{in
}T$

with the real coefficients $w(z):=v_{2}|_{T}(z)-v_{\mathrm{M}}(z)$ and
$w(E):=\fint_{E}\partial_{\nu_{E}}v_{2}|_{T}{\rm\,ds}-\fint_{E}\partial_{\nu_{E}}v_{\mathrm{M}}{\rm\,ds}$
for $z\in\mathcal{V}(T)$ and $E\in\operatorname{\mathcal{E}}(T)$. Notice that
the contributions of the piecewise quadratic $v_{2}$ are taken from
$T\in\mathcal{T}$ and this is written explicitly by $v_{2}|_{T}$ in the
coefficients, while the corresponding values of the Morley function
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ are independent of $T$ as long as
$z\in\mathcal{V}(T)$ or $E\in\operatorname{\mathcal{E}}(T)$. Given the
coefficients $w(z)$ and $w(E)$, the triangle inequality in $L^{2}(T)$ and the
scaling of the dual basis functions lead to

$\|v_{2}-v_{\mathrm{M}}\|_{L^{2}(T)}\lesssim
h_{T}\sum_{z\in\mathcal{V}(T)}|w(z)|+h_{T}^{2}\sum_{E\in\operatorname{\mathcal{E}}(T)}|w(E)|.$
(4.11)

The fourth step  analyses the coefficients in (4.11). Let the triangles
$\mathcal{T}(z):=\\{T\in\mathcal{T}:z\in\mathcal{V}(T)\\}=\\{T(1),\dots,T(J)\\}$
at the vertex $z\in\mathcal{V}$ be enumerated such that $T(j)$ and $T(j+1)$
share an edge $\partial T(j)\cap\partial
T(j+1)=:E(j)\in\operatorname{\mathcal{E}}(z)$ for $j=1,\dots,J$. For an
interior vertex $z\in\mathcal{V}(\Omega)$, the patch is closed and then $T(1)$
and $T(J)$ share an edge $\partial T(1)\cap\partial
T(J)=:E(J)\in\operatorname{\mathcal{E}}(z)$ as well. Define
$x_{j}:=((v_{2}-v_{\mathrm{M}})|_{T(j)})(z)$ for $j=1,\dots,J$ and observe for
an interior vertex $z\in\mathcal{V}(\Omega)$ that $\sum_{j=1}^{J}x_{j}=0$
(from the choice of $v_{\mathrm{M}}(z)$ as the arithmetic mean of the
${v_{2}}|_{T(j)}$) and that

$\sum_{E\in\operatorname{\mathcal{E}}(z)}|\left[v_{2}\right]_{E}(z)|=\sum_{j=1}^{J}|x_{j+1}-x_{j}|$

with $x_{J+1}:=x_{1}$ (recall $z\in\mathcal{V}(\Omega)$ here). Since the
arithmetic mean of the real numbers $x_{1},\dots,x_{J}$ vanishes, zero belongs
to their convex hull; whence $\underline{m}:=\min_{j=1,\dots,J}x_{j}\leq
0\leq\max_{j=1,\dots,J}x_{j}=:\overline{m}$. A triangle inequality in this
sequence $x_{1},\dots,x_{J}$ shows that
$\overline{m}-\underline{m}\leq\sum_{j=1}^{J}|x_{j+1}-x_{j}|$ (even with an
omitted factor $1/2$). It follows
$|x_{1}|,\dots,|x_{J}|\leq\sum_{j=1}^{J}|x_{j+1}-x_{j}|$ and so, for a
triangle $T\in\mathcal{T}(z)$ in the notation of (4.11),

$|w(z)|\leq\sum_{E\in\operatorname{\mathcal{E}}(z)}|\left[v_{2}\right]_{E}(z)|\leq
J^{1/2}\sqrt{\sum_{E\in\operatorname{\mathcal{E}}(z)}|\left[v_{2}\right]_{E}(z)|^{2}}$
(4.12)

follows (with a Cauchy inequality in $\mathbb{R}^{J}$ in the end). This is
suboptimal and the best constant in a squared version of this argument is
contained in [23, Appendix C]. Observe that $J\lesssim 1$ is bounded from
above by the shape regularity of the triangulation $\mathcal{T}$.

In the remaining case of a vertex $z\in\mathcal{V}(\partial\Omega)$ on the
boundary, $v_{\mathrm{M}}(z)=0$ and, in the above notation
$\mathcal{T}(z)=\\{T\in\mathcal{T}:z\in\mathcal{V}(T)\\}=\\{T(1),\dots,T(J)\\}$
and $x_{j}=((v_{2}-v_{\mathrm{M}})|_{T(j)})(z)=(v_{2}|_{T(j)})(z)$ for
$j=1,\dots,J$. The homogeneous boundary conditions enter in the jump terms for
$E(1):=T_{1}\cap\partial\Omega$ and $E(J):=T_{J}\cap\partial\Omega$ and

$\sum_{E\in\operatorname{\mathcal{E}}(z)}|\left[v_{2}\right]_{E}(z)|=|x_{1}|+|x_{J}|+\sum_{j=1}^{J-1}|x_{j+1}-x_{j}|.$

Triangle inequalities show
$|x_{1}|,\dots,|x_{J}|\leq|x_{1}|+|x_{J}|+\sum_{j=1}^{J-1}|x_{j+1}-x_{j}|$
(even with an omitted factor $1/2$) and the above arguments lead to (4.12) as
well. (The optimal constant for this argument may be found in [15, Lemma
4.2].) Recall the design of the Morley interpolation in Definition 3.2 with
the arithmetic mean
$\fint_{E}\partial_{\nu_{E}}v_{\mathrm{M}}{\rm\,ds}=\fint_{E}\left<\frac{\partial
v_{2}}{\partial\nu_{E}}\right>_{E}{\rm\,ds}$ of the two normal traces for an
interior edge $E=\partial T_{+}\cap\partial
T_{-}\in\operatorname{\mathcal{E}}(\Omega)$. This leads to the edge
contribution

$w(E)=\fint_{E}\partial_{\nu_{E}}v_{2}|_{T}{\rm\,ds}-\fint_{E}\left<\frac{\partial
v_{2}}{\partial\nu_{E}}\right>_{E}{\rm\,ds}=\pm\frac{1}{2}\fint_{E}\left[\partial_{\nu_{E}}v_{2}\right]_{E}{\rm\,ds}$

in (4.11) with a sign $\pm$ for $T=T_{\pm}$. The boundary conditions for a
boundary edge $E\in\operatorname{\mathcal{E}}(\partial\Omega)$ and the jump
convention for $[\bullet]_{E}$ (recall that $\nu_{E}$ points outwards for
$E\subset\partial\Omega$) directly show
$w(E)=\fint_{E}\left[\partial_{\nu_{E}}v_{2}\right]_{E}{\rm\,ds}$. It follows

$|w(E)|\leq|\fint_{E}\left[\partial_{\nu_{E}}v_{2}\right]_{E}{\rm\,ds}|\text{
for any }E\in\operatorname{\mathcal{E}}(T).$ (4.13)

The fitfh step  finishes the proof. Recall that the coefficients $w(z)$ for
$z\in\mathcal{V}(z)$ and $w(E)$ for any $E\in\operatorname{\mathcal{E}}(T)$ in
(4.11) satisfy (4.12)-(4.13). The resulting estimate reads

$h_{T}^{-4}\|v_{2}-v_{\mathrm{M}}\|_{L^{2}(T)}^{2}\lesssim
h_{T}^{-2}\sum_{z\in\mathcal{V}(T)}\sum_{E\in\operatorname{\mathcal{E}}(z)}|[v_{2}]_{E}(z)|^{2}+\sum_{E\in\operatorname{\mathcal{E}}(T)}|\fint_{E}[{\partial_{\nu_{E}}v_{2}}]_{E}{\rm\,ds}|^{2}\approx
j_{h}(v_{2},T)^{2}$

with the shape regularity $h_{F}\approx h_{T}$ for
$F\in\operatorname{\mathcal{E}}(z)$ and $z\in\mathcal{V}(T)$ in the end. This
concludes the proof of (4.10) and thus that of (a) as outlined at the end of
the second step. ∎

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

Given any $w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$, part (a) shows that the
first term $T_{1}$ in the equivalence (b) is $T_{1}\lesssim|\\!|\\!|v_{\rm
pw}-w_{\mathrm{M}}|\\!|\\!|_{\text{pw}}+j_{h}(v_{\rm
pw}-w_{\mathrm{M}})\leq\|v_{\rm pw}-w_{\mathrm{M}}\|_{h}=:T_{2}$ with
$j_{h}(v_{\rm pw})=j_{h}(v_{\rm pw}-w_{\mathrm{M}})$ in the last step. Theorem
4.1 applies to $v_{\rm pw}-w_{\mathrm{M}}\in H^{2}(\mathcal{T})$ and proves
$T_{2}\lesssim\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\rm
pw}-w_{\mathrm{M}})|_{H^{m}(\mathcal{T})}=:T_{3}$. The estimates
$T_{1}\lesssim T_{2}\lesssim T_{3}$ hold for all
$w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ and so for the respective minima as
well. Since $I_{\mathrm{M}}v_{\mathrm{pw}}\in\mathrm{M}(\mathcal{T})$, the
remaining estimate $\min_{w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})}T_{3}\leq
T_{1}$ is obvious. ∎

######  Remark 4.2 ($I_{\mathrm{M}}JI_{\mathrm{M}}=I_{\mathrm{M}}$ in
$H^{2}(\mathcal{T})$).

Let $J$ be any right-inverse of $I_{\mathrm{M}}$ in the sense of (3.9). Since
$I_{\mathrm{M}}J$ is identity in $\mathrm{M}(\mathcal{T})$,
$I_{\mathrm{M}}JI_{\mathrm{M}}v_{\mathrm{pw}}=I_{\mathrm{M}}v_{\mathrm{pw}}$
holds for any $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$.

###  4.3 Approximation errors

The subsequent theorem discusses the approximation properties of $J\circ
I_{\mathrm{M}}$ for piecewise smooth and piecewise quadratic functions. It is
formulated in terms of
$\|\bullet\|_{h}\approx\lVert\bullet\rVert_{\mathrm{dG}}$ and the norm
equivalence implies an (undisplayed) analog for
$\lVert\bullet\rVert_{\mathrm{dG}}$ as well.

######  Theorem 4.3 (approximation).

Any $v_{\mathrm{pw}}\in H^{2}(\mathcal{T})$ and $v_{2}\in P_{2}(\mathcal{T})$
satisfy (a)-(d).
(a)
$\|v_{\mathrm{pw}}-JI_{\mathrm{M}}v_{\mathrm{pw}}\|_{h}\lesssim\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\Omega)}+\min_{v\in
H^{2}_{0}(\Omega)}\|v_{\mathrm{pw}}-v\|_{h}$;
(b)
$\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\mathrm{pw}}-JI_{\mathrm{M}}v_{\mathrm{pw}})|_{H^{m}(\mathcal{T})}\lesssim\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\Omega)}$
$+\min_{v\in
H^{2}_{0}(\Omega)}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\mathrm{pw}}-v)|_{H^{m}(\mathcal{T})}$;
(c) $\|v_{2}-JI_{\mathrm{M}}v_{2}\|_{h}\approx\min_{v\in
H^{2}_{0}(\Omega)}\|v_{2}-v\|_{h}$
$\approx\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{2}-JI_{\mathrm{M}}v_{2})|_{H^{m}(\mathcal{T})}\approx\min_{v\in
H^{2}_{0}(\Omega)}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{2}-v)|_{H^{m}(\mathcal{T})}$;
(d) $\|v_{2}-JI_{\mathrm{M}}v_{2}\|_{H^{s}(\mathcal{T})}\lesssim
h_{\max}^{2-s}\min_{v\in H^{2}_{0}(\Omega)}\|v_{2}-v\|_{h}$ holds for any
$0\leq s\leq 2$.

######  Remark 4.3.

Theorem 4.3 implies that $P=Q=J\circ I_{\mathrm{M}}$ is a quasi-optimal
smoother with constant $\Lambda_{\rm P}=\Lambda_{\rm Q}$ that depends only on
the shape regularity of the triangulation.

######  Remark 4.4 (remainder in (a)-(b)).

The extra term
$\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\Omega)}$ in the
upper bound will vanish for piecewise quadratic functions but cannot be
omitted in (a)-(b). For a proof of the latter statement by contradiction
consider some $v_{\mathrm{pw}}\in
H^{2}_{0}(\Omega)\setminus(HCT(\mathcal{T})+P_{8}(\mathcal{T}))$. Since
$JI_{\mathrm{M}}v_{\mathrm{pw}}\in HCT(\mathcal{T})+P_{8}(\mathcal{T})$, the
left-hand side in (a)-(b) is positive, while $v=v_{\mathrm{pw}}\in
H^{2}_{0}(\Omega)$ leads to a right-hand side zero if the term
$\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}v_{\mathrm{pw}}\|_{L^{2}(\Omega)}$ was
neglected.

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

Theorem 4.1 implies the first estimate (4.14.a) below and Theorem 4.2.a
asserts the second (4.14.b) for the Morley interpolation
$v_{\mathrm{M}}:=I_{\mathrm{M}}v_{\mathrm{pw}}$ of $v_{\mathrm{pw}}\in
H^{2}(\mathcal{T})$ in

$\|v_{\mathrm{pw}}-v_{\mathrm{M}}\|_{h}\lesssim\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\rm
pw}-v_{\mathrm{M}})|_{H^{m}(\mathcal{T})}\lesssim\|(1-\Pi_{0})D^{2}_{\rm
pw}v_{\rm pw}\|_{L^{2}(\Omega)}+j_{h}(v_{\rm pw}).$ (4.14)

Notice that $j_{h}(v_{\mathrm{M}}-Jv_{\mathrm{M}})=0$ implies
$\|v_{\mathrm{M}}-Jv_{\mathrm{M}}\|_{h}=|\\!|\\!|v_{\mathrm{M}}-Jv_{\mathrm{M}}|\\!|\\!|_{\text{pw}}\lesssim|\\!|\\!|v_{\rm\mathrm{M}}-v|\\!|\\!|_{\text{pw}}$
for any $v\in V$ from Lemma 3.2.d in the last step. This and a triangle
inequality, (4.14), and
$|\\!|\\!|v_{\text{pw}}-v|\\!|\\!|_{\text{pw}}+j_{h}(v_{\rm
pw})\leq\sqrt{2}\|v_{\mathrm{pw}}-v\|_{h}$ show

$\displaystyle\|v_{\mathrm{M}}-Jv_{\mathrm{M}}\|_{h}\lesssim|\\!|\\!|v_{\mathrm{pw}}-v|\\!|\\!|_{\text{pw}}+|\\!|\\!|v_{\rm
pw}-v_{\mathrm{M}}|\\!|\\!|_{\text{pw}}\lesssim\|(1-\Pi_{0})D^{2}_{\rm
pw}v_{\rm pw}\|_{L^{2}(\Omega)}+\|v_{\mathrm{pw}}-v\|_{h}.$ (4.15)

This and a triangle inequality
$\|v_{\mathrm{pw}}-JI_{\mathrm{M}}v_{\mathrm{pw}}\|_{h}\leq\|v_{\mathrm{pw}}-v_{\mathrm{M}}\|_{h}+\|v_{\mathrm{M}}-Jv_{\mathrm{M}}\|_{h}$
and (4.14)-(4.15) conclude the proof of (a). ∎

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

Adapt the notation of part (a) and recall that Theorem 4.2.a provides
(4.14.b), the second estimate in (4.14). Since
$v_{\mathrm{M}}-Jv_{\mathrm{M}}=I_{\mathrm{M}}Jv_{\mathrm{M}}-Jv_{\mathrm{M}}$
(from (3.9)), Lemma 3.1 controls this interpolation error of
$Jv_{\mathrm{M}}\in H^{2}_{0}(\Omega)$ and shows

$\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{\mathrm{M}}-Jv_{\mathrm{M}})|_{H^{m}(\mathcal{T})}\leq
2|\\!|\\!|v_{\mathrm{M}}-Jv_{\mathrm{M}}|\\!|\\!|_{\text{pw}}\lesssim\|(1-\Pi_{0})D^{2}_{\rm
pw}v_{\rm pw}\|_{L^{2}(\Omega)}+\|v_{\mathrm{pw}}-v\|_{h}$

with (4.15) in the last step. Theorem 4.1 applies to $v_{\mathrm{pw}}-v\in
H^{2}(\mathcal{T})$. The combination of the resulting estimate with the
previous one concludes the proof of (b). ∎

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

The assertions (a)-(b) apply to $v_{\mathrm{pw}}:=v_{2}\in P_{2}(\mathcal{T})$
and the extra term $\|(1-\Pi_{0})D^{2}_{\rm pw}v_{\rm pw}\|_{L^{2}(\Omega)}$
vanishes. The resulting estimates allow for obvious converse inequalities and
so prove, for $v_{2}\in P_{2}(\mathcal{T})$ and
$v_{\mathrm{M}}:=I_{\mathrm{M}}v_{2}\in\mathrm{M}(\mathcal{T})$, that

$\displaystyle\|v_{2}-Jv_{\mathrm{M}}\|_{h}\approx\min_{v\in
H^{2}_{0}(\Omega)}\|v_{2}-v\|_{h}$
$\displaystyle\quad\lesssim\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{2}-Jv_{\mathrm{M}})|_{H^{m}(\mathcal{T})}\approx\min_{v\in
H^{2}_{0}(\Omega)}\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{2}-v)|_{H^{m}(\mathcal{T})}$

with Theorem 4.1 in between the two equivalences. A triangle inequality, the
estimate (4.14.b), the estimate for $(1-J)v_{\mathrm{M}}$ in the proof of
$(b)$ and (4.1) applies to $v_{\mathrm{pw}}:=v_{2}\in P_{2}(\mathcal{T})$ and
shows
$\sum_{m=0}^{2}|h_{\mathcal{T}}^{m-2}(v_{2}-Jv_{\mathrm{M}})|_{H^{m}(\mathcal{T})}\lesssim
j_{h}(v_{2})+\|v_{2}-v\|_{h}=j_{h}(v_{2}-v)+\|v_{2}-v\|_{h}\leq
2\|v_{2}-v\|_{h}$ for any $v\in H^{2}_{0}(\Omega)$. This concludes the proof
of (c). ∎

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

The equivalence of the Sobolev-Slobodeckii norm and the norm by interpolation
of Sobolev spaces [35, Remark 9.1], for instance for a fixed reference
triangle $T=T_{\rm ref}$ with $C_{2}{}(s)=C_{\ref{ccint3}}(s,T_{\rm ref})$,
provides for ${w}:=(v_{2}-JI_{\mathrm{M}}v_{2})|_{T}\in H^{2}(T)$ the estimate

$\|{w}\|_{H^{s}(T)}\leq
C_{\ref{ccint3}}(s)\,\|{w}\|_{H^{1}(T)}^{2-s}\|{w}\|_{H^{2}(T)}^{s-1}\quad\text{for
}1<s<2.$ (4.16)

A straightforward transformation of Sobolev norms [26, Theorem 3.1.2] show
(4.16) 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 [20] reveals that $C_{\ref{ccint3}}(s)$ exclusively depends on $s$
(but exploits singularities as $s$ approaches the end-points $0$ and $1$). The
estimate (4.16) shows the first inequality in

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

with the subadditivity $(a+b)^{p}\leq a^{p}+b^{p}$ for $a,b\geq 0$ and
$0<p=s-1<1$ (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 inequality
$ab\leq a^{p}/p+b^{q}/q$ for $p=(2-s)^{-1}$, $q=(s-1)^{-1}$,
$a=\|h_{T}^{-1}{w}\|_{H^{1}(T)}^{2(2-s)}$, and $b=|{w}|_{H^{2}(T)}^{2(s-1)}$
to prove

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

This and the trivial estimate ${h_{\max}^{2(s-1)}\leq\text{\rm
diam}(\Omega)^{2(s-1)}}$ leads to

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

for $C_{3}{}=C_{\ref{ccint3}}(s)^{2}(1+\text{\rm diam}(\Omega)^{2(s-1)})$. The
sum over all those contributions over $T\in\mathcal{T}$ proves

$\displaystyle\|v_{2}-JI_{\mathrm{M}}v_{2}\|_{H^{s}(\mathcal{T})}$
$\displaystyle\leq
C_{\ref{ccint4}}^{1/2}h_{\max}^{2-s}\left(\|h_{\mathcal{T}}^{-1}(v_{2}-JI_{\mathrm{M}}v_{2})\|_{H^{1}(\mathcal{T})}+|\\!|\\!|v_{2}-JI_{\mathrm{M}}v_{2}|\\!|\\!|_{\mathrm{pw}}\right)$
$\displaystyle\lesssim h_{\max}^{2-s}\min_{v\in V}\|v_{2}-v\|_{h}\ $

with Theorem 4.3.c in the last step. This concludes the proof of (d) for
$1<s<2$. The assertion (d) is included in Theorem 4.3.c for $s=0,1,2$. The
remaining case $0<s<1$ is similar to the above analysis with
$\|{w}\|_{H^{s}(T)}\leq
C_{\ref{ccint3}}(s)\|{w}\|_{L^{2}(T)}^{1-s}\|{w}\|_{H^{1}(T)}^{s}$ replacing
(4.16) and analogous arguments; hence further details are omitted. ∎

##  5 Abstract framework for best-approximation of lower-order methods

###  5.1 Discretisation

Suppose that $V_{h}\subset H^{2}(\mathcal{T})$ is the finite-dimensional trial
and test space of an abstract (discontinuous Galerkin) scheme with a bilinear
form

$A_{h}:(V_{h}+\mathrm{M}(\mathcal{T}))\times(V_{h}+\mathrm{M}(\mathcal{T}))\rightarrow{\mathbb{R}}$

that is coercive and continuous with respect to some norm $\|\bullet\|_{h}$ in
$H^{2}(\mathcal{T})$ in the sense that, for all  $v_{h},w_{h}\in V_{h}$,

$\alpha\|v_{h}\|_{h}^{2}\leq A_{h}(v_{h},v_{h})\text{ and
}A_{h}(v_{h},w_{h})\leq M\|v_{h}\|_{h}\|w_{h}\|_{h}$ (5.1)

hold for some universal constants $0<\alpha,M<\infty$. Suppose that
$\|\bullet\|_{h}$ is a norm in $H^{2}(\mathcal{T})$ and equal to the norm
$|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}:=a_{\mathrm{pw}}(\bullet,\bullet)^{1/2}$
in $V+\mathrm{M}(\mathcal{T})$ and stronger in general, i.e.,

$\text{ (a) }|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}\leq\|\bullet\|_{h}\text{
in }H^{2}(\mathcal{T})\quad\text{and}\quad\text{ (b)
}|\\!|\\!|\bullet|\\!|\\!|_{\mathrm{pw}}=\|\bullet\|_{h}\text{ in
}V+\mathrm{M}(\mathcal{T}).$ (5.2)

Given a linear operator $JI_{\mathrm{M}}:V_{h}\to V$ with the companion
operator $J$ from Lemma 3.2 and the (extended) linear interpolation operator
$I_{\mathrm{M}}$ from Subsection 3.3 the discrete problem reads: Given $F\in
V^{*}=H^{-2}(\Omega)$ seek the discrete solution $u_{h}\in V_{h}$ to

$A_{h}(u_{h},v_{h})=F(JI_{\mathrm{M}}v_{h})\quad\text{for all }v_{h}\in
V_{h}.$ (5.3)

The Lax-Milgram lemma assures the existence of a unique discrete solution
$u_{h}$ to (5.3).

######  Remark 5.1 ($\|\bullet\|_{h}$).

The examples of Sections 7 and 8 utilize $\|\bullet\|_{h}$ given in
(4.1)-(4.2), but the abstract framework allows more general $A_{h}$ and
$\|\bullet\|_{h}$ with (5.1)-(5.2) in (5.3).

###  5.2 First glance at the analysis

This subsection motivates the abstract conditions and emphasises the relevance
of the discrete consistency condition (dcc)

$\displaystyle
a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+b_{h}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})$
$\displaystyle\leq\Lambda_{\rm
dc}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}$ (5.4)

that leads to the best-approximation in terms of
$|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}=\min_{v_{2}\in
P_{2}(T)}|\\!|\\!|u-v_{2}|\\!|\\!|_{\mathrm{pw}}$ from (3.7). The test
function $e_{h}:=I_{h}I_{\mathrm{M}}u-u_{h}\in V_{h}\subset
H^{2}(\mathcal{T})$ is the discrete approximation of the error $u-u_{h}$ with
$I_{\mathrm{M}}:H^{2}(\mathcal{T})\rightarrow\mathrm{M}(\mathcal{T})$ from
Definition 3.2 and a transfer operator
$I_{h}:\mathrm{M}(\mathcal{T})\rightarrow V_{h}$ from Subsection 5.3 below.
For the dGFEM of Section 7 and the WOPSIP scheme of Section 10, $I_{h}$ is the
identity $1$ and otherwise it is controlled nicely (cf. (5.11) below for
details) [13]. So we may neglect the difference $1-I_{h}$ for the sake of this
first look at the analysis and suppose $I_{h}=1$. The key identity from the
continuous problem (3.1) and the discrete one (5.3) reads

$a(u,JI_{\mathrm{M}}e_{h})=F(JI_{\mathrm{M}}e_{h})=A_{h}(u_{h},e_{h}).$ (5.5)

The stability of the scheme $\alpha\|e_{h}\|^{2}_{h}\leq
A_{h}(e_{h},e_{h})\leq M\|e_{h}\|_{h}^{2}$ motivates the investigation of

$\displaystyle A_{h}(e_{h},e_{h})$
$\displaystyle=a_{\mathrm{pw}}(e_{h},e_{h})+b_{h}(e_{h},e_{h})+c_{h}(e_{h},e_{h})$
(5.6)

for the three bilinear forms that define the class of problems in (5.12)
displayed in Table LABEL:table:summary. The stability term
$c_{h}(\bullet,\bullet)$ is controlled nicely in harmony with the discrete
norm $\|\bullet\|_{h}$, while $b_{h}(\bullet,\bullet)$ drives the method and
completes the leading term $a_{\mathrm{pw}}(\bullet,\bullet).$

The definition of $e_{h}$ in (5.6) leads to

$\displaystyle A_{h}(e_{h},e_{h})$
$\displaystyle=a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h})-A_{h}(u_{h},e_{h})+b_{h}(I_{\mathrm{M}}u,e_{h})+c_{h}(I_{\mathrm{M}}u,e_{h})$
(5.7)

Since $J$ is a the right-inverse of $I_{\mathrm{M}}$, (3.5) implies
$a_{\mathrm{pw}}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h}-JI_{\mathrm{M}}e_{h})~{}=~{}0$.
This and elementary algebra show

$\displaystyle a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h})$
$\displaystyle=a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+a_{\mathrm{pw}}(I_{\mathrm{M}}u,JI_{\mathrm{M}}e_{h}).$

This in combination with (5.5) leads in (5.7) to

$\displaystyle A_{h}(e_{h},e_{h})$
$\displaystyle=a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+a_{\mathrm{pw}}(I_{\mathrm{M}}u-u,JI_{\mathrm{M}}e_{h})$
$\displaystyle\quad+b_{h}(I_{\mathrm{M}}u,e_{h})+c_{h}(I_{\mathrm{M}}u,e_{h}).$
(5.8)

The second term in the right-hand side of (5.2) is equal to
$a_{\mathrm{pw}}(I_{\mathrm{M}}u-u,JI_{\mathrm{M}}e_{h}-I_{\mathrm{M}}e_{h})$
and the stabilisation term is equal to $c_{h}(I_{\mathrm{M}}u-u,e_{h})$. They
are controlled by
$|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}$. The bilinear
form $b_{h}$ enjoys the miraculous property
$b_{h}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h})=0$ for the discontinuous Galerkin
schemes of this paper. The remaining term on the right-hand side of (5.2) is
$a_{\mathrm{pw}}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+b_{h}(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})$
and in fact controlled by the dcc (5.4). The proof of dcc in Section 8 is one
key argument in this paper.

######  Remark 5.2.

The arguments in this section applies to the case where
$A_{h}(\bullet,\bullet)$ satisfies an inf-sup condition; (and not the
coercivity condition). The key idea is to estimate the consistency error
$F(JI_{\mathrm{M}}v_{h})-A_{h}(I_{\mathrm{M}}u,v_{h})$ using (5.5)-(5.6) and
the orthogonality of the interpolation operator.

###  5.3 Transfer operators between $V_{h}$ and $\mathrm{M}(\mathcal{T})$

Recall $I_{\mathrm{M}}:H^{2}(\mathcal{T})\rightarrow\mathrm{M}(\mathcal{T})$
from Definition 3.2, and suppose the existence of some constant
$\Lambda_{\mathrm{M}}\geq 0$ with

$\displaystyle\|v_{h}-I_{\mathrm{M}}v_{h}\|_{h}$
$\displaystyle\leq\Lambda_{\mathrm{M}}\|v_{h}-v\|_{h}\qquad\text{for all
}v_{h}\in V_{h}\text{ and all }v\in V.$ (5.9)

Suppose the existence of constants
$\Lambda_{\mathrm{M}}^{\prime},M_{\mathrm{M}}\geq 0$, and boundedness in the
sense that

$\displaystyle\|v_{h}-I_{\mathrm{M}}v_{h}\|_{h}$
$\displaystyle\leq\Lambda_{\mathrm{M}}^{\prime}\|v_{h}\|_{h}\qquad\text{for
all }v_{h}\in V_{h},$
$\displaystyle|\\!|\\!|I_{\mathrm{M}}v_{h}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq M_{\mathrm{M}}\|v_{h}\|_{h}\;\quad\text{ for all }v_{h}\in
V_{h}.$ (5.10)

Apparently $\Lambda_{\mathrm{M}}^{\prime}\leq\Lambda_{\mathrm{M}}$ (with
$v=0$) and $M_{\mathrm{M}}\leq 1+\Lambda_{\mathrm{M}}^{\prime}$ (with (5.2.a)
and a triangle inequality). The possibly smaller constant $M_{\mathrm{M}}$
enters in Theorem 5.1.a., while $\Lambda_{\mathrm{M}}$ appears in Theorem
5.1.b. and Theorem 6.1 below. The above conditions control the transfer from
$V_{h}$ into $\mathrm{M}(\mathcal{T})$ via
$I_{\mathrm{M}}:V+V_{h}+\mathrm{M}(\mathcal{T})\to\mathrm{M}(\mathcal{T})$.

The transfer from $\mathrm{M}(\mathcal{T})$ into $V_{h}$ is modeled by some
linear map $I_{h}:\mathrm{M}(\mathcal{T})\rightarrow V_{h}$ that is bounded in
the sense that there exists some constant $\Lambda_{h}>0$ such that

$\|v_{\mathrm{M}}-I_{h}v_{\mathrm{M}}\|_{h}\leq\Lambda_{h}|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}\quad\text{for
all }v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})\text{ and for all }v\in V.$
(5.11)

The examples of this paper concern the discrete norm from (4.1)-(4.2) and then
the estimates of this subsection follow for piecewise quadratic discrete
spaces.

######  Example 5.1 ((5.9)-(5.10) hold for $V_{h}\subseteq P_{2}(\mathcal{T})$
and (4.1)-(4.2)).

Suppose that the discrete norm $\|\bullet\|_{h}$ is defined by (4.1)-(4.2) and
$V_{h}\subseteq P_{2}(\mathcal{T})$. Then (5.2) and (5.9)-(5.10) follow.

###### Proof of (5.9)-(5.10).

Given any $v_{2}\in P_{2}(\mathcal{T})$ and any $v\in V$, a triangle
inequality shows

$\|v_{2}-I_{\mathrm{M}}v_{2}\|_{h}\leq\|v_{2}-JI_{\mathrm{M}}v_{2}\|_{h}+\|I_{\mathrm{M}}v_{2}-JI_{\mathrm{M}}v_{2}\|_{h}=:t_{1}+t_{2}.$

Theorem 4.3.c controls the first term
$t_{1}:=\|v_{2}-JI_{\mathrm{M}}v_{2}\|_{h}\lesssim\|v_{2}-v\|_{h}$ on the
right-hand side. Since $j_{h}(v_{\mathrm{M}})=0$ in (4.2) vanishes for
$v_{\mathrm{M}}:=I_{\mathrm{M}}v_{2}\in\mathrm{M}(\mathcal{T})$, the second
term
$t_{2}:=|\\!|\\!|v_{\mathrm{M}}-Jv_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\lesssim|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}$
with Lemma 3.2.d in the last step. Since
$\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}v_{2}\|=0$ vanishes for $v_{2}\in
P_{2}(\mathcal{T})$, Theorem 4.2.a shows
$|\\!|\\!|v_{\mathrm{M}}-v_{2}|\\!|\\!|_{\mathrm{pw}}\lesssim
j_{h}(v_{2})=j_{h}(v_{2}-v)\leq\|v_{2}-v\|_{h}$ with (4.1)-(4.2) in the last
two steps. This and a triangle inequality prove
$t_{2}\lesssim\|v_{2}-v\|_{h}$. The combination of the estimates for
$t_{1}+t_{2}\lesssim\|v_{2}-v\|_{h}$ proves (5.9); and (5.9) immediately
implies (5.10) as discussed above. ∎

###  5.4 Sufficient conditions for best-approximation

The bilinear forms
$A_{h},a_{\mathrm{pw}},b_{h},c_{h}:(V_{h}+\mathrm{M}(\mathcal{T}))\times(V_{h}+\mathrm{M}(\mathcal{T}))\rightarrow{\mathbb{R}}$
in the discrete problem (5.3), (all bounded because
$V_{h}+\mathrm{M}(\mathcal{T})$ is finite dimensional) read

$\displaystyle
A_{h}(\widehat{v},\widehat{w}):=a_{\mathrm{pw}}(\widehat{v},\widehat{w})+b_{h}(\widehat{v},\widehat{w})+c_{h}(\widehat{v},\widehat{w})\quad\text{for
all }\widehat{v},\widehat{w}\in V_{h}+\mathrm{M}(\mathcal{T}).$ (5.12)

The key assumption in abstract form is the discrete consistency condition with
a constant $0<\Lambda_{\mathrm{dc}}<\infty$: All functions
$v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$, $w_{h}\in V_{h}$, and all $v,w\in
V$ satisfy

$a_{\mathrm{pw}}(v_{\mathrm{M}},w_{h}-I_{\mathrm{M}}w_{h})+b_{h}(v_{\mathrm{M}},w_{h}-I_{\mathrm{M}}w_{h})\leq\Lambda_{\mathrm{dc}}|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}\|w_{h}-w\|_{h}.$
(5.13)

(This is a straightforward generalization of (5.4) from Subsection 5.2.)
Assume that
$b_{h}:(V_{h}+\mathrm{M}(\mathcal{T}))\times(V_{h}+\mathrm{M}(\mathcal{T}))\to\mathbb{R}$
is bounded in $V_{h}+\mathrm{M}(\mathcal{T})$ by a constant $0<M_{\rm
b}<\infty$ and vanishes in
$\mathrm{M}(\mathcal{T})\times\mathrm{M}(\mathcal{T})$ in the sense that all
$v_{h},w_{h}\in V_{h}$ and all
$v_{\mathrm{M}},w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ satisfy

$\displaystyle b_{h}(v_{\mathrm{M}},w_{\mathrm{M}})$ $\displaystyle=0,$ (5.14)
$\displaystyle b_{h}(v_{h}+v_{\mathrm{M}},w_{h}+w_{\mathrm{M}})$
$\displaystyle\leq M_{\rm
b}\|v_{h}+v_{\mathrm{M}}\|_{h}\|w_{h}+w_{\mathrm{M}}\|_{h}.$ (5.15)

Suppose that the bilinear form
$c_{h}:(V_{h}+\mathrm{M}(\mathcal{T}))\times(V_{h}+\mathrm{M}(\mathcal{T}))\to\mathbb{R}$
and a constant $0<\Lambda_{\rm c}<\infty$ satisfy

$c_{h}(v_{h},w_{h})\leq\Lambda_{\rm
c}\|v-v_{h}\|_{h}\|w-w_{h}\|_{h}\quad\text{for all }v_{h},w_{h}\in V_{h}\text{
and }v,w\in V.$ (5.16)

######  Theorem 5.1 (best-approximation).

Suppose (5.1)-(5.2) and (5.9)- (5.16). Let $u\in V$ solve (3.1) and let
$u_{h}\in V_{h}$ solve (5.3). Then

$(a)\;\|u-u_{h}\|_{h}\leq
C_{\mathrm{qo}}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\text{ and
}(b)\;|\\!|\\!|u-JI_{\mathrm{M}}u_{h}|\\!|\\!|\leq(1+\Lambda_{\mathrm{M}})(1+\Lambda_{\rm
J})\|u-u_{h}\|_{h}$

hold with the constant
$C_{\mathrm{qo}}:=1+\Lambda_{h}+\alpha^{-1}(\Lambda_{0}M_{\mathrm{M}}+\Lambda_{h}(1+M_{\rm
b})+\Lambda_{\mathrm{dc}}+\Lambda_{\rm c}(1+\Lambda_{h}))$.

######  Remark 5.3.

The Morley FEM is included in the (non-symmetric) abstract framework of
Theorem 5.1 and leads to a sub-optimal best-approximation constant $C_{\rm
qo}=1+\Lambda_{0}$.

The error analysis of a post-processing dates back at least to [6] with a
design of an enrichment operator for C0IP functions replaced here by the
smoother $JI_{\mathrm{M}}$. For $F\in H^{-s}(\Omega)$ with $2-\sigma\leq s\leq
2$ (and $u\in H^{4-s}(\Omega)$ from elliptic regularity), Theorem 5.1 verifies

$\displaystyle|\\!|\\!|u-JI_{\mathrm{M}}u_{h}|\\!|\\!|\leq C_{\rm
qo}(1+\Lambda_{\mathrm{M}})(1+\Lambda_{\rm
J})|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\lesssim
h_{\max}^{2-s}\|F\|_{H^{-s}(\Omega)}.$

###  5.5 Proofs

Abbreviate $e_{h}:=I_{h}I_{\mathrm{M}}u-u_{h}\in V_{h}$ and
$(a_{\mathrm{pw}}+b_{h})(\bullet,\bullet):=a_{\mathrm{pw}}(\bullet,\bullet)+b_{h}(\bullet,\bullet)$
for the sum of the bilinear forms.

######  Lemma 5.2 (Key identity).

It holds

$\displaystyle A_{h}(e_{h},e_{h})$
$\displaystyle=a_{\mathrm{pw}}(u,(1-J)I_{\mathrm{M}}e_{h})+(a_{\mathrm{pw}}+b_{h})((I_{h}-1)I_{\mathrm{M}}u,e_{h})$
$\displaystyle\quad+(a_{\mathrm{pw}}+b_{h})(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+c_{h}(I_{h}I_{\mathrm{M}}u,e_{h}).$

###### Proof.

The test function $v:=JI_{\mathrm{M}}e_{h}\in V$ in (3.1) and the test
function $v_{h}:=e_{h}\in V_{h}$ in (5.3) lead to

$a(u,JI_{\mathrm{M}}e_{h})=F(JI_{\mathrm{M}}e_{h})=A_{h}(u_{h},e_{h}).$

This and the definition $e_{h}=I_{h}I_{\mathrm{M}}u-u_{h}$ result in

$A_{h}(e_{h},e_{h})=A_{h}(I_{h}I_{\mathrm{M}}u,e_{h})-a(u,JI_{\mathrm{M}}e_{h}).$

The identity
$a_{\mathrm{pw}}(u,I_{\mathrm{M}}e_{h})=a_{\mathrm{pw}}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h})$
from (3.5) shows that this is equal to

$a_{\mathrm{pw}}(u,(1-J)I_{\mathrm{M}}e_{h})+A_{h}(I_{h}I_{\mathrm{M}}u,e_{h})-a_{\mathrm{pw}}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h}).$

The last term $a_{\mathrm{pw}}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h})$ is part
of $A_{h}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h})$ by (5.12), while
$b_{h}(I_{\mathrm{M}}u,I_{\mathrm{M}}e_{h})=0$ owing to (5.14). This and
elementary algebra conclude the proof. ∎

######  Lemma 5.3.

The assumptions (3.5), (3.9), (3.11), and (5.10) imply

$a_{\mathrm{pw}}(u,(1-J)I_{\mathrm{M}}e_{h})\leq\Lambda_{0}M_{\mathrm{M}}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}.$

###### Proof.

Set $v_{\mathrm{M}}:=I_{\mathrm{M}}e_{h}$ and recall
$I_{\mathrm{M}}(v_{\mathrm{M}}-Jv_{\mathrm{M}})=0$ from (3.9). The
orthogonality (3.5) and the Cauchy inequality with respect to
$a_{\mathrm{pw}}(\bullet,\bullet)$ imply

$a_{\mathrm{pw}}(u,(1-J)v_{\mathrm{M}})=a_{\mathrm{pw}}(u-I_{\mathrm{M}}u,(1-J)v_{\mathrm{M}})\leq|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}|\\!|\\!|(1-J)v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}.$

The boundedness of $1-J$ in (3.11) and the boundedness of $I_{\mathrm{M}}$ in
(5.10) show

$|\\!|\\!|(1-J)v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}|\\!|\\!|v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{0}M_{\mathrm{M}}\|e_{h}\|_{h}.$

The combination of the two displayed inequalities concludes the proof. ∎

######  Lemma 5.4.

The assumptions (5.2), (5.11), and (5.15) imply

$(a_{\mathrm{pw}}+b_{h})((I_{h}-1)I_{\mathrm{M}}u,e_{h})\leq\Lambda_{h}(1+M_{\rm
b})|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}.$

###### Proof.

For $e_{h}\in V_{h}$ and $I_{\mathrm{M}}u\in\mathrm{M}(\mathcal{T})$, the
Cauchy inequality plus (5.2.a) show

$a_{\mathrm{pw}}((I_{h}-1)I_{\mathrm{M}}u,e_{h})\leq\|(1-I_{h})I_{\mathrm{M}}u\|_{h}\|e_{h}\|_{h}.$

This and the boundedness of $b_{h}$ in (5.15) result in

$\displaystyle(a_{\mathrm{pw}}+b_{h})((I_{h}-1)I_{\mathrm{M}}u,e_{h})\leq(1+M_{\rm
b})\|(1-I_{h})I_{\mathrm{M}}u\|_{h}\|e_{h}\|_{h}.$

The inequality (5.11) with $v_{\mathrm{M}}=I_{\mathrm{M}}u$ and $v=u$ reads

$\|(1-I_{h})I_{\mathrm{M}}u\|_{h}\leq\Lambda_{h}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}.$

The combination of the last two displayed inequalities concludes the proof. ∎

######  Lemma 5.5.

The assumptions (5.2), (5.11), (5.13), and (5.16) imply

$(a_{\mathrm{pw}}+b_{h})(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})+c_{h}(I_{h}I_{\mathrm{M}}u,e_{h})\leq\left(\Lambda_{\mathrm{dc}}+\Lambda_{\rm
c}(1+\Lambda_{h})\right)|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}.$

###### Proof.

The discrete consistency condition (5.13) for
$v_{\mathrm{M}}:=I_{\mathrm{M}}u$, $w_{h}:=e_{h}$, $v:=u$, and $w:=0$ lead to
the upper bound

$(a_{\mathrm{pw}}+b_{h})(I_{\mathrm{M}}u,e_{h}-I_{\mathrm{M}}e_{h})\leq\Lambda_{\mathrm{dc}}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}\|e_{h}\|_{h}$

for the first term on the left-hand side of the asserted estimate. The
remaining contribution $c_{h}(I_{h}I_{\mathrm{M}}u,e_{h})$ is controlled in
(5.16) with $v_{h}:=I_{h}I_{\mathrm{M}}u$, $w_{h}:=e_{h}$, $v=u$, and $w=0$ by

$c_{h}(I_{h}I_{\mathrm{M}}u,e_{h})\leq\Lambda_{\rm
c}\|u-I_{h}I_{\mathrm{M}}u\|_{h}\|e_{h}\|_{h}.$

A triangle inequality in $\|\bullet\|_{h}$, (5.11) with
$v_{\mathrm{M}}:=I_{\mathrm{M}}u$ and $v=u$, and (5.2.b) show

$\|u-I_{h}I_{\mathrm{M}}u\|_{h}\leq(1+\Lambda_{h})|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}.$

The combination of the resulting inequalities concludes the proof. ∎

###### Proof of best-approximation in Theorem 5.1.a.

The discrete ellipticity (5.1) is followed by Lemma 5.2 with terms controlled
in Lemmas 5.3-5.5. This leads (after a division by $\|e_{h}\|_{h}$, if
positive) to

$\alpha\|e_{h}\|_{h}\leq\left(\Lambda_{0}M_{\mathrm{M}}+\Lambda_{h}(1+M_{\rm
b})+\Lambda_{\mathrm{dc}}+\Lambda_{\rm
c}(1+\Lambda_{h})\right)|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}.$

On the other hand,
$\|(I_{h}-1)I_{\mathrm{M}}u\|_{h}\leq\Lambda_{h}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}$
from (5.11) for $v_{\mathrm{M}}:=I_{\mathrm{M}}u$ and $v:=u$. Triangle
inequalities in $\|\bullet\|_{h}$, (5.2.b), and the last two inequalities
result in

$\displaystyle\|u-u_{h}\|_{h}$
$\displaystyle\leq|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}+\|I_{\mathrm{M}}u-I_{h}I_{\mathrm{M}}u\|_{h}+\|e_{h}\|_{h}\leq
C_{\mathrm{qo}}|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}$

with the constant $C_{\mathrm{qo}}$ displayed in the assertion. ∎

Proof for post-processing in Theorem 5.1.b. The assertion (b) is formulated in
terms of $u$ and $u_{h}$ but holds for general $v_{h}\in V_{h}$ and $v\in V$
and the abbreviation $v_{\mathrm{M}}:=I_{\mathrm{M}}v_{h}\in M(\mathcal{T})$.
A triangle inequality and Lemma 3.2.d prove

$|\\!|\\!|v-Jv_{\mathrm{M}}|\\!|\\!|\leq|\\!|\\!|v-v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}+|\\!|\\!|(1-J)v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq(1+\Lambda_{\rm
J})|\\!|\\!|v-v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}.$

A triangle inequality, (5.2.a) twice, and (5.9) show

$|\\!|\\!|v-v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq|\\!|\\!|v-v_{h}|\\!|\\!|_{\mathrm{pw}}+\|v_{h}-v_{\mathrm{M}}\|_{h}\leq(1+\Lambda_{\mathrm{M}})\|v-v_{h}\|_{h}.$

The combination of the two displayed estimates reads

$|\\!|\\!|v-Jv_{\mathrm{M}}|\\!|\\!|\leq(1+\Lambda_{\rm
J})(1+\Lambda_{\mathrm{M}})\|v-v_{h}\|_{h}.\qquad\qquad\qquad\qed$

##  6 Weaker and piecewise Sobolev norm error estimates

###  6.1 Assumptions and result

This subsection presents one further condition sufficient for a lower-order a
priori error estimate for the discrete problem (5.3) beyond the hypotheses of
Subsections 5.1-5.4: The dual discrete consistency with a constant
$0\leq\Lambda_{\rm ddc}<\infty$ asserts that any $v_{h}\in V_{h}$,
$w_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$, and any $v,w\in V$ satisfy

$a_{\mathrm{pw}}(v_{h}-I_{\mathrm{M}}v_{h},w_{\mathrm{M}})+b_{h}(v_{h},w_{\mathrm{M}})\leq\Lambda_{\rm
ddc}\|v-v_{h}\|_{h}|\\!|\\!|w-w_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}.$ (6.1)

######  Remark 6.1 (symmetry).

If the bilinear form $(a_{\mathrm{pw}}+b_{h})(\bullet,\bullet)$ is symmetric,
then (5.13)-(5.14) imply (6.1). ( Rewrite the left-hand side in (6.1) with
(5.14) and symmetry into the left-hand side of (5.13) with $v_{h}$ and
$w_{\mathrm{M}}$ replacing $w_{h}$ and $v_{\mathrm{M}}$ to establish (6.1).)

Since $u_{h}\in V_{h}$ may not belong to $H^{s}(\Omega)$ for $2-\sigma\leq
s\leq 2$ in general, the post-processing $JI_{\mathrm{M}}u_{h}\in V$ arises in
the duality argument with $0<\sigma\leq 1$ from Example 3.1.

######  Theorem 6.1 (lower-order error estimates).

Suppose the assumptions of Theorem 5.1, (6.1), and $2-\sigma\leq s\leq 2$.
Then there exist constants $C_{4}{}(s),C_{5}{}(s)>0$ such that (a)-(b) hold
for any $F\in H^{-s}(\Omega)$ with solution $u\in V$ to (3.1) and the solution
$u_{h}\in V_{h}$ to (5.3). (a) $\|u-JI_{\mathrm{M}}u_{h}\|_{H^{s}(\Omega)}\leq
C_{\ref{2}}(s)h_{\max}^{2-s}\|u-u_{h}\|_{h}$ and (b), if $u_{h}\in
P_{2}(\mathcal{T})$, then $\|u-u_{h}\|_{{H^{s}}(\mathcal{T})}\leq
C_{\ref{3}}(s)h_{\max}^{2-s}\|u-u_{h}\|_{h}$.

###  6.2 Duality and algebra

The duality of $H^{-s}(\Omega)$ and $H^{s}_{0}(\Omega)$ reveals for the exact
solution $u\in V$ to (3.1) and the post-processing $v:=JI_{\mathrm{M}}u_{h}\in
V$ of the discrete solution $u_{h}\in V_{h}$ to (5.3) that

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

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 $a(z,\bullet)=G\in V^{*}$ has a unique Riesz representation
$z\in V$ in the Hilbert space $(V,a)$; $z\in V$ is the weak solution to the
PDE $\Delta^{2}z=G$. The elliptic regularity (as in Example 3.1) leads to
$z\in V\cap H^{4-s}(\Omega)$ with $2\leq 4-s\leq 2+\sigma$ and (3.2); hence

$\|u-v\|_{H^{s}(\Omega)}=a(u-v,z)\quad\text{and}\quad\|z\|_{H^{4-s}(\Omega)}\leq
C_{\rm reg}.$

The proof of Theorem 6.1 consists of a series of lemmas to establish an upper
bound of $a(u-v,z)$ for the above $z\in V\cap H^{4-s}(\Omega)$. The notation

$v:=JI_{\mathrm{M}}u_{h}\in V,\qquad z_{h}:=I_{h}I_{\mathrm{M}}z\in
V_{h}\quad\text{and}\quad\zeta:=JI_{\mathrm{M}}z_{h}\in V$

for the discrete, exact, and dual solution $u_{h},u,$ and $z$ applies
throughout this section.

######  Lemma 6.2 (Key identity).

It holds

$\displaystyle a(u-v,z)=a(u-v,z-\zeta)+a_{\mathrm{pw}}(u_{h}-v,\zeta-
z_{h})+a_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-v,z_{h}-I_{\mathrm{M}}z_{h})$
$\displaystyle\quad+a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h}-\zeta)+A_{h}(u_{h},z_{h})-a_{\mathrm{pw}}(u_{h},I_{\mathrm{M}}z_{h})+a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},z_{h}).$

###### Proof.

Let $\zeta\equiv JI_{\mathrm{M}}z_{h}\in V$ substitute the test function $v$
in (3.1). This and the test function $v_{h}:=z_{h}$ in (5.3) lead to

$a(u,\zeta)=F(\zeta)=A_{h}(u_{h},z_{h}).$

This identity and elementary algebra result in

$\displaystyle a(u-v,\zeta)$ $\displaystyle=a_{\mathrm{pw}}(u_{h}-v,\zeta-
z_{h})+a_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-v,z_{h})+a_{\mathrm{pw}}(u_{h},(1-J)I_{\mathrm{M}}z_{h})$
$\displaystyle\quad+A_{h}(u_{h},z_{h})-a_{\mathrm{pw}}(u_{h},I_{\mathrm{M}}z_{h})+a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},z_{h}).$

The formulas
$a_{\mathrm{pw}}((1-J)I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h})=0=a_{\mathrm{pw}}(I_{\mathrm{M}}u_{h},(1-J)I_{\mathrm{M}}z_{h})$
(from (3.5) and (3.9)) and elementary algebra conclude the proof. ∎

###  6.3 Elementary bounds

######  Lemma 6.3.

Each of the following terms (a) $a(u-v,z-\zeta)$, (b)
$a_{\mathrm{pw}}(u_{h}-v,\zeta-z_{h})$, (c)
$a_{\mathrm{pw}}(I_{\mathrm{M}}u_{h}-v,z_{h}-I_{\mathrm{M}}z_{h})$, and (d)
$a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h}-\zeta)$ is
bounded in modulus by a constant
$\leq{\left(1+(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}})\right)^{2}}(1+\Lambda_{h})$
times $\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}$.

###### Proof.

The assumption (5.9) (with $(v,v_{h})$ replaced by $(u,u_{h})$ and
$(z,z_{h})$) implies

$\displaystyle\|u_{h}-I_{\mathrm{M}}u_{h}\|_{h}$
$\displaystyle\leq\Lambda_{\mathrm{M}}\|u-u_{h}\|_{h}\quad\text{and}\quad\|z_{h}-I_{\mathrm{M}}z_{h}\|_{h}\leq\Lambda_{\mathrm{M}}\|z-z_{h}\|_{h}.$
(6.2)

Recall $v\equiv JI_{\mathrm{M}}u_{h}$ and $\zeta\equiv JI_{\mathrm{M}}z_{h}\in
V$ to deduce from Lemma 3.2.d that

$\displaystyle|\\!|\\!|v-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq\Lambda_{\mathrm{J}}|\\!|\\!|u-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}\quad\text{and}\quad|\\!|\\!|\zeta-
I_{\mathrm{M}}z_{h}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|z-I_{\mathrm{M}}z_{h}|\\!|\\!|_{\mathrm{pw}}.$
(6.3)

The combination of (6.2)-(6.3) with (5.2.a) and triangle inequalities lead to

$\displaystyle|\\!|\\!|u-v|\\!|\\!|$
$\displaystyle\leq|\\!|\\!|u-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}+|\\!|\\!|v-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}\leq(1+\Lambda_{\mathrm{J}})|\\!|\\!|u-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}$
$\displaystyle\leq(1+\Lambda_{\mathrm{J}})(|\\!|\\!|u-u_{h}|\\!|\\!|_{\mathrm{pw}}+\|u_{h}-I_{\mathrm{M}}u_{h}\|_{h})\leq(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}})\|u-u_{h}\|_{h}.$
(6.4)

The above arguments have not utilized any solution property and hence also
apply for $(z,\zeta,z_{h})$ replacing $(u,v,u_{h})$ to reveal (instead of
(6.3))

$\displaystyle|\\!|\\!|z-\zeta|\\!|\\!|$
$\displaystyle\leq(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}})\|z-z_{h}\|_{h}.$
(6.5)

Consider $v_{\mathrm{M}}:=I_{\mathrm{M}}z\in\mathrm{M}(\mathcal{T})$ with
$z_{h}\equiv I_{h}I_{\mathrm{M}}z=I_{h}v_{\mathrm{M}}$ in (5.11) to show

$\|z_{h}-I_{\mathrm{M}}z\|_{h}\leq\Lambda_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$
(6.6)

This, a triangle inequality, and (5.2.b) result in

$\displaystyle\|z-z_{h}\|_{h}\leq(1+\Lambda_{h})|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$
(6.7)

The combination of (6.5) and (6.7) proves

$\displaystyle|\\!|\\!|z-\zeta|\\!|\\!|\leq(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}})(1+\Lambda_{h})|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$
(6.8)

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

This follows from a Cauchy inequality plus (6.3) and (6.8). ∎

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

A triangle inequality, (5.2.a), and (6.3) verify

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

The triangle inequality with (5.2.a) and (6.7)-(6.8) show

$|\\!|\\!|\zeta-z_{h}|\\!|\\!|_{\mathrm{pw}}\leq\|\zeta-
z_{h}\|_{h}\leq(1+\Lambda_{h})\left(1+(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}})\right)|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

A Cauchy inequality and the preceding estimates conclude the proof of (b). ∎

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

The estimate (6.3), a triangle inequality, (5.2.a), and (6.2) show

$|\\!|\\!|v-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\rm
J}|\\!|\\!|u-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}(1+\Lambda_{\mathrm{M}})\|u-u_{h}\|_{h}.$
(6.9)

The combination of (6.2) and (6.7) after (5.2.a) leads to

$\displaystyle|\\!|\\!|z_{h}-I_{\mathrm{M}}z_{h}|\\!|\\!|_{\mathrm{pw}}\leq\|z_{h}-I_{\mathrm{M}}z_{h}\|_{h}\leq\Lambda_{\mathrm{M}}(1+\Lambda_{h})|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$
(6.10)

A Cauchy inequality and the preceding estimates conclude the proof of (c). ∎

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

This follows from a Cauchy inequality with (5.2.a) and the estimates for
$\|u_{h}-I_{\mathrm{M}}u_{h}\|_{h}$ in (6.2) and $|\\!|\\!|\zeta-
I_{\mathrm{M}}z_{h}|\\!|\\!|_{\mathrm{pw}}$ in (6.3). ∎

###  6.4 Discrete consistency bounds

######  Lemma 6.4.

It holds

$\displaystyle
A_{h}(u_{h},z_{h})-a_{\mathrm{pw}}(u_{h},I_{\mathrm{M}}z_{h})+a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},z_{h})$
$\displaystyle\qquad\leq(1+\Lambda_{h})((2+M_{\rm
b})\Lambda_{\mathrm{M}}^{2}+(\Lambda_{\mathrm{dc}}+\Lambda_{\rm
ddc})(1+\Lambda_{\mathrm{M}})\Lambda_{\rm J}+\Lambda_{\rm
c})\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

###### Proof.

Recall $b_{h}(I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h})=0$ from (5.14) and
exploit (5.12) with elementary (but lengthy) algebra to check that the left-
hand side LHS of the assertion is equal to

LHS
$\displaystyle=(2a_{\mathrm{pw}}+b_{h})(u_{h}-I_{\mathrm{M}}u_{h},z_{h}-I_{\mathrm{M}}z_{h})+(a_{\mathrm{pw}}+b_{h})(I_{\mathrm{M}}u_{h},z_{h}-I_{\mathrm{M}}z_{h})$
(6.11)
$\displaystyle\quad+(a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h})+b_{h}(u_{h},I_{\mathrm{M}}z_{h}))+c_{h}(u_{h},z_{h})$
(6.12)

with the short notation, e.g.,
$(2a_{\mathrm{pw}}+b_{h})(\bullet,\bullet):=2a_{\mathrm{pw}}(\bullet,\bullet)+b_{h}(\bullet,\bullet)$,
for the sum of the bilinear forms announced in Subsection 5.5. The two lines
(6.11)-(6.12) of expressions for the LHS give rise to four estimates. The
continuity of $a_{\mathrm{pw}}(\bullet,\bullet)$ and $b_{h}(\bullet,\bullet)$
in (5.15), (6.2), and (6.10) prove

$\displaystyle(2a_{\mathrm{pw}}+b_{h})(u_{h}-I_{\mathrm{M}}u_{h},z_{h}-I_{\mathrm{M}}z_{h})\leq(2+M_{\rm
b})\Lambda_{\mathrm{M}}^{2}(1+\Lambda_{h})\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

The discrete consistency (5.13) leads in the last term in (6.11) to a product
of $|\\!|\\!|v-I_{\mathrm{M}}u_{h}|\\!|\\!|_{\mathrm{pw}}$ controlled in (6.9)
and $\|z-z_{h}\|_{h}$ controlled in (6.7). This results in

$\displaystyle(a_{\mathrm{pw}}+b_{h})(I_{\mathrm{M}}u_{h},z_{h}-I_{\mathrm{M}}z_{h})\leq\Lambda_{\mathrm{dc}}\Lambda_{\rm
J}(1+\Lambda_{\mathrm{M}})(1+\Lambda_{h})\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

The dual discrete consistency in (6.1) applies to the first two terms in
(6.12) and leads to $\Lambda_{\rm ddc}\|u-u_{h}\|_{h}$ times $|\\!|\\!|\zeta-
I_{\mathrm{M}}z_{h}|\\!|\\!|_{\mathrm{pw}}$ controlled with (5.2.b) in (6.3).
This with (6.2.b) and (6.7) result in

$\displaystyle
a_{\mathrm{pw}}(u_{h}-I_{\mathrm{M}}u_{h},I_{\mathrm{M}}z_{h})+b_{h}(u_{h},I_{\mathrm{M}}z_{h})\leq\Lambda_{\rm
ddc}\Lambda_{\rm
J}(1+\Lambda_{\mathrm{M}})(1+\Lambda_{h})\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

The last term in (6.12) is controlled in (5.16). This and (6.7) show

$c_{h}(u_{h},z_{h})\leq\Lambda_{\rm
c}\|u-u_{h}\|_{h}\|z-z_{h}\|_{h}\leq\Lambda_{\rm
c}(1+\Lambda_{h})\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}.$

A combination of the preceding four estimates with (6.11)-(6.12) concludes the
proof. ∎

###  6.5 Proof of Theorem 6.1

Given $2-\sigma\leq s\leq 2$, there exists a constant $0<C_{\rm
int}(s)<\infty$ (which exclusively depends on the shape regularity of
$\mathcal{T}$ and $s$) such that the solution $z\in V$ of the dual problem in
Section 6.2 satisfies (with Lemma 3.1.c) that

$|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}\leq C_{\rm int}(s)h_{\rm
max}^{2-s}\|z\|_{H^{4-s}(\Omega)}\leq C_{\rm int}(s)C_{\rm reg}(s)h_{\rm
max}^{2-s}\|G\|_{H^{-s}(\Omega)}.$ (6.13)

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

Recall $\|u-v\|_{H^{s}(\Omega)}=a(u-v,z)$ from Subsection 6.2 and its formula
in Lemma 6.2. Lemma 6.3 applies to the first four terms and Lemma 6.4 to the
remaining three. The resulting estimate reads

$\|u-v\|_{H^{s}(\Omega)}\leq
C_{6}{}\|u-u_{h}\|_{h}|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}$

with
$C_{\ref{theoremlowerorder}}=(1+\Lambda_{h}){\big{(}4(1+(1+\Lambda_{\mathrm{J}})(1+\Lambda_{\mathrm{M}}))^{2}}+(2+M_{\rm
b})\Lambda_{\mathrm{M}}^{2}+(\Lambda_{\mathrm{dc}}+\Lambda_{\rm
ddc})(1+\Lambda_{\mathrm{M}})\Lambda_{\rm J}+\Lambda_{\rm c})\big{)}.$ This
and $|\\!|\\!|z-I_{\mathrm{M}}z|\\!|\\!|_{\mathrm{pw}}\leq C_{\rm
int}(s)C_{\rm reg}(s)h_{\max}^{2-s}$ from (6.13) prove Theorem 6.1.a. ∎

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

The norm in ${H^{s}(\mathcal{T})}=\prod_{T\in\mathcal{T}}H^{s}(T)$ is the
$\ell^{2}$ norm of those contributions $\|\bullet\|_{H^{s}(T)}$ for all
$T\in\mathcal{T}$. 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 contributions 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 $1<s<2$. Hence Theorem 6.1.a implies

$\|u-JI_{\mathrm{M}}u_{h}\|_{H^{s}(\mathcal{T})}\leq
C_{\ref{2}}(s)\,h_{\max}^{2-s}\|u-u_{h}\|_{h}\quad\text{for all }s\text{ with
}2-\sigma\leq s\leq 2.$ (6.14)

Since $u_{h}\in P_{2}(\mathcal{T})$, Theorem 4.3.d provides the estimate

$\|u_{h}-JI_{\mathrm{M}}u_{h}\|_{H^{s}(\mathcal{T})}\lesssim
h_{\max}^{2-s}\|u-u_{h}\|_{h}.$

The triangle inequality in the norm of $H^{s}(\mathcal{T})$ concludes the
proof of Theorem 6.1.b. ∎

###  6.6 Verification of (H) and $\widehat{\text{\bf(H)}}$

For the choice $P=Q=J\circ I_{\mathrm{M}}$, Theorem 4.3.c. shows that
(2.3)-(2.3) hold for all the lowest-order schemes $V_{h}\subseteq
P_{2}(\mathcal{T})$ considered in this paper. This subsection verifies
(H)-${\widehat{\text{\bf(H)}}}$.

######  Lemma 6.5 (Verification of (H)-${\widehat{\text{\bf(H)}}}$).

Suppose the assumptions of Theorems 5.1 and 6.1. Any $v_{h}\in V_{h}$,
$v:=JI_{\mathrm{M}}v_{h}\in V$, $w_{h}\in V_{h}$, and $w=JI_{\rm M}w_{h}\in V$
satisfy

$\displaystyle{\text{\bf(H)}}\;A_{h}(v_{h},w_{h})-a(v,w)$
$\displaystyle\leq\Lambda_{1}\|v_{h}-v\|_{h}\|w_{h}\|_{h},$
$\displaystyle{\widehat{\text{\bf(H)}}}\;A_{h}(v_{h},w_{h})-a(v,w)$
$\displaystyle\leq\Lambda_{2}\|v_{h}-v\|_{h}\|w_{h}-w\|_{h}$

with $\Lambda_{1}:=(1+\Lambda_{\mathrm{M}})(\Lambda_{\rm
dc}+\Lambda_{\text{\rm J}}M_{\rm M}\|J\|)+(1+M_{\rm b})\Lambda_{\rm
M}+\Lambda_{\rm c}$, and $\Lambda_{2}:=\Lambda_{\rm ddc}+(1+M_{\rm
b})\Lambda_{\rm M}^{2}+(1+\Lambda_{\rm M})(1+\Lambda_{\rm M}+\Lambda_{\rm
dc})+\Lambda_{\rm c}$.

###### Proof of (H).

For $w_{\rm M}:=I_{\rm M}w,$ and $v_{\rm M}:=I_{\rm M}v$, (3.9) implies
$w_{\rm M}=I_{\rm M}w_{h}$, and $v_{\rm M}=I_{\rm M}v_{h}$. The definition of
$A_{h}(\bullet,\bullet)$, algebraic manipulations, and (5.14) result in

$\displaystyle A_{h}(v_{h},w_{h})-a(v,w)={}$ $\displaystyle a_{\rm
pw}(v_{h},w_{h})+b_{h}(v_{h},w_{h})+c_{h}(v_{h},w_{h})-a(v,w)$
$\displaystyle={}a_{\rm pw}(v_{h}-v_{\rm
M},w_{h})+b_{h}(v_{h}-v_{\mathrm{M}},w_{h})+a_{\rm pw}(v_{\rm M},w_{h}-w_{\rm
M})+b_{h}(v_{\rm M},w_{h}-w_{\rm M})$
$\displaystyle+c_{h}(v_{h},w_{h})+a_{\mathrm{pw}}(v_{\mathrm{M}},w_{\mathrm{M}})-a(v,w).$
(6.15)

The boundedness of $a_{\rm pw}(\bullet,\bullet)$, (5.15), (5.9), and (5.2)
prove

$\displaystyle a_{\rm pw}(v_{h}-v_{\rm M},w_{h})+b_{h}(v_{h}-v_{\rm
M},w_{h})\leq{}$ $\displaystyle(1+M_{\rm b})\Lambda_{\rm
M}\|v_{h}-v\|_{h}\|w_{h}\|_{h}.$

The discrete consistency condition (5.13) (with $w=0$), a triangle inequality,
(5.9), and (5.2) show

$\displaystyle a_{\rm pw}(v_{\rm M},w_{h}-w_{\rm M})+b_{h}(v_{\rm
M},w_{h}-w_{\rm M})\leq\Lambda_{\rm
dc}(1+\Lambda_{\mathrm{M}})\|v_{h}-v\|_{h}\|w_{h}\|_{h}.$

The bound in (5.16) with the choice $w=0$ implies

$\displaystyle c_{h}(v_{h},w_{h})\leq\Lambda_{\rm
c}\|v_{h}-v\|_{h}\|w_{h}\|_{h}.$

The orthogonality condition (3.5), Lemma 3.2.d., (5.9), and (5.10), result in

$\displaystyle a_{\rm pw}(v_{\rm M},w_{\rm M})-a(v,w)$
$\displaystyle=a_{\mathrm{pw}}((1-J)v_{\mathrm{M}},Jw_{\mathrm{M}})\leq\Lambda_{\text{J}}(1+\Lambda_{\mathrm{M}})M_{\rm
M}\|J\|\|v_{h}-v\|_{h}\|w_{h}\|_{h}.$

A combination of the last four displayed estimates in (6.15) leads to the
desired result. ∎

###### Proof of ${\widehat{\text{\bf(H)}}}$.

An alternate split of the left-hand side of the desired estimate leads to

$\displaystyle A_{h}(v_{h},w_{h})-a(v,w)$ $\displaystyle={}a_{\rm
pw}(v_{h}-v_{\rm M},w_{\rm M})+b_{h}(v_{h},w_{\rm M})+a_{\rm pw}(v_{h}-v_{\rm
M},w_{h}-w_{\rm M})$ $\displaystyle+b_{h}(v_{h}-v_{\rm M},w_{h}-w_{\rm
M})+a_{\rm pw}(v_{\rm M},w_{h}-w_{\rm M})+b_{h}(v_{\rm M},w_{h}-w_{\rm M})$
$\displaystyle+a_{\rm pw}(v_{\rm M},w_{\rm M})-a(v,w)+c_{h}(v_{h},w_{h}).$
(6.16)

The discrete consistency condition (6.1) shows

$\displaystyle a_{\rm pw}(v_{h}-v_{\rm M},w_{\rm M})+b_{h}(v_{h},w_{\rm
M})\leq\Lambda_{\rm ddc}\|v-v_{h}\|_{h}|\\!|\\!|w-w_{\rm M}|\\!|\\!|_{\rm
pw}.$

The boundedness of $a_{\rm pw}(\bullet,\bullet)$, (5.15), and (5.9) prove

$\displaystyle a_{\rm pw}(v_{h}-v_{\rm M},w_{h}-w_{\rm M})+b_{h}(v_{h}-v_{\rm
M},w_{h}-w_{\rm M})\leq{}$ $\displaystyle(1+M_{\rm b})\Lambda_{\rm
M}^{2}\|v_{h}-v\|_{h}\|w_{h}-w\|_{h}.$

The discrete consistency condition (5.13) and (5.2) lead to

$\displaystyle a_{\rm pw}(v_{\rm M},w_{h}-w_{\rm M})+b_{h}(v_{\rm
M},w_{h}-w_{\rm M})\leq\Lambda_{\rm
dc}(1+\Lambda_{\mathrm{M}})\|v_{h}-v\|_{h}\|w_{h}-w\|_{h}.$

The orthogonality condition (3.5) and (5.9) result in

$\displaystyle a_{\rm pw}(v_{\rm M},w_{\rm M})-a(v,w)$ $\displaystyle=-a_{\rm
pw}(v-v_{\rm M},w-w_{\rm M})\leq(1+\Lambda_{\rm
M})^{2}\|v_{h}-v\|_{h}\|w_{h}-w\|_{h}.$

The bound in (5.16) implies

$\displaystyle c_{h}(v_{h},w_{h})\leq\Lambda_{\rm
c}\|v_{h}-v\|_{h}\|w_{h}-w\|_{h}.$

A substitution of the last five displayed estimates in (6.16) leads to the
desired result. ∎

######  Remark 6.2 (Theorem 2.4 implies Theorem 6.1).

For $v:=JI_{\mathrm{M}}u_{h}\in V$, recall $\|u-v\|_{H^{s}(\Omega)}=a(u-v,z)$
from Subsection 6.2. Theorem 2.4 applies as Lemma 6.5 holds and with (5.2)
leads to $a(u-v,z)\leq\widehat{C_{\rm
qo}}\|u-u_{h}\|_{h}\|z-I_{\mathrm{M}}z\|_{h}$.

##  7 Modified dGFEM

The bilinear form $A_{h}(\bullet,\bullet):=A_{\rm dG}(\bullet,\bullet)$ [2,
28] is defined, for all $v_{2},w_{2}\in V_{h}:=P_{2}(\mathcal{T})$, by

$\displaystyle A_{\rm dG}(v_{2},w_{2})$
$\displaystyle:=a_{\text{pw}}(v_{2},w_{2})+b_{h}(v_{2},w_{2})+c_{\rm
dG}(v_{2},w_{2}),$ (7.1) $\displaystyle b_{h}(v_{2},w_{2})$
$\displaystyle:=-\Theta{\cal J}(v_{2},w_{2})-{\cal J}(w_{2},v_{2}),$ (7.2a)
$\displaystyle{\cal J}(v_{2},w_{2})$
$\displaystyle:=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left[\nabla_{\text{pw}}v_{2}\right]_{E}\cdot\left<D^{2}_{\text{pw}}w_{2}\right>_{E}\nu_{E}{\rm\,ds}$
(7.2b)

with $c_{\rm dG}(\bullet,\bullet)$ from (4.3) and given $-1\leq\Theta\leq 1$.
Let the jumps $[\bullet]_{E}$ across and the averages
$\left<\bullet\right>_{E}$ at an edge $E\in\operatorname{\mathcal{E}}$ from
Subsection 4.1 act componentwise. Recall from Theorem 4.1 the equivalent
discrete norms $\|\bullet\|_{h}\approx\|\bullet\|_{\rm dG}$ in
$H^{2}(\mathcal{T})$ defined in (4.1)-(4.2) and (4.4). Set $\Theta=1$ (resp.
$\Theta=-1$) to obtain the symmetric (resp. non symmetric) interior penalty
Galerkin formulation; see [39] for an alternative formulation. Appropriate
positive parameters $\sigma_{1},\sigma_{2}$ in (4.3) guarantee (5.1).

######  Lemma 7.1 (Boundedness and ellipticity of $A_{{\rm
dG}}(\bullet,\bullet)$).

[28, 39] (a) Any $v_{2},w_{2}\in P_{2}(\mathcal{T})$ satisfy $A_{{\rm
dG}}(v_{2},w_{2})\lesssim\|v_{2}\|_{\rm dG}\|w_{2}\|_{\rm dG}.$ (b) For
$\Theta=-1$ and any $\sigma_{\rm dG}=\sigma_{1}=\sigma_{2}>0$,
$\|v_{2}\|_{{\rm dG}}^{2}\leq A_{{\rm dG}}(v_{2},v_{2})$ holds for all
$v_{2}\in P_{2}(\mathcal{T})$. (c) For $-1<\Theta\leq 1$ and a sufficiently
large parameter $\sigma_{\rm dG}=\sigma_{1}=\sigma_{2}>0$, there exists
$\alpha>0$ (which depends on $\sigma_{\rm dG}$ and the shape regularity of
$\mathcal{T}$) such that $\alpha\|v_{2}\|_{{\rm dG}}^{2}\leq A_{{\rm
dG}}(v_{2},v_{2})$ for all $v_{2}\in P_{2}(\mathcal{T}).$

Throughout this paper, the parameter $\sigma_{\rm dG}$ is chosen to guarantee
the ellipticity of $A_{\rm dG}(\bullet,\bullet)$ in Lemma 7.1 with the short
notation $\sigma_{\rm dG}\approx 1\approx\alpha$. The modified dGFEM (5.3)
seeks the solution $u_{\rm dG}\in P_{2}(\mathcal{T})$ to

$A_{\rm dG}(u_{\rm dG},v_{2})=F(JI_{\mathrm{M}}v_{2})\quad\text{for all
}v_{2}\in P_{2}(\mathcal{T}).$ (7.3)

######  Theorem 7.2 (error estimates).

The solution $u\in V$ to (3.1) and the solution $u_{\rm dG}\in
P_{2}(\mathcal{T})$ to (7.3) satisfy (a) $\|u-u_{\rm
dG}\|_{h}\lesssim|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}$ and (b) if
$\Theta=1$ and $F\in H^{-s}(\Omega)$ for $2-\sigma\leq s\leq 2$, then
$\|u-JI_{\mathrm{M}}u_{\rm dG}\|_{H^{s}(\Omega)}+\|u-u_{\rm
dG}\|_{H^{s}(\mathcal{T})}\lesssim h_{\max}^{2-s}\|u-u_{\rm dG}\|_{h}$.

Overview of the proof. The assertion (a) follows from Theorem 5.1 for the
particular spaces, operators, norms, and bilinear forms defined below. The
application of Theorem 5.1 requires the proof of the abstract conditions
(5.1)-(5.2), (5.9) -(5.16). The assertion (b) follows from Theorem 6.1
provided (6.1) holds.

Setting and first consequences. Recall $V_{h}:=P_{2}(\mathcal{T})$ and the
norms $\|\bullet\|_{h}$ and $\|\bullet\|_{\rm dG}$ in (4.1)-(4.2) and (4.4).
Recall the Morley interpolation operator $I_{\mathrm{M}}$ from Definition 3.2
and the companion operator $J$ from Lemma 3.2. Recall that Lemma 7.1
guarantees (5.1)-(5.2). The dGFEM in (7.3) corresponds to (5.3) with the
solution $u_{h}:=u_{\rm dG}$. Example 5.1 implies (5.9)-(5.10). Set
$I_{h}:=\text{id}$ and observe (5.11) holds for $\Lambda_{h}=0$. Recall
$A_{h}(\bullet,\bullet):=A_{\rm dG}(\bullet,\bullet)$,
$b_{h}(\bullet,\bullet):=-\Theta{\cal J}(\bullet,\bullet)-{\cal
J}^{*}(\bullet,\bullet)$, and $c_{h}(\bullet,\bullet):=c_{\rm
dG}(\bullet,\bullet)$ in (5.12).

###### Proof of (5.13).

Since the integral
$\int_{E}\left[\nabla_{\mathrm{pw}}v_{\mathrm{M}}\right]_{E}{\rm\,ds}=0$
vanishes for $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ and since
$\left<D^{2}_{\mathrm{pw}}w_{2}\right>_{E}$ is constant on any edge
$E\in\operatorname{\mathcal{E}}$ for any $w_{2}\in P_{2}(\mathcal{T})$,

$\displaystyle\mathcal{J}(v_{\mathrm{M}},w_{2})=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left[\nabla_{\mathrm{pw}}v_{\mathrm{M}}\right]_{E}\cdot\left<D^{2}_{\mathrm{pw}}w_{2}\right>_{E}\nu_{E}{\rm\,ds}=0.$
(7.4)

Hence the term $\Theta\mathcal{J}(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})$
disappears below in definitions of $(a_{\mathrm{pw}}+b_{h})(\bullet,\bullet)$,
written in the short notation of Subsection 5.5;
$(a_{\mathrm{pw}}+b_{h})(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})$ is equal
to

$\sum_{T\in\mathcal{T}}\int_{K}D^{2}_{\mathrm{pw}}v_{\mathrm{M}}:D^{2}_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2}){\rm\,dx}-\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left[\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\right]_{E}\cdot\left<D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\right>_{E}\nu_{E}{{\rm\,ds}}.$

A piecewise integration by parts of the term
$a_{\mathrm{pw}}(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})$ shows equality to

$\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left(\left[\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\cdot(D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\>\nu_{E})\right]_{E}-\left[\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\right]_{E}\cdot\left<D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\right>_{E}\nu_{E}\right){\rm\,ds}.$

The product rule for the jump terms results in

$\displaystyle(a_{\mathrm{pw}}+b_{h})(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left<\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\right>_{E}\cdot\left[D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\right]_{E}\nu_{E}{\rm\,ds}.$
(7.5)

The further analysis concerns the split of the vector
$\left<\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\right>_{E}$$\in
P_{1}(E;{\mathbb{R}}^{2})\equiv P_{1}(E)^{2}$ into normal and tangential
components,

$\left<\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\right>_{E}=\langle\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial\nu_{E}\rangle_{E}\nu_{E}+\langle\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial
s\rangle_{E}\tau_{E}.$

The integral of the normal component
$\langle\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial\nu_{E}\rangle_{E}$ over
an edge $E\in\operatorname{\mathcal{E}}$ vanishes by definition of
$I_{\mathrm{M}}w_{2}$ in Definition 3.2. Since the jump
$\left[\partial^{2}_{\nu_{E}\nu_{E}}v_{\mathrm{M}}\right]_{E}:=\nu_{E}\cdot\left[D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\right]_{E}\nu_{E}$
is constant along $E$, the integral
$\int_{E}\langle\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial\nu_{E}\rangle_{E}\left[\partial^{2}_{\nu_{E}\nu_{E}}v_{\mathrm{M}}\right]_{E}{\rm\,ds}=0$
vanishes. The tangential components with
$\left[\partial^{2}_{\tau_{E}\nu_{E}}v_{\mathrm{M}}\right]_{E}:=\tau_{E}\cdot\left[D^{2}v_{\mathrm{M}}\right]_{E}\nu_{E}$
remain in

$(a_{\mathrm{pw}}+b_{h})(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left<\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial
s\right>_{E}\left[\partial^{2}_{\tau_{E}\nu_{E}}v_{\mathrm{M}}\right]_{E}{\rm\,ds}.$

The Hadamard jump condition asserts that the jump in the derivative of a
globally continuous function that is smooth up to the boundary on either side
of an interface $E$ points merely in the normal direction $\nu_{E}$ only. The
function $Jv_{\mathrm{M}}$ has a continuous gradient $\nabla Jv_{\mathrm{M}}$
and $\nabla Jv_{\mathrm{M}}$ is smooth on each triangle $\widehat{T}$ in the
HCT refinement of $\mathcal{T}$. Hence
$\left[\partial^{2}_{\tau_{E}\nu_{E}}Jv_{\mathrm{M}}\right]_{E}=0$ along $E$.
Consequently,

$\displaystyle(a_{\mathrm{pw}}+b_{h})(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})=\sum_{E\in\operatorname{\mathcal{E}}}\int_{E}\left<\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial
s\right>_{E}\left[\partial^{2}_{\tau_{E}\nu_{E}}(1-J)v_{\mathrm{M}}\right]_{E}{\rm\,ds}.$
(7.6)

For an interior edge $E=\partial T_{+}\cap\partial
T_{-}=\partial\widehat{T}_{+}\cap\partial\widehat{T}_{-}$ with the
neighbouring triangles $T_{\pm}\in\mathcal{T}$ and the two neighbouring sub-
triangles $\widehat{T}_{\pm}:=\text{\rm conv}(E,\operatorname{mid}(T_{\pm}))$
from the HCT refinement of $\mathcal{T}$ with patches
$\widehat{\omega}(E)=\text{int}(\widehat{T}_{+}\cup\widehat{T}_{-})\subset\omega(E)=\text{int}(T_{+}\cup
T_{-})$, Cauchy and triangle inequalities show

$\displaystyle I(E)$
$\displaystyle:=\int_{E}\left<\partial(w_{2}-I_{\mathrm{M}}w_{2})/\partial
s\right>_{E}\left[\partial^{2}_{\tau_{E}\nu_{E}}(1-J)v_{\mathrm{M}}\right]_{E}{\rm\,ds}$
$\displaystyle\leq\frac{1}{2}\left(\|\nabla(w_{2}-I_{\mathrm{M}}w_{2})|_{T_{+}}\|_{L^{2}(E)}+\|\nabla(w_{2}-I_{\mathrm{M}}w_{2})|_{T_{-}}\|_{L^{2}(E)}\right)$
$\displaystyle\hskip
8.53581pt\times\left(\|D^{2}(1-J)v_{\mathrm{M}}|_{\widehat{T}_{+}}\|_{L^{2}(E)}+\|D^{2}(1-J)v_{\mathrm{M}}|_{\widehat{T}_{-}}\|_{L^{2}(E)}\right).$

Since $(w_{2}-I_{\mathrm{M}}w_{2})|_{\widehat{T}_{\pm}}$ resp.
$(1-J)v_{\mathrm{M}}|_{\widehat{T}_{\pm}}$ is a polynomial of degree at most
$2$ resp. $3$ in the triangle $\widehat{T}_{\pm}$, the discrete trace
inequalities

$\displaystyle\|\nabla(w_{2}-I_{\mathrm{M}}w_{2})|_{T_{\pm}}\|_{L^{2}(E)}$
$\displaystyle\leq
h_{E}^{1/2}C_{\ref{standardshaperegularityconstantpart2}}\|h_{T_{\pm}}^{-1}\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})\|_{L^{2}(\widehat{T}_{\pm})}$
$\displaystyle\|D^{2}(1-J)v_{\mathrm{M}}|_{\widehat{T}_{\pm}}\|_{L^{2}(E)}$
$\displaystyle\leq
h_{E}^{-1/2}C_{\ref{standardshaperegularityconstantpart2}}|(1-J)v_{\mathrm{M}}|_{H^{2}(\widehat{T}_{\pm})}$

hold for a constant $C_{7}{}\approx 1$ that solely depends on the shape
regularity of $\widehat{T}_{\pm}$ (and so on the shape regularity of
$\mathcal{T}$). This leads to

$I(E)\leq
C_{\ref{standardshaperegularityconstantpart2}}^{2}\|h_{\mathcal{T}}^{-1}\nabla_{\mathrm{pw}}(w_{2}-I_{\mathrm{M}}w_{2})||_{L^{2}(\widehat{\omega}(E))}\|D^{2}_{\mathrm{pw}}(1-J)v_{\mathrm{M}}\|_{L^{2}(\widehat{\omega}(E))}$

for any interior edge $E\in\operatorname{\mathcal{E}}(\Omega)$ with the
reduced edge-patch $\widehat{\omega}(E)$. The same estimate follows for a
boundary edge $E\in\operatorname{\mathcal{E}}(\partial\Omega)$ (the proof
omits $T_{-}$, $\widehat{T}_{-}$, and some factor $1/2$ above). Since the
reduced edge-patches $(\widehat{\omega}(E):E\in\operatorname{\mathcal{E}})$
have no overlap, the sum of all the above estimates of $I(E)$ in (7.6) and
Cauchy inequalities prove

$\displaystyle(a_{\mathrm{pw}}+b_{h})(v_{\mathrm{M}},w_{2}-I_{\mathrm{M}}w_{2})\leq
C_{\ref{standardshaperegularityconstantpart2}}^{2}|h_{\mathcal{T}}^{-1}(w_{2}-I_{\mathrm{M}}w_{2})|_{H^{1}(\Omega)}|\\!|\\!|(1-J)v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}.$
(7.7)

Recall from Theorem 4.2.a. (with $\|(1-\Pi_{0})D^{2}_{\mathrm{pw}}w_{2}\|=0$
for $w_{2}\in P_{2}(\mathcal{T})$) that

$\displaystyle|h_{\mathcal{T}}^{-1}(w_{2}-I_{\mathrm{M}}w_{2})|_{H^{1}(\mathcal{T})}\lesssim
j_{h}(w_{2}-w)\lesssim\|w_{2}-w\|_{h}$

for all $w\in V$. Lemma 3.2.d shows
$|\\!|\\!|(1-J)v_{\mathrm{M}}|\\!|\\!|_{\mathrm{pw}}\leq\Lambda_{\mathrm{J}}|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}$
for any $v\in V$. The combination of this with (7.7) concludes the proof of
(5.13). ∎

###### Proof of (5.14).

This follows from (7.4). ∎

###### Proof of (5.15).

This follows from the boundedness of $b_{h}(\bullet,\bullet)$ (see [28, 39]).
∎

###### Proof of (5.16).

The jump contributions in (4.3) vanish for arguments in $V$, $c_{\rm
dG}(v_{2},w_{2})=c_{\rm dG}(v-v_{2},w-w_{2})$ on the left-hand side of (5.16).
Recall that $c_{\rm dG}(\bullet,\bullet)$ is a semi-norm scalar product and
the Cauchy inequality with the induced semi-norm $|\bullet|_{c_{\rm
dG}}:=c_{\rm dG}(\bullet,\bullet)^{1/2}\leq\|\bullet\|_{\rm dG}$ is a part of
the discrete norm $\|\bullet\|_{\rm dG}$. This leads to (5.16) with
$\Lambda_{\rm c}=1$. ∎

###### Proof of (6.1) for $\Theta=1$.

This follows from Remark 6.1. ∎

##  8 Modified $C^{0}$IP method

For the right-hand side $F\in H^{-2}(\Omega)$, the modified $C^{0}$IP method
is based on the continuous Lagrange $P_{2}$ finite element space
$V_{h}:=S^{2}_{0}(\mathcal{T}):=P_{2}(\mathcal{T})\cap H^{1}_{0}(\Omega)$ and
penalty terms along edges. The scheme is a modification of the dGFEM in
Section 7 but with trial and test functions restricted to
$S^{2}_{0}(\mathcal{T}):=P_{2}(\mathcal{T})\cap H^{1}_{0}(\Omega)$. The norm
$\lVert\bullet\rVert_{\mathrm{IP}}$ is $\|\bullet\|_{\rm dG}$ with restriction
to $S^{2}_{0}(\mathcal{T})$ and excludes one of the penalty parameters of the
modified dGFEM.

Given $\sigma_{\mathrm{IP}}>0$, the bilinear forms [6, 17] for
$v_{\mathrm{IP}},w_{\mathrm{IP}}\in S^{2}_{0}(\mathcal{T})$ are defined by

$\displaystyle A_{\mathrm{IP}}(v_{\mathrm{IP}},w_{\mathrm{IP}}):=$
$\displaystyle
a_{\text{pw}}(v_{\mathrm{IP}},w_{\mathrm{IP}})+b_{h}(v_{\mathrm{IP}},w_{\mathrm{IP}})+c_{\mathrm{IP}}(v_{\mathrm{IP}},w_{\mathrm{IP}}),\text{
where }$ (8.1) $\displaystyle
c_{\mathrm{IP}}(v_{\mathrm{IP}},w_{\mathrm{IP}})$
$\displaystyle:=\sum_{E\in\operatorname{\mathcal{E}}}\frac{\sigma_{\mathrm{IP}}}{h_{E}}\int_{E}\left[\frac{\partial
v_{\mathrm{IP}}}{\partial\nu_{E}}\right]_{E}\left[\frac{\partial
w_{\mathrm{IP}}}{\partial\nu_{E}}\right]_{E}{\rm\,ds},$ (8.2)

and $b_{h}(\bullet,\bullet)=b_{h}(\bullet,\bullet)|_{S^{2}_{0}(\mathcal{T})}$
from (7.2a).

The modified $C^{0}$IP method is of the form (5.3) and seeks
$u_{\mathrm{IP}}\in S^{2}_{0}(\mathcal{T})$ such that

$\displaystyle
A_{\mathrm{IP}}(u_{\mathrm{IP}},v_{\mathrm{IP}})=F(JI_{\mathrm{M}}v_{\mathrm{IP}})\quad\text{for
all }v_{\mathrm{IP}}\in S^{2}_{0}(\mathcal{T}).$ (8.3)

For all $v+v_{\mathrm{IP}}\in V+S_{0}^{2}(\mathcal{T})$, the discrete norm
reads $\displaystyle\lVert
v+v_{{\mathrm{IP}}}\rVert_{\mathrm{IP}}:=(|\\!|\\!|{v+v_{\mathrm{IP}}}|\\!|\\!|_{\mathrm{pw}}^{2}+c_{\mathrm{IP}}(v_{\mathrm{IP}},v_{\mathrm{IP}}))^{1/2}$
and
$j_{h}(v_{\mathrm{IP}})=(\sum_{E\in\operatorname{\mathcal{E}}}(\fint_{E}\left[{\partial
v_{{\mathrm{IP}}}}/{\partial\nu_{E}}\right]_{E}{\rm\,ds})^{2})^{1/2}$. Theorem
4.1 shows $\|v_{{\mathrm{IP}}}\|_{h}\approx\|v_{\mathrm{IP}}\|_{\mathrm{IP}}$.
The coercivity $\lVert\cdot\rVert_{\mathrm{IP}}^{2}\lesssim
A_{\mathrm{IP}}(\cdot,\cdot)$ on $S^{2}_{0}(\mathcal{T})$ holds provided
$\sigma_{\mathrm{IP}}$ is sufficiently large [6, 17]. The boundedness
$A_{{\mathrm{IP}}}(v_{{\mathrm{IP}}},w_{{\mathrm{IP}}})\lesssim\|v_{{\mathrm{IP}}}\|_{\mathrm{IP}}\|w_{{\mathrm{IP}}}\|_{\mathrm{IP}}$
holds for all $v_{\mathrm{IP}},w_{\mathrm{IP}}$ on $S^{2}_{0}(\mathcal{T})$
and (8.3) has a unique solution $u_{\mathrm{IP}}\in S^{2}_{0}(\mathcal{T})$.

######  Theorem 8.1 (error estimates).

The solution $u\in V$ to (3.1) and the solution $u_{\mathrm{IP}}\in
S^{2}_{0}(\mathcal{T})$ to (8.3) satisfy (a)
$\|u-u_{\mathrm{IP}}\|_{h}\lesssim|\\!|\\!|u-I_{\mathrm{M}}u|\\!|\\!|_{\mathrm{pw}}$
and (b) if $\Theta=1$ and $F\in H^{-s}(\Omega)$ for $2-\sigma\leq s\leq 2$,
then
$\|u-JI_{\mathrm{M}}u_{\mathrm{IP}}\|_{H^{s}(\Omega)}+\|u-u_{\mathrm{IP}}\|_{H^{s}(\mathcal{T})}\lesssim
h_{\max}^{2-s}\|u-u_{\mathrm{IP}}\|_{h}$.

######  Remark 8.1.

A $C^{0}$IP discrete scheme is analysed in [6] for a general $F\in
H^{\sigma-2}(\Omega)$. The consistency of the scheme allows a best
approximation [6, Lemma 8] (since $V_{h}$ subset $H^{2-\sigma}(\Omega)$ in the
pure Dirichlet problem for $C^{0}$IP). For $F\in H^{-2}(\Omega)$, a modifed
scheme and error estimates for the post-processed solution are derived in [6,
(4.17) and Theorem 4].

Overview of the proof of Theorem 8.1. The proof follows the lines of that of
Theorem 7.2 and partly from the analysis provided there. The bilinear forms in
the $C^{0}$IP are exactly the respective bilinear forms of the dGFEM when
restricted to the subspace $S^{2}_{0}(\mathcal{T})+M(\mathcal{T})$. With the
single exception of (5.11), all the estimates in (5.9)-(5.16) and (6.1) for
$\Theta=1$ follow for $V_{h}=S^{2}_{0}(\mathcal{T})$ in the $C^{0}$IP from the
respective properties verified in Section 7 for $V_{h}=P_{2}(\mathcal{T})$ in
the dGFEM. The remaining detail is the analysis of the operator $I_{h}\equiv
I_{\mathrm{C}}:\mathrm{M}(\mathcal{T})\rightarrow S^{2}_{0}(\mathcal{T})$
(denoted by $I_{2}^{*}$ in [13, Lemma 3.2]) defined by averaging the values of
a function $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ at the midpoint of an
interior edge $E$,

$\displaystyle(I_{\mathrm{C}}v_{\mathrm{M}})(z)=\begin{cases}v_{\mathrm{M}}(z)\quad\text{for
all }z\in\mathcal{V},\\\ \left<v_{\mathrm{M}}\right>_{E}(z)\quad\text{for
}z=\text{mid}(E),\;E\in\operatorname{\mathcal{E}}(\Omega),\\\ 0\quad\text{for
}z=\text{mid}(E),\;E\in\operatorname{\mathcal{E}}(\partial\Omega).\end{cases}$
(8.4)

###### Proof of (5.11).

This is included in [13, Lemma 3.2f] in a slightly different notation. In the
notation of this paper, Lemma 3.2 of [13] shows

$\|v_{\mathrm{M}}-I_{\mathrm{C}}v_{\mathrm{M}}\|^{2}_{h}\lesssim\sum_{E\in\operatorname{\mathcal{E}}}h_{E}^{-2}|\left[v_{\mathrm{M}}\right]_{E}({\text{mid}(E)})|^{2}$
(8.5)

for any $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$. Lemma 3.3 of [13] controls
the upper bound of (8.5) by the a posteriori terms
$\sum_{E\in\operatorname{\mathcal{E}}}h_{E}\|\left[D^{2}_{\mathrm{pw}}v_{\mathrm{M}}\right]_{E}\tau_{E}\|^{2}_{L^{2}(E)}$.
The latter is efficient, i.e.,
$\lesssim|\\!|\\!|v_{\mathrm{M}}-v|\\!|\\!|_{\mathrm{pw}}$ for any $v\in
H^{2}_{0}(\Omega)$. This leads to (5.11). Theorem 4.3 allows for an
alternative proof that departs at (8.5) with the quadratic function
$\left[v_{\mathrm{M}}\right]_{E}$ along the edge
$E\in\operatorname{\mathcal{E}}$. Since $\left[v_{\mathrm{M}}\right]_{E}$
vanishes at each end point $z\in\mathcal{V}(E)$ (owing to the continuity of
the Morley function $v_{\mathrm{M}}\in\mathrm{M}(\mathcal{T})$ at the
vertices), the (exact) Simpson’s quadrature rule asserts

$\left[v_{\mathrm{M}}\right]_{E}({\text{mid}(E)})=3/2\fint_{E}\left[v_{\mathrm{M}}\right]_{E}{\rm\,ds}.$

A Cauchy inequality, the continuity of $Jv_{\mathrm{M}}\in V$, triangle and
trace inequalities lead to

$h_{E}^{-2}|\left[v_{\mathrm{M}}\right]_{E}({\text{mid}(E)})|^{2}\leq\frac{9}{4}h_{E}^{-3}\|\left[v_{\mathrm{M}}-Jv_{\mathrm{M}}\right]_{E}\|_{L^{2}(E)}^{2}\lesssim\sum_{\ell=0}^{1}|h_{\mathcal{T}}^{\ell-2}(v_{\mathrm{M}}-Jv_{M})|_{H^{\ell}(\omega(E))}^{2}.$

This estimate and the finite overlap of the edge-patches
$(\omega(E):E\in\operatorname{\mathcal{E}})$ lead to an upper bound in (8.5)
as in Theorem 4.3.c and so to