# Error estimates for completely discrete FEM in energy-type and weaker norms

Lutz Angermann Dept. of Mathematics, Clausthal University of Technology,
Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany, lutz.angermann@tu-
clausthal.de Peter Knabner Dept. of Mathematics, University of Erlangen-
Nuremberg, Cauerstrasse 11, D-91058 Erlangen, Germany<EMAIL_ADDRESS>Andreas Rupp School of Engineering Science, Lappeenranta-Lahti University of
Technology, P.O.Box 20, FI-53851 Lappeenranta, Finland<EMAIL_ADDRESS>
###### Abstract

The paper presents error estimates within a unified abstract framework for the
analysis of FEM for boundary value problems with linear diffusion-convection-
reaction equations and boundary conditions of mixed type. Since neither
conformity nor consistency properties are assumed, the method is called
completely discrete. We investigate two different stabilized discretizations
and obtain stability and optimal error estimates in energy-type norms and, by
generalizing the Aubin-Nitsche technique, optimal error estimates in weaker
norms.

Keywords: Strang lemma, consistency, error estimate, Aubin-Nitsche technique,
discontinuous Galerkin method

AMS Subject Classification (2022): 65 N 12, 65 N 30, 46 B 10

##  1 Introduction

In this paper we present a unified approach to the analysis of FEM for
boundary value problems with linear elliptic differential equations of the
second order, where, in addition to the diffusion-convection-reaction
structure of the differential equation, boundary conditions of mixed type
(first, second, third kind) are taken into account. We allow completely
discrete formulations in the sense that the discrete FE spaces are not
necessarily embedded into the spaces of the weak formulation of the boundary
value problem and – based on a suitable notion of consistency – that the
discrete problems do not have to be consistent. In addition to stability and
error estimates in energy-type norms, a generalization of the Aubin-Nitsche
technique for obtaining error estimates in weaker norms is discussed.

Selected aspects of our exposition are not entirely new or even not really
profound; this applies, for example, to our version of Strang’s second lemma,
which is circulating in the literature in several, slightly different
versions. There are also other variants of the generalization of the duality
argument; we refer here – also for an overview – to [DPD18]. The abstract
results obtained are applied to two concrete discretizations – a Crouzeix-
Raviart discretization of order one and more general discontinuous Galerkin
methods of the IPG type. For both cases we discuss the stability, consistency
and convergence properties that result from the general theory.

As for the theoretical aspects, the work already mentioned [DPD18] and its
extension [DPD21] are perhaps the most closely related publications to our
work. Compared to these, we prefer not to include an interpolation (or quasi-
interpolation) operator in the consistency definition, but rather require that
the discrete bilinear form can be extended in such a manner that the solution
of the continuous problem (which often has more regularity than the elements
of spaces in which the weak formulation of the boundary value problem is
given), belongs to the extended domain of definition (as in [DPE12, Def.
1.31]). Furthermore, the extended paper [DPD21] applies its theory only to a
pure diffusion problem with homogeneous Dirichlet boundary conditions.

The paper is structured as follows. In the subsequent section we present, in
an abstract framework, error estimates for the solution of discretized
variational equations in both energy-type norms and weaker norms, whereby
neither conformity nor consistency are assumed. Then, in section 3, we
describe the model problem, the solution of which is to be approximated, and
the most important prerequisites. The model problem is a boundary value
problem for a scalar diffusion-convection-reaction equation with boundary
conditions of mixed type. Sections 4 and 5 describe the specific
discretizations, including their stabilization mechanisms, and the theoretical
results from Section 2 are applied. In both situations and under reasonable
assumptions, optimal error estimates are obtained.

##  2 General Variational Equations

Given two real Banach spaces $U,V$, this paper is concerned with the
application of finite element methods to approximate the solution to the
following problem:

$\text{Find }u\in U\text{ such that}\quad a(u,v)=\ell(v)\quad\text{for all
}v\in V,$ (1)

where throughout the paper $\ell:\;V\to\mathbb{R}$ is a continuous linear
functional, and $a:\;U\times V\to\mathbb{R}$ is a continuous bilinear form.

In the above setting, the following result about the existence of a unique
solution is known.

######  Theorem 1.

Let $V$ be reflexive. Then the problem (1) is uniquely solvable for every
right-hand side $\ell\in V^{\prime}$ if and only if the following two
conditions are satisfied:

$\displaystyle\alpha:=\inf_{{u\in U\setminus\\{0\\}}}\sup_{{v\in
V\setminus\\{0\\}}}\frac{a(u,v)}{\|u\|_{U}\|v\|_{V}}>0,$ (2) $\displaystyle
a(u,v)=0\quad\text{for all }u\in U\quad\Longrightarrow\quad v=0\quad\text{for
}v\in V.$ (3)

If both conditions are fulfilled, the problem (1) is stable in the sense that
the solution $u\in U$ of (1) satisfies the estimate

$\|u\|_{U}\leq\frac{1}{\alpha}\|\ell\|_{V^{\prime}}.$

###### Proof.

First we note that the variational equation (1) can be reformulated as an
operator equation:

$Au=\ell,$

where $A:\;U\to V^{\prime}$ is defined by means of the relationship
$(Au)(v):=a(u,v)$ for all $u\in U$, $v\in V$. The continuity of $a$ implies
the continuity of $A$ (notation: $A\in L[U,V^{\prime}]$).

The assertion follows from the following chain of arguments, but we omit their
detailed proofs.

  1. 1)

Let $U,V$ be normed spaces only. Then:

$\eqref{eq:vfg1}\quad\Longleftrightarrow\quad
A^{-1}:\;{\operatorname{im}}{(A)}\to U\text{ exists and is continuous}.$

  2. 2)

If, in addition to the assumption in 1), $U$ is complete, i. e. a Banach
space, then $\overline{{\operatorname{im}}{(A)}}={\operatorname{im}}{(A)}$.

  3. 3)

Let, in addition to the assumptions in 2), $V$ be a reflexive Banach space.
Then:

$\eqref{eq:vfg2}\quad\Longleftrightarrow\quad{\operatorname{im}}{(A)}=V^{\prime}.$

∎

We now turn to Galerkin approximations of (1). Let $U_{h},V_{h}$ be finite-
dimensional Hilbert spaces (which is not a restriction, since all finite-
dimensional real normed spaces of same dimension, say $m$, are isometrically
isomorphic each to the other, in particular to the Hilbert space
$\mathbb{R}^{m}$ equipped with the Euclidean scalar product) of equal
dimension and $a_{h}:\;U_{h}\times V_{h}\to\mathbb{R}$,
$\ell_{h}:\;V_{h}\to\mathbb{R}$ be bilinear and linear forms, respectively.
Then the discrete problem is:

$\text{Find }u_{h}\in U_{h}\text{ such that }\quad
a_{h}(u_{h},v_{h})=\ell_{h}(v_{h})\quad\text{for all }v_{h}\in V_{h}.$ (4)

For $U_{h}=V_{h}$, we speak of _(standard) Galerkin methods_ (sometimes also
referred as _Bubnov-Galerkin methods_), otherwise of _Petrov-Galerkin methods_
(even if $U=V$).

We call the discretization (4) _conforming_ , if $U_{h}\subset U$ and
$V_{h}\subset V$, and _nonconforming_ otherwise. In the conforming case, we
set $\|\cdot\|_{U_{h}}:=\|\cdot\|_{U}$, $\|\cdot\|_{V_{h}}:=\|\cdot\|_{V}$
unless differently specified.

To ensure that elements of $U$ or elements of $V$ can also be compared with
elements of $U_{h}$ or $V_{h}$, respectively, in the nonconforming case, we
introduce the _augmented spaces_

$U(h):=U+U_{h},\ V(h):=V+V_{h}.$

This implicitly assumes the existence of linear superspaces for $U,U_{h}$ and
$V,V_{h}$, respectively. In this framework we assume that the spaces $U(h)$,
$V(h)$ are equipped with norms $\|\cdot\|_{U(h)}$, $\|\cdot\|_{V(h)}$,
respectively. Furthermore, in some cases it can be useful to reduce the
original solution space $U$ to a proper subspace, say $W\subset U$ with $W\neq
U$, for example by incorporating additional knowledge about the regularity of
the solution.

######  Definition 2.

The discrete formulation (4) is called _(exactly) consistent_ on the solution
space $W\subset U$, if the bilinear form $a_{h}$ can be extended to
$(W+U_{h})\times V_{h}$ such that the exact solution $u\in W$ of (1) satisfies
the approximate problem, i. e.,

$a_{h}(u,v_{h})=\ell_{h}(v_{h})\quad\text{for all }v_{h}\in V_{h},$

and _nonconsistent_ otherwise.

In the consistent case, the _Galerkin orthogonality_ trivially holds true, i.
e.,

$a_{h}(u-u_{h},v_{h})=0\quad\text{for all }v_{h}\in V_{h}.$

In the nonconsistent case, additionally a _consistency error_ occurs.

An extension of the standard convergence analysis for the consistent
conforming case is given by the following generalization of Strang’s second
lemma. Extensions of this kind can be found in the literature (e. g. [DPE12],
[CDGH17]) but in fact all of these results (including our subsequent Theorem
3) are not very deep and rather technical in nature, but allow the analysis of
more general finite element (and related) methods.

######  Theorem 3.

Assume that the solution $u$ of (1) belongs to some subspace $W\subset U$, the
norm $\|\cdot\|_{U_{h}}$ extends to a norm on $W+U_{h}$ (keeping the notation
$\|\cdot\|_{U_{h}}$), and the condition

$\alpha_{h}:=\inf_{u_{h}\in U_{h}\setminus\\{0\\}}\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{a_{h}(u_{h},v_{h})}{\|u_{h}\|_{U_{h}}\|v_{h}\|_{V_{h}}}>0$
(5)

is satisfied.

Further, assume that the bilinear form $a_{h}$ can be continuously extended
onto
$\left(W+U_{h},\|\cdot\|_{U(h)}\right)\times\left(V_{h},\|\cdot\|_{V_{h}}\right)$,
i. e., there exists a constant $\widetilde{M}_{h}\geq 0$ such that

$|a_{h}(w,v_{h})|\leq\widetilde{M}_{h}\|w\|_{U(h)}\|v_{h}\|_{V_{h}}\quad\text{for
all }w\in W+U_{h},\ v_{h}\in V_{h}.$

Then, we have that

$\displaystyle\|u-u_{h}\|_{U_{h}}$ $\displaystyle\leq\inf_{w_{h}\in
U_{h}}\left(\frac{\widetilde{M}_{h}}{\alpha_{h}}\|u-w_{h}\|_{U(h)}+\|u-w_{h}\|_{U_{h}}\right)$
$\displaystyle\qquad+\frac{1}{\alpha_{h}}\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{|a_{h}(u,v_{h})-\ell_{h}(v_{h})|}{\|v_{h}\|_{V_{h}}}\,.$

###### Proof.

Since

$\displaystyle a_{h}(u_{h}-w_{h},v_{h})$
$\displaystyle=a_{h}(u_{h}-u,v_{h})+a_{h}(u-w_{h},v_{h})$
$\displaystyle=\ell_{h}(v_{h})-a_{h}(u,v_{h})+a_{h}(u-w_{h},v_{h})\quad\text{for
any }w_{h}\in U_{h},$

it follows from (5) that

$\displaystyle\alpha_{h}\|u_{h}-w_{h}\|_{U_{h}}$
$\displaystyle\leq\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{a_{h}(u_{h}-w_{h},v_{h})}{\|v_{h}\|_{V_{h}}}$
$\displaystyle\leq\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{|a_{h}(u,v_{h})-\ell_{h}(v_{h})|}{\|v_{h}\|_{V_{h}}}+\widetilde{M}_{h}\|u-w_{h}\|_{U(h)}.$

The triangle inequality
$\|u-u_{h}\|_{U_{h}}\leq\|u-w_{h}\|_{U_{h}}+\|w_{h}-u_{h}\|_{U_{h}}$ yields
the result. ∎

######  Remark 4.

  1. 1)

Under the assumptions of Theorem 3, if there exists a constant
$\widetilde{C}_{h}>0$ such that

$\|w_{h}\|_{U(h)}\leq\widetilde{C}_{h}\|w_{h}\|_{U_{h}}\quad\text{for all
}w_{h}\in U_{h},$

we apply the triangle inequality w.r.t. the $U(h)$-norm and conclude

$\|u-u_{h}\|_{U(h)}\leq\left(1+\frac{\widetilde{C}_{h}\widetilde{M}_{h}}{\alpha_{h}}\right)\inf_{w_{h}\in
U_{h}}\|u-w_{h}\|_{U(h)}+\frac{\widetilde{C}_{h}}{\alpha_{h}}\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{|a_{h}(u,v_{h})-\ell_{h}(v_{h})|}{\|v_{h}\|_{V_{h}}}\,.$

  2. 2)

Under the assumptions of Theorem 3, if there exists a constant
$\widetilde{C}_{h}>0$ such that

$\|w_{h}\|_{U_{h}}\leq\widetilde{C}_{h}\|w_{h}\|_{U(h)}\quad\text{for all
}w\in W+U_{h},$

then we have

$\|u-u_{h}\|_{U_{h}}\leq\left(\widetilde{C}_{h}+\frac{\widetilde{M}_{h}}{\alpha_{h}}\right)\inf_{w_{h}\in
U_{h}}\|u-w_{h}\|_{U(h)}+\frac{1}{\alpha_{h}}\sup_{v_{h}\in
V_{h}\setminus\\{0\\}}\frac{|a_{h}(u,v_{h})-\ell_{h}(v_{h})|}{\|v_{h}\|_{V_{h}}}\,.$

### Error Estimates in Weaker Norms

In the standard finite element literature for second-order linear elliptic
boundary value problems, the so-called Aubin-Nitsche duality argument [Aub67],
[Nit68] is used to establish error estimates in norms which are weaker than
the natural energy norm (or equivalent norms), typically in the $L^{2}$-norm.
Considering the adjoint variational problem

Find $v\in V$ such that

$a(w,v)=\tilde{\ell}(w)\quad\text{for all }w\in U$ (6)

for specially chosen right-hand sides $\tilde{\ell}\in U^{\prime}$, it is
often possible to improve the order of convergence in norms that are weaker
than $\|\cdot\|_{U}$, provided the solution of the adjoint problem is
sufficiently regular. This requirement can be expressed by the assumption that
there exists a suitably chosen subspace $Y\subset V$ in which the solution $v$
of (6) lies.

Here we extend this setting to a more general framework, in which the
occurring discrete spaces no longer have to be subspaces of the “continuous”
spaces $U,V$ of the weak formulation (1), and the discretization does not
necessarily have to be consistent.

This abstract framework is applied to two examples of nonconforming FEM for
diffusion-convection-reaction equations in Sections 4 and 5. However, the
range of models and numerical techniques covered by the analytical framework
goes well beyond these examples.

We basically assume here that there exists a reflexive Banach space $Z$ such
that $U+U_{h}\subset Z^{\prime}$. By
$\left\langle\cdot,\cdot\right\rangle_{Z^{\prime}\times
Z""}:\;Z^{\prime}\times Z^{\prime\prime}\to\mathbb{R}$ we denote the duality
pairing on $Z^{\prime}\times Z^{\prime\prime}$ and identify $Z^{\prime\prime}$
with $Z$: $Z^{\prime\prime}\cong Z$.

Now let an arbitrary element $g\in Z$ be given and consider the adjoint
problem (6) with the right-hand side

$\tilde{\ell}(w):=\tilde{\ell}_{g}(w):=\left\langle
w,\,g\right\rangle_{Z^{\prime}\times Z""},\ w\in U.$ (7)

In addition, it is assumed that the Galerkin approximation $v_{gh}\in V_{h}$
to the solution $v_{g}\in Y$ of (6), (7) exists uniquely, i. e., $v_{gh}$
satisfies

$a_{h}(w_{h},v_{gh})=\left\langle w_{h},g\right\rangle_{Z^{\prime}\times
Z""}\quad\text{for all }w_{h}\in U_{h}.$ (8)

######  Theorem 5.

Consider a general Galerkin approximation (4) of the variational form (1) with
solutions $u_{h}\in U_{h}$ and $u\in W$, respectively, where $W$ is a certain
subspace of $U$. Assume that the bilinear form $a_{h}$ can be extended to
$(W+U_{h})\times(Y+V_{h})$ such that it holds, for a constant
$\widetilde{M}_{h}\geq 0$,

$|a_{h}(w,z)|\leq\widetilde{M}_{h}\|w\|_{U(h)}\|z\|_{V(h)}\quad\text{for all
}(w,z)\in(W+U_{h})\times(Y+V_{h}),$ (9)

and that $\ell_{h}$ can be extended to $Y+V_{h}$. Then:

$\displaystyle\|u-u_{h}\|_{Z^{\prime}}$ $\displaystyle\leq\sup_{g\in
Z\setminus\\{0\\}}\frac{1}{\|g\|_{Z}}\Big{\\{}\widetilde{M}_{h}\|u-u_{h}\|_{U(h)}\|v_{g}-v_{gh}\|_{V(h)}$
(10) $\displaystyle\quad-\left[a_{h}(u-u_{h},v_{g})-\left\langle
u-u_{h},g\right\rangle_{Z^{\prime}\times Z""}\right]$
$\displaystyle\quad-\left[a_{h}(u,v_{g}-v_{gh})-\ell_{h}(v_{g}-v_{gh})\right]$
$\displaystyle\quad+\left[(a_{h}-a)(u,v_{g})-(\ell_{h}-\ell)(v_{g})\right]\Big{\\}}.$

###### Proof.

Since

$\|u-u_{h}\|_{Z^{\prime}}=\sup_{g\in
Z\setminus\\{0\\}}\frac{1}{\|g\|_{Z}}\left\langle
u-u_{h},g\right\rangle_{Z^{\prime}\times Z""}$

holds it is sufficient to consider the expression

$\displaystyle\left\langle u-u_{h},g\right\rangle_{Z^{\prime}\times Z""}$
$\displaystyle=a(u,v_{g})-a_{h}(u_{h},v_{gh})$
$\displaystyle=a_{h}(u,v_{g})-a_{h}(u_{h},v_{gh})+(a-a_{h})(u,v_{g})$
$\displaystyle=:I_{1}+(a-a_{h})(u,v_{g}).$

The term $I_{1}$ can be decomposed as

$I_{1}=a_{h}(u-u_{h},v_{g}-v_{gh})+a_{h}(u_{h},v_{g}-v_{gh})+a_{h}(u-u_{h},v_{gh})=:I_{2}+I_{3}+I{{}_{4}},$

where $I_{2}$ can be estimated immediately by means of (9):

$|I_{2}|\leq\widetilde{M}_{h}\|u-u_{h}\|_{U(h)}\|v_{g}-v_{gh}\|_{V(h)}.$

The terms $I_{3},I{{}_{4}}$ can be rewritten as follows:

$\displaystyle I_{3}$
$\displaystyle=-a_{h}(u-u_{h},v_{g})+a_{h}(u,v_{g})-a_{h}(u_{h},v_{gh})$
$\displaystyle=-a_{h}(u-u_{h},v_{g})+a(u,v_{g})-a_{h}(u_{h},v_{gh})+(a_{h}-a)(u,v_{g})$
$\displaystyle=-[a_{h}(u-u_{h},v_{g})-\left\langle
u,g\right\rangle_{Z^{\prime}\times Z""}+\left\langle
u_{h},g\right\rangle_{Z^{\prime}\times Z""}]+(a_{h}-a)(u,v_{g}),$
$\displaystyle=-[a_{h}(u-u_{h},v_{g})-\left\langle
u-u_{h},g\right\rangle_{Z^{\prime}\times Z""}]+(a_{h}-a)(u,v_{g}),$
$\displaystyle I_{4}$
$\displaystyle=-a_{h}(u,v_{g}-v_{gh})+a_{h}(u,v_{g})-a_{h}(u_{h},v_{gh})$
$\displaystyle=-[a_{h}(u,v_{g}-v_{gh})-a(u,v_{g})+\ell_{h}(v_{gh})]+(a_{h}-a)(u,v_{g})$
$\displaystyle=-[a_{h}(u,v_{g}-v_{gh})-\ell_{h}(v_{g}-v_{gh})]+(a_{h}-a)(u,v_{g})-(\ell_{h}-\ell)(v_{g}).$

In summary, the statement follows. ∎

In the following we want to discuss the four contributions in the estimate
(10). All terms must allow an estimate with the factor $\|g\|_{Z}$ to achieve
an improved order of convergence result. The order of the first term is the
product of the convergence order of the original problem (4) according to
Theorem 3 and the corresponding order of convergence for the adjoint problem
(6), (7).

To be more specific we assume that $Y\subset V$ is a Banach space equipped
with a (semi)norm $\|\cdot\|_{Y}$ such that the following regularity and
convergence properties hold:

  * •

The solution $v_{g}$ of (6), (7) even belongs to the space $Y$ and there is a
constant $C_{s}\geq 0$ independent of $v_{g}$ such that

$\|v_{g}\|_{Y}\leq C_{s}\|g\|_{Z}.$ (11)

  * •

There are constants $C_{a}>0$, $q>0$ such that the Galerkin approximation
$v_{gh}\in V_{h}$ from (8) to the solution $v_{g}\in Y$ of (6), (7) satisfies

$\|v_{g}-v_{gh}\|_{V(h)}\leq C_{a}h^{q}\|v_{g}\|_{Y}.$ (12)

The estimate (12) is in turn the consequence of an a priori estimate, as it is
given, for example, by Theorem 3, and reflects the order of the approximation
of $v_{g}$ by elements in $V_{h}$ and of the consistency errors. Under the
assumptions (11), (12) the first term of (10) obviously can be estimated by

$\widetilde{M}_{h}C_{a}C_{s}\|u-u_{h}\|_{U(h)}h^{q}.$

The second term is a consistency error for the adjoint problem with the test
function $u-u_{h}$. We aim at a consistency error estimate for $v_{g}\in Y$ of
the type

$|a_{h}(w,v_{g})-\left\langle w,g\right\rangle_{Z^{\prime}\times Z""}|\leq
C_{ca}\|w\|_{U(h)}\|v_{g}\|_{Y}h^{\alpha}\leq
C_{ca}C_{s}\|w\|_{U(h)}\|g\|_{Z}h^{\alpha}$ (13)

with constants $C_{ca}\geq 0$, $\alpha>0$, giving the following estimate for
the second term:

$C_{ca}C_{s}\|u-u_{h}\|_{U(h)}h^{\alpha}.$

Analogously the third term is a consistency error for the original problem in
the test function $v_{g}-v_{gh}$. Assuming analogously to (13) an estimate of
the type

$|a_{h}(u,z)-\ell_{h}(z)|\leq C_{cp}\|z\|_{V(h)}\|u\|_{W}h^{\beta}$

with constants $C_{cp}\geq 0$, $\beta>0$, in particular $u\in W$, then the
third term together with (12) allows the following estimate:

$C_{cp}C_{\alpha}C_{s}h^{q+\beta}\|u\|_{W}.$

Finally the last term describes approximation errors. Here we assume an
estimate of the type

$|(a_{h}-a)(u,v_{g})-(\ell_{h}-\ell)(v_{g})|\leq
C_{q}\|v_{g}\|_{Y}h^{\gamma}\leq C_{q}C_{s}\|g\|_{Z}h^{\gamma}$

with constants $C_{q}\geq 0$, $\gamma>0$. If finally the order of convergence
of the original problem is $p>0$, then the achieved order of the bound of
$\|\cdot\|_{Z^{\prime}}$ is

$\min\\{p+q,\,p+\alpha,\,q+\beta,\,\gamma\\}.$

In the conforming case (both for the original and the adjoint problem, see the
following table) the last three terms in (10) can be rearranged to

$a(u-u_{h},v_{g}-v_{gh})-(a-a_{h})(u_{h},v_{gh})-(\ell_{h}-\ell)(v_{gh}).$
(14)

Concerning the choice of $Z$, we concentrate here on formulations with
$U\subset H^{1}(\Omega)$, and $U=V$, $U_{h}=V_{h}$ being conforming
$\mathcal{P}_{k}$-elements. Then an obvious choice is

$Z:=L^{2}(\Omega),$

and (11), (12) can be satisfied with $Y:=H^{2}(\Omega)$, $q=1$, for sufficient
smooth data. However, $Z:=H^{k-1}(\Omega)$ can also be selected, which may
enable the choice $Y:=H^{k+1}(\Omega)$ and also $q=k$, and thus leads to the
optimal case of order doubling (provided the consistency errors behave
correspondingly) in the norm $\|\cdot\|_{-k+1}$.

Such negative norm estimates then also allow for error estimates of
functionals $\varphi\in U(h)^{\prime}$: Assuming that $\varphi$ is given as a
smoother function in the sense that even

$\varphi\in Z\ (\cong Z^{\prime\prime})$

holds, then

$\varphi(u-u_{h})\leq\|\varphi\|_{Z}\|u-u_{h}\|_{Z^{\prime}}\,.$

In the following table we list the properties of the variational formulations
used, whereby the properties of the second and third columns apply in addition
to the first column. Here as above $u\in U,v_{g}\in V$ are the weak solutions,
$u_{h}\in U_{h},v_{gh}\in V_{h}$ are their numerical approximations, and $v\in
V,v_{h}\in V_{h},w\in U,w_{h}\in U_{h}$ are the test functions.

General case | Conforming case | Consistent case
---|---|---
$a(u,v)=\ell(v)$ |  |
| $a(u,v_{h})=\ell(v_{h})$ |
$a_{h}(u_{h},v_{h})=\ell_{h}(v_{h})$ |  |
|  | $a_{h}(u,v_{h})=\ell_{h}(v_{h})$
$a(w,v_{g})=\left\langle w,g\right\rangle_{Z^{\prime}\times Z""}$ |  |
| $a(w_{h},v_{g})=\left\langle w_{h},g\right\rangle_{Z^{\prime}\times Z""}$ |
$a_{h}(w_{h},v_{gh})=\left\langle w_{h},g\right\rangle_{Z^{\prime}\times Z""}$ |  |
|  | $a_{h}(w_{h},v_{g})=\left\langle w_{h},g\right\rangle_{Z^{\prime}\times Z""}$

######  Remark 6.

In the conforming case with $U=V$ an alternative estimate is possible. Under
the assumption of Theorem 5 we have (using the notation of the proof):

$\displaystyle\left\langle u-u_{h},g\right\rangle_{Z^{\prime}\times Z""}$
$\displaystyle=a(u-u_{h},v_{g})$
$\displaystyle=a(u-u_{h},v_{g}-\Pi_{h}v_{g})+\ell(\Pi_{h}v_{g})-a(u_{h},\Pi_{h}v_{g})$

with $\Pi_{h}v_{g}\in V_{h}$, leading to the estimate

$\displaystyle\|u-u_{h}\|_{Z^{\prime}}\leq\sup_{g\in
Z\setminus\\{0\\}}\frac{1}{\|g\|_{Z}}\Big{\\{}\widetilde{M}\|u-u_{h}\|_{V}\|v_{g}-\Pi_{h}v_{g}\|_{V}$
$\displaystyle-[a(u_{h},\Pi_{h}v_{g})-\ell(\Pi_{h}v_{g})]\Big{\\}}.$

For $Z:=V$ and $\Pi_{h}$ being the orthogonal projection of $V_{h}$ to $V$,
the second term, a “unvariational” consistency error term, can be estimated by

$|\left\langle
a(u_{h},\cdot)-a_{h}(u_{h},\cdot),\Pi_{h}v_{g}\right\rangle_{V}|+\|v_{g}\|_{V}\|\ell_{h}-\Pi_{h}\ell\|_{V^{\prime}},$

as by decomposing the term by introducing $\ell_{h}:=a_{h}(u_{h},\cdot)$, the
second summand reads by the symmetry of $\Pi_{h}$

$\left\langle
a_{h}(u_{h},\cdot)-\ell,\Pi_{h}v_{g}\right\rangle_{V}=\left\langle\Pi_{h}(a_{h}(u_{h},\cdot)-\ell),v_{g}\right\rangle_{V}=\left\langle\ell_{h}-\Pi_{h}\ell,v_{g}\right\rangle.$

##  3 The Model Problem

In this and the subsequent section we will apply the theoretical results to
finite element discretizations of the following diffusion-convection-reaction
problem. Given a bounded polyhedral Lipschitz domain
$\Omega\subset\mathbb{R}^{d}$, $d\in\\{2,3\\}$, we consider the differential
equation in divergence form

$-\nabla\cdot(\boldsymbol{K}\nabla u-\boldsymbol{c}\,u)+r\,u=f\quad\text{in
}\Omega$ (15)

with the data

$\displaystyle\boldsymbol{K}=\boldsymbol{K}^{\top}\in
L^{\infty}(\Omega)^{d,d},\quad\boldsymbol{c}\in
L^{\infty}(\Omega)^{d},\quad\partial_{j}c_{j}\in L^{3/2}(\Omega),\
j\in\\{1,\ldots,d\\},$ $\displaystyle r\in L^{\infty}(\Omega),\quad f\in
L^{2}(\Omega),$

where, for some constant $k_{0}>0$,

$\boldsymbol{\xi}\cdot(\boldsymbol{K}(x)\boldsymbol{\xi})\geq
k_{0}|\boldsymbol{\xi}|^{2}\quad\text{for all
}\boldsymbol{\xi}\in\mathbb{R}^{d}\text{ and almost all }x\in\Omega.$ (16)

To formulate the boundary conditions we assume that the boundary
${\partial\Omega}$ is decomposed into disjoint subsets $\Gamma_{j},$
$j\in\\{1,2,3\\}$:
${\partial\Omega}=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\,,$ where
$\Gamma_{3}$ is supposed to be a relatively closed subset of
${\partial\Omega}$. Given the functions $g_{j}\in L^{2}(\Gamma_{j})$,
$j\in\\{1,2\\}$, and $\tilde{\alpha}\in L^{\infty}(\Gamma_{2})$, the boundary
conditions are (the symbol ${\mathfrak{n}}$ denotes the unit outer normal on
${\partial\Omega}$):

$\displaystyle(\boldsymbol{K}\nabla
u-\boldsymbol{c}\,u)\cdot\boldsymbol{{\mathfrak{n}}}$ $\displaystyle=g_{1}$
$\displaystyle\text{on }\Gamma_{1},$ (17) $\displaystyle(\boldsymbol{K}\nabla
u-\boldsymbol{c}\,u)\cdot\boldsymbol{{\mathfrak{n}}}+\tilde{\alpha}u$
$\displaystyle=g_{2}$ $\displaystyle\text{on }\Gamma_{2},$ $\displaystyle u$
$\displaystyle=0$ $\displaystyle\text{on }\Gamma_{3},$

i. e., we restrict our investigations to homogeneous Dirichlet boundary
conditions.

The variational formulation (1) is specified as follows:

$\displaystyle U:=V$ $\displaystyle:=\big{\\{}v\in H^{1}(\Omega)\;|\;v=0\text{
on }\Gamma_{3}\text{ in the sense of traces}\big{\\}},$ $\displaystyle a(u,v)$
$\displaystyle:=(\boldsymbol{K}\nabla u-\boldsymbol{c}u,\nabla
v)+(ru,v)+(\tilde{\alpha}u,v)_{\Gamma_{2}}\quad\text{for all }u,v\in V,$
$\displaystyle\ell(v)$
$\displaystyle:=(f,v)+(g_{1},v)_{\Gamma_{1}}+(g_{2},v)_{\Gamma_{2}}\quad\text{for
all }v\in V.$

Since we will also have to deal with the adjoint boundary value problem, we
formulate the additional requirements to the data in a form that slightly
differs from the usual one:

1) $\displaystyle r+\frac{1}{2}\nabla\cdot\boldsymbol{c}\geq 0\quad\text{in
}\Omega\,,$ (18) 2)
$\displaystyle\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}\geq
0\quad\text{on
}\Gamma_{2,1}:=\\{x\in\Gamma_{2}\,|\,\tilde{\alpha}(x)=\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}\\}\,,\hfill\rule{99.02747pt}{0.0pt}$
3)
$\displaystyle\tilde{\alpha}-\frac{1}{2}\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}\geq
0\quad\text{on
}\Gamma_{2}\setminus\Gamma_{2,1}\,,\quad\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}\leq
0\quad\text{on }\Gamma_{1}\,.$

These assumptions together with (16) ensure that the bilinear form $a$ is at
least positively semidefinite on $V$, as the following identity shows:

$\displaystyle a(v,v)$ $\displaystyle=(\boldsymbol{K}\nabla v,\nabla
v)+(r+\frac{1}{2}\nabla\cdot\boldsymbol{c},v^{2})$
$\displaystyle-(\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c},v^{2})_{\Gamma_{1}}+\frac{1}{2}(\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c},v^{2})_{\Gamma_{2,1}}+(\tilde{\alpha}-\frac{1}{2}\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c},v^{2})_{\Gamma_{2}\setminus\Gamma_{2,1}}\quad\text{for
all }v\in V.$

######  Remark 7.

The above choice of boundary conditions (17) together with the requirements
(18), especially the sign condition to
$\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}$ on $\Gamma_{1}$ in (18),3),
does not include the possibility to prescribe boundary data for
$(\boldsymbol{K}\nabla u-\boldsymbol{c}\,u)\cdot\boldsymbol{{\mathfrak{n}}}$
at an outflow boundary.

At first glance this seems physically questionable, but that there are
arguments underlying this fact in two extreme situations, namely the
diffusion-dominated and the convection-dominated regimes. First we note that
the boundary term on $\Gamma_{1}$ in the above identity can be controlled
thanks to the boundedness of the trace operator $V\to L^{2}(\Gamma_{1})$
[BS08, Thm. 1.6.6]. That is, if $k_{0}$ in (16) is sufficiently large in
comparison with the $L^{\infty}$-norm of
$\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}$ on $\Gamma_{1}$ (“diffusion-
dominated regime near $\Gamma_{1}$”), the positive definiteness of the
bilinear form $a$ on $V$ can be preserved even if
$\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}>0$ on $\Gamma_{1}$. In the
contrary case, if the $L^{\infty}$-norm of $\boldsymbol{K}$ on $\Omega$ is
very small in comparison with the $L^{\infty}$-norm of
$\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}$ on $\Gamma_{1}$ (“convection-
dominated regime near $\Gamma_{1}$”), the problem is almost elliptically
degenerate, and in such a case it is not appropriate to prescribe boundary
data at an outflow (“noncharacteristic”) boundary. In practice, a so-called
do-nothing boundary condition, which is more or less artificial, is often
prescribed at an outflow boundary in order to avoid boundary layer effects.
Therefore, in the convection-dominated case, outflow boundary conditions can
and have to be be modeled via $\Gamma_{2}$.

In the next step, to formulate the needed regularity conditions to the problem
(15)–(17), and to describe the discretization, we introduce a family
$(\mathcal{T}_{h})_{h}$ of consistent partitions of the domain $\Omega$. Given
an _admissible partition_ $\mathcal{T}:=\mathcal{T}_{h}$ of $\Omega$ (in the
sense of [Cia02, (FEM 1)], where we omit the subscript $h$ for simplicity of
notation), it is called _consistent_ , if the following additional properties
are met:

  * •

Every face $F$ of an element $K\in\mathcal{T}$ is either a subset of the
boundary ${\partial\Omega}$ of $\Omega$ or identical to a face of another
element $K^{\prime}\in\mathcal{T}$.

  * •

Each of the boundary subsets $\Gamma_{j}$ is related to a set of faces
$\mathcal{F}_{j}$ as follows:

$\text{cl}_{\text{\,rel}}\big{(}\Gamma_{j}\big{)}=\bigcup_{F\in\mathcal{F}_{j}}F,\quad
j\in\\{1,2,3\\},$

where $\text{cl}_{\text{\,rel}}$ denotes the closure of a boundary subset in
the relative topology of ${\partial\Omega}$.

To simplify the notation in the further analysis, we denote the set of all
faces of all elements of $\mathcal{T}$ by $\overline{\mathcal{F}}$, the set of
those faces that are lying on the boundary ${\partial\Omega}$ by
$\partial\mathcal{F}$, and the set of all faces of an element
$K\in\mathcal{T}$ by $\mathcal{F}_{K}$. Hence
$\mathcal{F}:=\overline{\mathcal{F}}\setminus\partial\mathcal{F}$ is the set
of all interior faces.

Furthermore we introduce _jumps_ and _averages_ of piecewise defined functions
as follows. Let $F\in\mathcal{F}_{K}\cap\mathcal{F}_{K^{\prime}}\neq\emptyset$
be an interior face of the partition $\mathcal{T}$, $K$,
$K^{\prime}\in\mathcal{T}$, $K\neq K^{\prime}$. By
$\boldsymbol{{\mathfrak{n}}}_{K}$ we denote the unit outer normal on the
boundary $\partial K$ of an element $K\in\mathcal{T}$. For scalar functions
$v:\;\overline{\Omega}\to\mathbb{R}$ such that $v|_{K}\in H^{1}(K)$,
$v|_{K^{\prime}}\in H^{1}(K^{\prime})$, we set

$\displaystyle{\llbracket v\rrbracket}$ $\displaystyle:={\llbracket
v\rrbracket}_{F}$ $\displaystyle\,:=$ $\displaystyle
v|_{K}\,\boldsymbol{{\mathfrak{n}}}_{K}+v|_{K^{\prime}}\,\boldsymbol{{\mathfrak{n}}}_{K^{\prime}}\,,$
$\displaystyle\\{|{v}|\\}$ $\displaystyle:=\\{|{v}|\\}_{F}$
$\displaystyle\,:=$ $\displaystyle\frac{1}{2}(v|_{K}+v|_{K^{\prime}})\,,$

where here and in the subsequent relationships (19)–(21) the terms on right-
hand sides are to be understood in the sense of traces on the face
$\mathcal{F}$. For vector fields
$\boldsymbol{p}:\;\overline{\Omega}\to\mathbb{R}^{d}$ such that
$\boldsymbol{p}|_{K}\in H(\operatorname{div};K)$,
$\boldsymbol{p}|_{K^{\prime}}\in H(\operatorname{div};K^{\prime})$, we define

$\displaystyle{\llbracket\boldsymbol{p}\rrbracket}$
$\displaystyle:={\llbracket\boldsymbol{p}\rrbracket}_{F}$ $\displaystyle\,:=$
$\displaystyle\boldsymbol{p}|_{K}\cdot\boldsymbol{{\mathfrak{n}}}_{K}+\boldsymbol{p}|_{K^{\prime}}\cdot\boldsymbol{{\mathfrak{n}}}_{K^{\prime}}\,,$
(19) $\displaystyle\\{|{\boldsymbol{p}}|\\}$
$\displaystyle:=\\{|{\boldsymbol{p}}|\\}_{F}$ $\displaystyle\,:=$
$\displaystyle\frac{1}{2}(\boldsymbol{p}|_{K}+\boldsymbol{p}|_{K^{\prime}})\,.$

For a boundary face $F\in\partial\mathcal{F}\cap\mathcal{F}_{K}$,
$K\in\mathcal{T}$, we define

${\llbracket
v\rrbracket}_{F}:=v|_{K}\,\boldsymbol{{\mathfrak{n}}}_{K},\quad\\{|{v}|\\}_{F}:=v|_{K},\quad{\llbracket\boldsymbol{p}\rrbracket}_{F}:=\boldsymbol{p}|_{K}\cdot\boldsymbol{{\mathfrak{n}}}_{K},\quad\\{|{\boldsymbol{p}}|\\}_{F}:=\boldsymbol{p}|_{K}.$
(20)

Note that jumps of scalar functions are vector fields and vice versa, but
averages keep the type. A very useful relation between jumps and averages on
interior faces is the so-alled _magic formula_ :

$\\{|{v}|\\}{\llbracket\boldsymbol{p}\rrbracket}+{\llbracket
v\rrbracket}\\{|{\boldsymbol{p}}|\\}=v|_{K}\,\boldsymbol{p}|_{K}\cdot\boldsymbol{{\mathfrak{n}}}_{K}+v|_{K^{\prime}}\,\boldsymbol{p}|_{K^{\prime}}\cdot\boldsymbol{{\mathfrak{n}}}_{K^{\prime}}.$
(21)

Finally, given a family of partitions of the domain $\Omega$, we define by
$W^{k,q}(\mathcal{T})$, $k\in\mathbb{N}$, $q\geq 1$, the _broken Sobolev
space_

$W^{k,q}(\mathcal{T}):=\\{v\in L^{2}(\Omega)\;|\;v|_{K}\in
W^{k,q}(K)\quad\text{for all }K\in\mathcal{T}\\}$

equipped with the norm

$\|v\|_{k,q,\mathcal{T}}:=\Big{(}\sum_{K\in\mathcal{T}}\|v\|_{k,q,K}^{q}\Big{)}^{1/q}\quad\text{for
}q\in[1,\infty)$

and

$\|v\|_{k,\infty,\mathcal{T}}:=\max_{K\in\mathcal{T}}\|v\|_{k,\infty,K}\quad\text{for
}q=\infty.$

As usual, by

$|v|_{k,q,\mathcal{T}}:=\Big{(}\sum_{K\in\mathcal{T}}|v|_{k,q,K}^{q}\Big{)}^{1/q}\quad\text{for
}q\in[1,\infty)$

and

$|v|_{k,\infty,\mathcal{T}}:=\max_{K\in\mathcal{T}}|v|_{k,\infty,K}\quad\text{for
}q=\infty$

the corresponding seminorms are denoted. In the case $q=2$ we use the standard
notations $H^{k}(\mathcal{T}):=W^{k,2}(\mathcal{T})$,
$\|v\|_{k,\mathcal{T}}:=\|v\|_{k,2,\mathcal{T}}$,
$|v|_{k,\mathcal{T}}:=|v|_{k,2,\mathcal{T}}$.

### Regularity assumptions

Basically, we assume that the problem has a unique weak solution $u\in
H^{1}(\Omega)$. This can be guaranteed by making additional assumptions to
(18), see, e. g., [KA21, Thm. 3.16]. To investigate the consistency error,
however, we need additional regularity assumptions (which are also additional
requirements to the coefficients and source terms of (15)–(17)):

$\boldsymbol{K}\nabla u\in H(\operatorname{div};\Omega),\ cu\in
H^{1}(\Omega).$ (22)

Note that, as consequence of (22),

$\displaystyle{\llbracket\boldsymbol{K}\nabla
u-\boldsymbol{c}u\rrbracket}_{F}=0\quad\text{for all }F\in\mathcal{F},$ (23)

i. e., there is the inter-element continuity of the normal flux densities.

##  4 Example I: The Crouzeix-Raviart Discretization

In this section we consider _shape-regular_ (i. e., _regular_ in the sense of
[Cia02, Sect. 3.1.1]) families of consistent simplicial partitions of
$\Omega$. For the approximation spaces we choose the global _Crouzeix-Raviart
space_ of degree one:

$CR_{1}(\Omega):=\\{v\in L^{1}(\Omega)\;|\;v|_{K}\in\mathcal{P}_{1}(K)\text{
and }\int_{F}{\llbracket v\rrbracket}\,d\sigma=\boldsymbol{0}\text{ for all
}F\in\mathcal{F}\\},$ (24)

where $\mathcal{P}_{1}(K)$ denotes the set of polynomials of degree one on
$K$. Note that, in general, $v\in V_{h}$ is bi-valued on $\mathcal{F}$. With
that we define

$U_{h}:=V_{h}:=\Big{\\{}v\in
CR_{1}(\Omega)\;|\;\int_{F}v\,d\sigma=0\quad\text{for all
}F\in\mathcal{F}_{3}\Big{\\}}.$ (25)

Now, to formulate a (nonconsistent) finite element method for (15)–(17), we
introduce the forms

$\begin{split}a_{h}(w,v)&:=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla
w-\boldsymbol{c}w,\nabla
v)_{K}+(rw,v)+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(w),{\llbracket
v\rrbracket})_{F}+(\tilde{\alpha}w,v)_{\Gamma_{2}},\\\
\ell_{h}(v)&:=\ell(v)\qquad\text{for all }w,v\in
U(h):=V(h):=H^{1}(\mathcal{T}),\end{split}$ (26)

where the _upwind evaluation_ $\boldsymbol{c}_{\mathrm{upw}}(w)$ of
$\boldsymbol{c}w$ at interior faces
$F\in\mathcal{F}_{K}\cap\mathcal{F}_{K^{\prime}}\neq\emptyset$ is defined
pointwise as follows:

$\displaystyle\boldsymbol{c}_{\mathrm{upw}}(w):=\begin{cases}\boldsymbol{c}w|_{K}&\text{for
}\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}_{K}>0,\\\
\boldsymbol{c}w|_{K^{\prime}}&\text{for
}\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}_{K}\leq 0,\end{cases}$ (27)

and at boundary faces $F\in\partial\mathcal{F}$ we set

$\displaystyle\boldsymbol{c}_{\mathrm{upw}}(w):=\begin{cases}\boldsymbol{c}w|_{K}&\text{for
}\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}_{K}>0,\\\ 0&\text{for
}\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}_{K}\leq 0.\end{cases}$ (28)

Then the discrete method reads as follows:

$\text{Find }u_{h}\in V_{h}\text{ such that }\quad
a_{h}(u_{h},v_{h})=\ell_{h}(v_{h})\quad\text{for all }v_{h}\in V_{h}\,.$ (29)

At first we study the coercivity of the bilinear form $a_{h}$. So let $v\in
H^{1}(\mathcal{T})$. Starting from

$\displaystyle a_{h}(v,v)$
$\displaystyle=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla v,\nabla
v)_{K}+(rv,v)+(\tilde{\alpha}v,v)_{\Gamma_{2}}$
$\displaystyle\qquad-\sum_{K\in\mathcal{T}}(\boldsymbol{c}v,\nabla
v)_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F},$

we treat the fourth term as follows :

$\displaystyle-\sum_{K\in\mathcal{T}}(\boldsymbol{c}v,\nabla v)_{K}$
$\displaystyle=-\frac{1}{2}\sum_{K\in\mathcal{T}}(\boldsymbol{c},\nabla
v^{2})_{K}$
$\displaystyle=\frac{1}{2}\sum_{K\in\mathcal{T}}((\nabla\cdot\boldsymbol{c})v,v)_{K}-\frac{1}{2}\sum_{F\in\partial\mathcal{F}}(\boldsymbol{c}v,\boldsymbol{{\mathfrak{n}}}v)_{F}-\frac{1}{2}\sum_{F\in\mathcal{F}}(\boldsymbol{c},{\llbracket
v^{2}\rrbracket})_{F}.$

This gives

$\displaystyle a_{h}(v,v)$
$\displaystyle=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla v,\nabla
v)_{K}+\Big{(}\Big{(}r+\frac{1}{2}\nabla\cdot\boldsymbol{c}\Big{)}v,v\Big{)}+(\tilde{\alpha}v,v)_{\Gamma_{2}}-\frac{1}{2}\sum_{F\in\partial\mathcal{F}}(\boldsymbol{c}v,\boldsymbol{{\mathfrak{n}}}v)_{F}$
$\displaystyle\quad+\sum_{F\in\mathcal{F}}\Big{[}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}-\frac{1}{2}(\boldsymbol{c},{\llbracket
v^{2}\rrbracket})_{F}\Big{]}+\sum_{F\in\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}$ $\displaystyle\geq\sum_{K\in\mathcal{T}}k_{0}\|\nabla
v\|_{K}^{2}-\frac{1}{2}\sum_{F\in\mathcal{F}_{1}}(\boldsymbol{c}v,\boldsymbol{{\mathfrak{n}}}v)_{F}+\sum_{F\in\mathcal{F}_{2}}\Big{(}\Big{(}\tilde{\alpha}-\frac{1}{2}\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}\Big{)}v,v\Big{)}_{F}$
$\displaystyle\quad+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}-\frac{1}{2}(\boldsymbol{c},{\llbracket
v^{2}\rrbracket})_{F}\Big{]}$
$\displaystyle\geq\sum_{K\in\mathcal{T}}k_{0}\|\nabla
v\|_{K}^{2}+\frac{1}{2}\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(|\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}|,{\llbracket
v\rrbracket}^{2})_{F}\geq k_{0}|v|_{1,\mathcal{T}}^{2},$

where we have used the properties (18), (27), and (28).

If we now include additional conditions (which are similar to the conditions
mentioned at the beginning of the subsection about the regularity assumptions,
but a little more stringent), we obtain the $(V+V_{h})$-coercivity of $a_{h}$
uniform in $h$. Namely, we assume that, in addition to the conditions (18),
one of the following conditions is satisfied:

a) $\displaystyle|\Gamma_{3}|_{d-1}>0.$ (30) b) $\displaystyle\text{There
exists some constant }r_{0}>0\text{ such that
}r+\frac{1}{2}\nabla\cdot\boldsymbol{c}\geq r_{0}\text{ on }\Omega.$ (31)

Indeed, the case b) immediately implies the estimate $a_{h}(v,v)\geq
k_{0}|v|_{1,\mathcal{T}}^{2}+r_{0}\|v\|_{0,\Omega}^{2}\geq\min\\{k_{0};r_{0}\\}\|v\|_{1,\mathcal{T}}^{2}$
for all $v\in H^{1}(\mathcal{T})$.

To verify the case a) for $v\in V$ we make use of the Poincaré-Friedrichs
inequality [BS08, Exercise 5.x.13], i. e., there exists a constant
$C_{\mathrm{PF}}>0$ such that

$\|v\|_{0}\leq
C_{\mathrm{PF}}|v|_{1}=C_{\mathrm{PF}}|v|_{1,\mathcal{T}}\quad\text{for all
}v\in V.$

If $v_{h}\in V_{h}$, we make use of a discrete version of this inequality
which can be proven analogously to the proof of [BS08, Thm. (10.6.12)], i. e.,

$\|v_{h}\|_{0}\leq\tilde{C}_{\mathrm{PF}}|v_{h}|_{1,\mathcal{T}}\quad\text{for
all }v_{h}\in V_{h}$

with some constant $\tilde{C}_{\mathrm{PF}}>0$ independent of $h$. Since
$\|v+v_{h}\|_{V+V(h)}:=\inf\\{\|v\|_{0,\Omega}+\|v_{h}\|_{0,\Omega}\\}$ is a
norm of $v+v_{h}$ considered as an element of the subspace $V+V_{h}\subset
L^{2}(\Omega)$, the combination of the above estimates shows that
$|v|_{1,\mathcal{T}}$ itself is already a norm on $V+V_{h}$. Hence
$a_{h}(v,v)\geq k_{0}|v|_{1,\mathcal{T}}^{2}$ is the required coercivity
estimate.

Next we investigate the consistency error. For $u\in V\cap
H^{2}(\mathcal{T})$, satisfying the regularity conditions (22) (this defines
the regularity space $W$), and $v_{h}\in V_{h}$, we have:

$\displaystyle a_{h}(u,v_{h})-\ell_{h}(v_{h})$ (32)
$\displaystyle=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla
u-\boldsymbol{c}u,\nabla
v_{h})_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u),{\llbracket
v_{h}\rrbracket})_{F}+(ru-f,v_{h})$
$\displaystyle\quad-(g_{1},v_{h})_{\Gamma_{1}}+(\tilde{\alpha}u-g_{2},v_{h})_{\Gamma_{2}}.$

By integration by parts, the diffusion term can be rewritten as

$\displaystyle\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla u,\nabla v_{h})_{K}$
$\displaystyle=-\sum_{K\in\mathcal{T}}(\nabla\cdot(\boldsymbol{K}\nabla
u),v_{h})_{K}$
$\displaystyle+\sum_{F\in\mathcal{F}}\big{[}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket v_{h}\rrbracket})_{F}+({\llbracket\boldsymbol{K}\nabla
u\rrbracket},\\{|{v_{h}}|\\})_{F}\big{]}+\sum_{F\in\partial\mathcal{F}}(\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{K}\nabla
u,v_{h})_{F},$

where we have used (21). A rearrangement of the last three terms, taking into
consideration the definition (20) of the boundary jumps and averages, yields

$\displaystyle\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla u,\nabla v_{h})_{K}$
$\displaystyle=-\sum_{K\in\mathcal{T}}(\nabla\cdot(\boldsymbol{K}\nabla
u),v_{h})_{K}$ (33)
$\displaystyle\quad+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{1}\cup\mathcal{F}_{2}}({\llbracket\boldsymbol{K}\nabla
u\rrbracket},\\{|{v_{h}}|\\})_{F}.$

The next terms to consider are the convection terms. Using integration by
parts in the terms over $K\in\mathcal{T}$ together with (21), we get

$\displaystyle-\sum_{K\in\mathcal{T}}(\boldsymbol{c}u,\nabla
v_{h})_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u),{\llbracket
v_{h}\rrbracket})_{F}$ (34)
$\displaystyle=(\nabla\cdot(\boldsymbol{c}u),v_{h})+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u),{\llbracket
v_{h}\rrbracket})_{F}$
$\displaystyle\quad-\sum_{F\in\mathcal{F}}\big{[}(\\{|{\boldsymbol{c}u}|\\},{\llbracket
v_{h}\rrbracket})_{F}+({\llbracket\boldsymbol{c}u\rrbracket},\\{|{v_{h}}|\\})_{F}\big{]}-\sum_{F\in\partial{F}}(\boldsymbol{c}u,\boldsymbol{{\mathfrak{n}}}v_{h})_{F}$
$\displaystyle=(\nabla\cdot(\boldsymbol{c}u),v_{h})+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u)-\\{|{\boldsymbol{c}u}|\\},{\llbracket
v_{h}\rrbracket})_{F}$
$\displaystyle\quad-\sum_{F\in\mathcal{F}}({\llbracket\boldsymbol{c}u\rrbracket},\\{|{v_{h}}|\\})_{F}-\sum_{F\in\mathcal{F}_{1}\cup\mathcal{F}_{2}}(\boldsymbol{c}u,\boldsymbol{{\mathfrak{n}}}v_{h})_{F}.$

Since ${\llbracket u\rrbracket}=\boldsymbol{0}$ on $\mathcal{F}$ thanks to
$u\in H^{1}(\Omega)$, see [DPE12, Lemma 1.23], we have

$\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u)-\\{|{\boldsymbol{c}u}|\\},{\llbracket
v_{h}\rrbracket})_{F}=\sum_{F\in\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(u)-\boldsymbol{c}u,\boldsymbol{{\mathfrak{n}}}v_{h})_{F}\,.$

The latter term vanishes because of $u=0$ on $\Gamma_{3}$. Inserting (33),
(34) into (32), we arrive at

$\displaystyle a_{h}(u,v_{h})-\ell_{h}(v_{h})$
$\displaystyle=-\sum_{K\in\mathcal{T}}(\nabla\cdot(\boldsymbol{K}\nabla
u),v_{h})_{K}+\sum_{F\in\mathcal{F}}({\llbracket\boldsymbol{K}\nabla
u-\boldsymbol{c}u\rrbracket},\\{|{v_{h}}|\\})_{F}$
$\displaystyle\quad+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{1}\cup\mathcal{F}_{2}}({\llbracket\boldsymbol{K}\nabla
u\rrbracket},\\{|{v_{h}}|\\})_{F}$
$\displaystyle\quad+(\nabla\cdot(\boldsymbol{c}u),v_{h})-\sum_{F\in\mathcal{F}_{1}\cup\mathcal{F}_{2}}(\boldsymbol{c}u,\boldsymbol{{\mathfrak{n}}}v_{h})_{F}$
$\displaystyle\quad+(ru-f,v_{h})-(g_{1},v_{h})_{\Gamma_{1}}+(\tilde{\alpha}u-g_{2},v_{h})_{\Gamma_{2}}$
$\displaystyle=(-\nabla\cdot(\boldsymbol{K}\nabla
u-\boldsymbol{c}u)+ru-f,v_{h})+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket v_{h}\rrbracket})_{F}$
$\displaystyle\quad+\sum_{F\in\mathcal{F}_{1}\cup\mathcal{F}_{2}}(\boldsymbol{K}\nabla
u-\boldsymbol{c}u,\boldsymbol{{\mathfrak{n}}}v_{h})_{F}-(g_{1},v_{h})_{\Gamma_{1}}+(\tilde{\alpha}u-g_{2},v_{h})_{\Gamma_{2}},$

where we have used (23). Because the differential equation and the boundary
conditions on $\Gamma_{1}\cup\Gamma_{2}$ are satisfied in the sense of
$L^{2}(\Omega)$ and $L^{2}(\Gamma_{1}\cup\Gamma_{2})$, respectively, the first
term and the sum of the last three terms vanish, hence

$\displaystyle a_{h}(u,v_{h})-\ell_{h}(v_{h})$
$\displaystyle=\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket v_{h}\rrbracket})_{F}$
$\displaystyle=\sum_{F\in\mathcal{F}}\Big{(}\\{|{\boldsymbol{K}\nabla
u}|\\}-\frac{1}{2}\big{[}\Pi_{K}(\boldsymbol{K}\nabla
u)+\Pi_{K^{\prime}}(\boldsymbol{K}\nabla u)\big{]},{\llbracket
v_{h}\rrbracket}\Big{)}_{F}$
$\displaystyle\quad+\sum_{F\in\mathcal{F}_{3}}\Big{(}\boldsymbol{K}\nabla
u-\Pi_{K}(\boldsymbol{K}\nabla u),v_{h}\Big{)}_{F},$

where each $\Pi_{K}(\boldsymbol{K}\nabla
u):=|K|^{-1}\int\limits_{K}\boldsymbol{K}\nabla u\,d\sigma$, $K\supset F$, is
a constant vector, and thus the added contribution vanishes due to the weak
continuity condition in the definition of $CR_{1}(\Omega)$ and the weak
homogeneous Dirichlet boundary condition (see (24), (25)).

Therefore, by the Cauchy–Schwarz inequality,

$|a_{h}(u,v_{h})-\ell_{h}(v_{h})|\leq\Big{(}\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}h_{F}\big{\|}\boldsymbol{K}\nabla
u-\Pi_{K}(\boldsymbol{K}\nabla
u)\big{\|}_{0,F}^{2}\Big{)}^{1/2}\Big{(}\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}h_{F}^{-1}\big{\|}{\llbracket
v_{h}\rrbracket}\big{\|}_{0,F}^{2}\Big{)}^{1/2}.$ (35)

According to the multiplicative trace inequality [DF15, Lemma 2.19]

$\|v\|_{0,\partial K}^{2}\leq
C\big{[}\|v\|_{0,K}|v|_{1,K}+h_{K}^{-1}\|v\|_{0,K}^{2}\big{]}\quad\text{for
all }v\in H^{1}(K),\ K\in\mathcal{T},$

a standard error estimate for $L^{2}$-projections (see, e. g., [DF15, Lemma
2.24]), and

$C_{1}h_{K}\leq h_{F}\leq C_{2}h_{K}\quad\text{for all }F\in\mathcal{F}_{K},\
K\in\mathcal{T},$ (36)

with constants $C_{1},C_{2}>0$ independent of $h_{K}$, the first factor in
(35) allows the estimate

$Ch|\boldsymbol{K}\nabla u|_{1,\mathcal{T}}.$

Denoting by $a_{S,F}$ the barycentre of the face $F$, the summand of the
second factor can be written as

$h_{F}^{-1}\big{\|}{\llbracket v_{h}-v(a_{S,F})\rrbracket}\big{\|}^{2}_{0,F},$

so that $\big{[}v_{h}-v(a_{S,F})\big{]}\big{|}_{K}$ and
$\big{[}v_{h}-v(a_{S,F})\big{]}\big{|}_{K^{\prime}}$ vanish at the same point
in $F\in\mathcal{F}_{K}\cap\mathcal{F}_{K^{\prime}}\neq\emptyset$. Thus the
scaled trace inequality (see, e. g., [KA21, Lemma 7.5])

$\|v\|_{0,F}\leq Ch_{K}^{1/2}|v|_{1,K}\quad\text{for all }v\in H^{1}(K)$

is applicable, leading together with (36) to the following estimate of the
second factor:

$\Big{(}\sum_{F=K\cap
K^{\prime}\in\mathcal{F}}\big{[}h_{K}^{-1}h_{K}|v_{h}|_{1,K}^{2}+h_{K^{\prime}}^{-1}h_{K^{\prime}}|v_{h}|_{1,K}^{2}\big{]}+\sum_{F=K\cap\Gamma_{3}\in\mathcal{F}_{3}}h_{K}^{-1}h_{K}|v_{h}|_{1,K}^{2}\Big{)}^{1/2}\leq
C|v_{h}|_{1,\mathcal{T}}.$

Putting the obtained estimates together, we arrive at the following estimate
of the consistency error:

$|a_{h}(u,v_{h})-\ell_{h}(v_{h})|\leq Ch|\boldsymbol{K}\nabla
u|_{1,\mathcal{T}}|v_{h}|_{1,\mathcal{T}}.$ (37)

In order to apply Theorem 3, we still have to estimate the approximation order
of $V_{h}$. Since the space of conforming $\mathcal{P}_{1}$-elements is a
subspace of $V_{h}$, it follows, for a sufficiently smooth solution $u\in W\
(\subset H^{2}(\mathcal{T}))$, that

$\inf_{v_{h}\in V_{h}}\|u-v_{h}\|_{1,\mathcal{T}}\leq
Ch|u|_{2,\mathcal{T}}\,.$

In summary, we have proved the following result.

######  Theorem 8.

Let the family of triangulations be shape-regular, the weak solution $u\in
V\cap H^{2}(\mathcal{T})$ be such that (22) is satisfied, and $\boldsymbol{K}$
so smooth that $\boldsymbol{K}\nabla u\in H^{1}(\mathcal{T})^{d}$. Then, under
the conditions (18), (30) or (18), (31), the Crouzeix-Raviart discretization
of order $1$, i. e., the solution $u_{h}$ of the discrete problem (24)–(29),
satisfies the error estimate

$\|u-u_{h}\|_{1,\mathcal{T}}\leq
Ch\big{[}|u|_{2,\mathcal{T}}+|\boldsymbol{K}\nabla u|_{1,\mathcal{T}}\big{]},$

where $C>0$ is a constant independent of $h$.

The error bound can be simplified if additional smoothness of $\boldsymbol{K}$
is assumed.

######  Corollary 9.

Let the family of triangulations be shape-regular, $\boldsymbol{K}\in
W^{1,\infty}(\mathcal{T})$ and the weak solution $u\in V\cap
H^{2}(\mathcal{T})$ be such that (22) is satisfied. Then, under the conditions
(18), (30) or (18), (31), the Crouzeix-Raviart discretization of order $1$, i.
e., the solution $u_{h}$ of the discrete problem (24)–(29), satisfies the
error estimate

$\|u-u_{h}\|_{1,\mathcal{T}}\leq Ch|u|_{2,\mathcal{T}},$

where $C>0$ is a constant independent of $h$.

So we have seen that the effect of including inter-element boundary terms in
the variational formulation is to guarantee the coercivity. They have no
influence on the consistency error.

### Order of Convergence Estimates in Weaker Norms

To apply Theorem 5, the adjoint problem and its discretization have to be
studied. The adjoint problem (6) with $g\in Z:=L^{2}(\Omega)$ is given by the
forms

$\displaystyle a^{*}(v,w)$ $\displaystyle:=a(w,v)=(\boldsymbol{K}\nabla
w-\boldsymbol{c}w,\nabla v)+(rw,v)+(\tilde{\alpha}w,v)_{\Gamma_{2}}$ (38)
$\displaystyle=(\boldsymbol{K}\nabla v,\nabla w)-(\boldsymbol{c}\cdot\nabla
v,w)+(rv,w)+(\tilde{\alpha}v,w)_{\Gamma_{2}},$ $\displaystyle\tilde{\ell}(w)$
$\displaystyle:=(w,g)\quad\text{for all }v,w\in V,$

hence the adjoint problem corresponds to the following formal boundary value
problem in nondivergence form:

$\displaystyle-\nabla\cdot\left(\boldsymbol{K}\nabla
v\right)-\boldsymbol{c}\cdot\nabla v+rv$ $\displaystyle=g\quad\text{in
}\Omega,$ (39) $\displaystyle\boldsymbol{K}\nabla
v\cdot\boldsymbol{{\mathfrak{n}}}$ $\displaystyle=0\quad\text{on
}\Gamma_{1}\,,$ $\displaystyle\boldsymbol{K}\nabla
v\cdot\boldsymbol{{\mathfrak{n}}}+\tilde{\alpha}v$
$\displaystyle=0\quad\text{on }\Gamma_{2}\,,$ $\displaystyle v$
$\displaystyle=0\quad\text{on }\Gamma_{3}\,.$

Analogous to the continuous case, we define the discrete adjoint problem as
the adjoint of the discrete problem with the bilinear form (26):

$a_{h}^{*}(v,w):=a_{h}(w,v):=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla
w-\boldsymbol{c}w,\nabla
v)_{K}+(rw,v)+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(w),{\llbracket
v\rrbracket})_{F}+(\tilde{\alpha}w,v)_{\Gamma_{2}}.$

Obviously, the $V_{h}$-coercivity constant of $a_{h}^{*}$ is the same as that
of $a_{h}$.

The consistency error

$\displaystyle\quad a_{h}^{*}(v,w_{h})-\tilde{\ell}(w_{h})$
$\displaystyle=\sum_{K\in\mathcal{T}}(\boldsymbol{K}\nabla
w_{h}-\boldsymbol{c}w_{h},\nabla
v)_{K}+(rw_{h},v)+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(w_{h}),{\llbracket
v\rrbracket})_{F}+(\tilde{\alpha}w_{h},v)_{\Gamma_{2}}-(g,w_{h})$

can be splitted into the consistency error of the symmetric part (cf. (33) for
the relevant diffusion term) and the nonsymmetric part

$-\sum_{K\in\mathcal{T}}(\boldsymbol{c}w_{h},\nabla
v)_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(w_{h}),{\llbracket
v\rrbracket})_{F}=-(\boldsymbol{c}w_{h},\nabla
v)+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{c}_{\mathrm{upw}}(w_{h}),{\llbracket
v\rrbracket})_{F}.$

The first term contributes directly to the strong form of the differential
equation, whereas the second term vanishes. Therefore the consistency error
estimation is reduced to the estimation of the consistency error of the
symmetric part. Thanks to symmetry, the estimate can be taken directly from
(37) in the proof of Theorem 8. Thus the error estimate of Theorem 8 also
holds true for the adjoint problem.

In order to apply Theorem 5 we have to assume that the adjoint problem is
regular in the sense that for every $g\in L^{2}(\Omega)$ there exists a unique
solution $v_{g}\in V\cap H^{2}(\mathcal{T})$ of the adjoint boundary value
problem (6), (7) such that, for some constant $\tilde{C}>0$, a stability
estimate of the form

$|v_{g}|_{2,\mathcal{T}}\leq\tilde{C}\|g\|_{0}\quad\text{for given }g\in
L^{2}(\Omega),$ (40)

is satisfied (i. e., (11) holds with $Y:=V\cap H^{2}(\mathcal{T})$), the first
term in the estimate of Theorem 5 can be estimated by
$Ch^{2}|u|_{2,\mathcal{T}}$, and the fourth term vanishes. The third term
being a consistency error for the original problem, can be estimated similar
to (35) and the following considerations, but with $v_{h}$ substituted by
$v_{g}-v_{gh}$, the approximation error of the adjoint problems (6), (8).

Thus the final estimate reads (compare (37) in the proof of Theorem 8):

$Ch|\boldsymbol{K}\nabla
u|_{1,\mathcal{T}}\,|v_{g}-v_{gh}|_{1,\mathcal{T}}\leq
Ch^{2}|\boldsymbol{K}\nabla u|_{1,\mathcal{T}}\,|v_{g}|_{2,\mathcal{T}}\leq
Ch^{2}|\boldsymbol{K}\nabla u|_{1,\mathcal{T}}\|g\|_{0,\Omega},$

giving the desired estimate.

Finally the second term, being a consistency error of the adjoint problem, can
be handled similarly with exchanged roles $u\leftrightarrow v_{g}$,
$v_{g}-v_{gh}\leftrightarrow u-u_{h}$, leading to the bound

$Ch|\boldsymbol{K}\nabla v_{g}|_{1,\mathcal{T}}\,|u-u_{h}|_{1,\mathcal{T}}.$

But since we only have the estimate (40) of the
$|\cdot|_{2,\mathcal{T}}$-seminorm of $v_{g}$, we need an additional
assumption, for instance a regularity requirement to $\boldsymbol{K}$ as in
Corollary 9. Under this assumption we can apply Corollary 9 to estimate both
consistency errors. In summary, we can formulate the following result.

######  Theorem 10.

Let the family of triangulations be shape-regular, $\boldsymbol{K}\in
W^{1,\infty}(\mathcal{T})$, the weak solution $u\in V\cap H^{2}(\mathcal{T})$
be such that (22) is satisfied, and the solution of the adjoint problem (6) be
regular such that (40) is satisfied. Then, under the conditions (18), (30) or
(18), (31), the Crouzeix-Raviart discretization of order $1$ satisfies the
error estimate

$\|u-u_{h}\|_{0,\Omega}\leq Ch^{2}|u|_{2,\mathcal{T}},$

where $C>0$ is a constant independent of $h$.

##  5 Example II: Discontinuous Galerkin Methods

Based on the setting of Section 3, here we consider more general consistent
partitions of $\Omega$ (not necessarily consisting of $d$-simplices only),
namely _shape- and contact-regular_ families of partitions, see [DPE12, Def.
1.38]. We also assume that all partitions are _compatible_ with the structure
of the boundary piece $\Gamma_{2}$ in the sense that there are subsets
$\mathcal{F}_{2,1},\mathcal{F}_{2,2}\subset\mathcal{F}_{2}$ such that the
following representation holds:

$\Gamma_{2,1}=\bigcup_{F\in\mathcal{F}_{2,1}}F,\quad\Gamma_{2}\setminus\Gamma_{2,1}=\bigcup_{F\in\mathcal{F}_{2,2}}F\setminus\Gamma_{2,1}.$

This requirement is for clarity of presentation only. In principle, it can be
omitted if the correct integration regions, which then do not have to be
complete faces, are specified for the corresponding integrations.

We use the finite element spaces

$U_{h}:=V_{h}:=\mathbb{P}_{k}(\mathcal{T}):=\\{w_{h}\in
L^{2}(\Omega)\;|\;w_{h}|_{K}\in\mathbb{P}_{k}(K)\quad\text{for all
}K\in\mathcal{T}\\},$

where $\mathbb{P}_{k}(K):=\mathcal{P}_{k}(K)$ or
$\mathbb{P}_{k}(K):=\mathcal{Q}_{k}(K)$, i. e., the _broken polynomial spaces_
of maximum degree $k$. We also allow for inhomogeneous Dirichlet boundary
conditions $u=g_{3}$ on $\Gamma_{3}$ with a nontrivial function $g_{3}$. For a
fixed _symmetrization parameter_ $\theta\in\\{0,\pm 1\\}$, the discrete method
reads as follows:

Find $u_{h}\in V_{h}$ such that

$a_{h}(u_{h},v_{h})=\ell_{h}(v_{h})\quad\text{for all }v_{h}\in V_{h},$ (41)

where

$\displaystyle a_{h}(u_{h},v_{h})$
$\displaystyle:=\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla
u_{h}-\boldsymbol{c}u_{h},\nabla v_{h})_{K}+(ru_{h},v_{h})_{K}\big{]}$ (42)
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}\theta(\\{|{\boldsymbol{K}\nabla
v_{h}}|\\},{\llbracket u_{h}\rrbracket})_{F}-\Big{(}\\{|{\boldsymbol{K}\nabla
u_{h}}|\\}-\frac{\eta}{h_{F}}{\llbracket u_{h}\rrbracket},{\llbracket
v_{h}\rrbracket}\Big{)}_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}_{\mathrm{upw}}(u_{h}),{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}u_{h},v_{h})_{F},$
$\displaystyle\ell_{h}(v_{h})$
$\displaystyle:=(f,v_{h})+\sum_{F\in\mathcal{F}_{1}}(g_{1},v_{h})_{F}+\sum_{F\in\mathcal{F}_{2}}(g_{2},v_{h})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{3}}\Big{[}\frac{\eta}{h_{F}}(g_{3},v_{h})_{F}+(\theta\boldsymbol{K}\nabla
v_{h}-\boldsymbol{c}v_{h},\boldsymbol{{\mathfrak{n}}}g_{3})_{F}\Big{]}.$

The choice $\theta=0$ gives the _incomplete IPG_ (_IIPG_) method, while
$\theta=1$ yields the _nonsymmetric IPG_ (_NIPG_) and $\theta=-1$ the
_symmetric IPG_ (_SIPG_) method. The symmetrization term neither destroys the
consistency (cf. the subsequent Lemma 11) of the method, nor introduces an
artificial numerical flux on interior faces. Note that the Dirichlet boundary
conditions are weakly imposed due to an idea by Nitsche.

Now we want to demonstrate a basic result that characterizes the IPG methods
as consistent but clearly nonconforming methods (cf. Section 2). Similar to
Section 4 to this end we assume sufficient regularity for the solution
expressed by (22).

######  Lemma 11 (Consistency of IPG methods).

If $u\in V\cap H^{2}(\mathcal{T})$ is the weak solution of the problem (15)
satisfying the regularity conditions (22) (this defines the regularity space
$W$), the IPG methods are consistent for $k\in\mathbb{N}$ in the sense that

$a_{h}(u,v_{h})=\ell_{h}(v_{h})\quad\text{for all }v_{h}\in V_{h}.$

###### Proof.

For $v_{h}\in V_{h}$ we have that

$\displaystyle a_{h}(u,v_{h})-\ell_{h}(v_{h})$ $\displaystyle=$
$\displaystyle\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla
u-\boldsymbol{c}u,\nabla
v_{h})_{K}+(ru,v_{h})_{K}\big{]}-\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket v_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{3}}\Big{[}\theta(\boldsymbol{K}\nabla
v_{h},\boldsymbol{{\mathfrak{n}}}u)_{F}+\Big{(}\frac{\eta}{h_{F}}u,v_{h}\Big{)}_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}u,{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}u,v_{h})_{F}$
$\displaystyle-(f,v_{h})-\sum_{F\in\mathcal{F}_{1}}(g_{1},v_{h})_{F}-\sum_{F\in\mathcal{F}_{2}}(g_{2},v_{h})_{F}$
$\displaystyle-\sum_{F\in\mathcal{F}_{3}}\Big{[}\frac{\eta}{h_{F}}(g_{3},v_{h})_{F}-(\theta\boldsymbol{K}\nabla
v_{h}-\boldsymbol{c}v_{h},\boldsymbol{{\mathfrak{n}}}g_{3})_{F}\Big{]}$
$\displaystyle=$
$\displaystyle\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla
u-\boldsymbol{c}u,\nabla
v_{h})_{K}+(ru,v_{h})_{K}\big{]}-\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u}|\\},{\llbracket v_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}u,{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}u,v_{h})_{F}$
$\displaystyle-(f,v_{h})-\sum_{F\in\mathcal{F}_{1}}(g_{1},v_{h})_{F}-\sum_{F\in\mathcal{F}_{2}}(g_{2},v_{h})_{F}+\sum_{F\in\mathcal{F}_{3}}(\boldsymbol{c}v_{h},\boldsymbol{{\mathfrak{n}}}g_{3})_{F},$

where the first equality uses that ${\llbracket u\rrbracket}=\boldsymbol{0}$
on $\mathcal{F}$ and the second equality holds due to the fact that $u=g_{3}$
on $F\in\mathcal{F}_{3}$. Now, elementwise integration by parts, as in (33)
and (34), and the use of the boundary conditions in (15) result in

$\displaystyle a_{h}(u,v_{h})$
$\displaystyle-\ell_{h}(v_{h})=(-\nabla\cdot\left(\boldsymbol{K}\nabla
u-\boldsymbol{c}u\right)+ru-f,v_{h})$ $\displaystyle+$
$\displaystyle(\left(\boldsymbol{K}\nabla
u-\boldsymbol{c}u\right)\cdot\boldsymbol{{\mathfrak{n}}}-g_{1},v_{h})_{\Gamma_{1}}+(\left(\boldsymbol{K}\nabla
u-\boldsymbol{c}u\right)\cdot\boldsymbol{{\mathfrak{n}}}+\tilde{\alpha}u-g_{2},v_{h})_{\Gamma_{2}}.$

The right-hand side vanishes since the differential equation in (15) is
satisfied in the sense of $L^{2}(\Omega)$, and the boundary conditions in the
sense of the corresponding trace spaces. ∎

######  Remark 12 (Relation to FEM and CR-elements).

  1. 1)

Using the IPG bilinear and linear forms and the standard (continuous) finite
element space $V_{h}\cap C^{0}(\overline{\Omega})$ results in a classical
finite element method with a different treatment of boundary conditions, since
all jumps on interfaces vanish due to the continuity of test and trial
functions.

  2. 2)

Compared to Section 4, the term

$\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\boldsymbol{K}\nabla u,{\llbracket
v_{h}\rrbracket})_{F}$

has been removed from the consistency error by including

$-\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u_{h}}|\\},{\llbracket v_{h}\rrbracket})_{F}$

to the bilinear form.

### Stability of IPG methods

Next, we have to ensure that the problem (41) allows for a unique solution we
can solve for in a stable manner. We try to show the uniform coercivity and
the boundedness of the bilinear form $a_{h}$ in terms of a certain energy-type
norm $\|\cdot\|_{V_{h}}$. To build this norm, we consider the NIPG bilinear
form with $\theta=1$, since it contains the minimum amount of summands which
all can be rewritten to be obviously coercive (remember that $\eta>0$). For
$v\in H^{1}(\mathcal{T})$, we have the identity

$\displaystyle a_{h}^{\text{NIPG}}(v,v)=$
$\displaystyle\sum_{K\in\mathcal{T}}\|\boldsymbol{K}^{1/2}\nabla
v\|^{2}_{0,K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\frac{\eta}{h_{F}}\|{\llbracket
v\rrbracket}\|^{2}_{0,F}$ (43)
$\displaystyle+\frac{1}{2}(2r+\nabla\cdot\boldsymbol{c},v^{2})+\frac{1}{2}\sum_{F\in\overline{\mathcal{F}}\setminus\mathcal{F}_{2,2}}(|\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}|,{\llbracket
v\rrbracket}^{2})_{F}$
$\displaystyle+\frac{1}{2}\sum_{F\in\mathcal{F}_{2,2}}(2\tilde{\alpha}-\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c},v^{2})_{F}=:\|v\|^{2}_{V_{h}},$

as can be seen as follows:

$\displaystyle-\sum_{K\in\mathcal{T}}(v,\boldsymbol{c}\cdot\nabla
v)_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{K\in\mathcal{T}}(\boldsymbol{c},\nabla
v^{2})_{K}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K\in\mathcal{T}}(\nabla\cdot\boldsymbol{c},v^{2})_{K}-\frac{1}{2}\sum_{F\in\mathcal{F}_{1}\cup\mathcal{F}_{2,2}\cup\mathcal{F}_{3}}(\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}},v^{2})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}\Big{[}(\boldsymbol{c}_{\mathrm{upw}}(v),{\llbracket
v\rrbracket})_{F}-\frac{1}{2}(\boldsymbol{c},{\llbracket
v^{2}\rrbracket})_{F}\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{K\in\mathcal{T}}(\nabla\cdot\boldsymbol{c},v^{2})_{K}+\frac{1}{2}\sum_{F\in\overline{\mathcal{F}}\setminus\mathcal{F}_{2,2}}(|\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}|,{\llbracket
v\rrbracket}^{2})_{F}-\frac{1}{2}\sum_{F\in\mathcal{F}_{2,2}}(\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}},v^{2})_{F}.$

The last step requires the same manipulations as in (34), the sign conditions
(18), and the additional condition

$\boldsymbol{c}\cdot\boldsymbol{{\mathfrak{n}}}\leq 0\quad\text{on
}\Gamma_{3}.$ (44)

######  Lemma 13.

Assume that, in addition to the conditions (18) and (44), one of the
conditions (30) or (31) is satisfied. Then, if $h>0$ is sufficiently small,
(43) defines a norm on $V+V_{h}$, and there exists a constant $C>0$
independent of $h$ such that

$\|v_{h}\|_{1,\mathcal{T}}\leq C\|v_{h}\|_{V_{h}}\quad\text{for all }v_{h}\in
V+V_{h}.$

###### Proof.

By assumptions, all terms in (43) are nonnegative, and it holds

$\|v\|_{V_{h}}^{2}\geq\sum_{K\in\mathcal{T}}\|\boldsymbol{K}^{1/2}\nabla
v\|^{2}_{0,K}\geq k_{0}|v|^{2}_{1,\mathcal{T}}.$

Thus, $\|\cdot\|_{V_{h}}$ is a seminorm on $H^{1}(\mathcal{T})$ and
$\|v\|_{V_{h}}=0$ renders $v$ to be elementwise constant. The additional
conditions force such functions to be zero.

Indeed, as in Section 4, the case b) yields the estimate
$\|v\|_{V_{h}}^{2}\geq\min\\{k_{0};r_{0}\\}\|v\|_{1,\mathcal{T}}^{2}$ for all
$v\in H^{1}(\mathcal{T})$. To prove a) we combine , as in Section 4, the
Poincaré-Friedrichs inequality [BS08, Exercise 5.x.13] on $V$ with a discrete
inequality on $V_{h}$. Namely, as a consequence of [DPE12, Thm. 5.3], there is
a constant $C_{\mathrm{Poin}}>0$ independent of $h>0$ such that following
refined Poincaré inequality holds:

$\|v_{h}\|_{0}\leq C_{\mathrm{Poin}}\|v_{h}\|_{V_{h}}\quad\text{for all
}v_{h}\in V_{h}.$

∎

Now we are prepared to summarize the essential properties of the linear and
bilinear forms.

######  Lemma 14.

Assume that $\boldsymbol{K}$ is piecewise continuous, i. e., continuous on
each $K\in\mathcal{T}$. The linear form $\ell_{h}$ and bilinear form $a_{h}$
are bounded on $V_{h}$ with respect to $\|\cdot\|_{V_{h}}$, possibly depending
on $h$. The bilinear form $a_{h}$ is uniformly coercive with respect to
$\|\cdot\|_{V_{h}}$ with parameter $\alpha=\alpha_{h}$ independent of $h$
provided that

$4\eta>(1-\theta)|\mathcal{F}_{K}|C_{\mathrm{tr}}^{2}\|\boldsymbol{K}\|_{\infty},\quad\text{where}\quad|\mathcal{F}_{K}|:=\begin{cases}d+1,&\mathbb{P}_{k}(K)=\mathcal{P}_{k}(K),\\\
2d,&\mathbb{P}_{k}(K)=\mathcal{Q}_{k}(K).\end{cases}$ (45)

The NIPG method is uniformly coercive with $\alpha=1$ if $\eta>0$.

###### Proof.

To verify the coercivity of $a_{h}$ consider
$u_{h}\in\mathbb{P}_{k}(\mathcal{T})$ and observe that

$a_{h}(u_{h},u_{h})=\|u_{h}\|^{2}_{V_{h}}-(1-\theta)\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
u_{h}}|\\},{\llbracket u_{h}\rrbracket})_{F}.$

The last statement follows immediately from this representation. For
$\theta\neq 1$, we proceed as follows. The properties of $\boldsymbol{K}$, a
discrete trace inequality [DPE12, Lemma 1.46] (here we need the shape- and
contact-regularity of the family of partitions) and the so-called
$\varepsilon$-inequality (here with some $\delta>0$ instead of $\varepsilon$)
allow the estimation

$\displaystyle|(\\{|{\boldsymbol{K}\nabla u_{h}}|\\},{\llbracket
u_{h}\rrbracket})_{F}|$ $\displaystyle\leq\|\\{|{\boldsymbol{K}\nabla
u_{h}}|\\}\|_{0,F}\|{\llbracket u_{h}\rrbracket}\|_{0,F}$
$\displaystyle\leq\|\boldsymbol{K}\|_{\infty}^{1/2}C_{\mathrm{tr}}h_{F}^{-1/2}\|\boldsymbol{K}^{1/2}\nabla
u_{h}\|_{0,\Delta(F)}\|{\llbracket u_{h}\rrbracket}\|_{0,F}$
$\displaystyle\leq\delta\|\boldsymbol{K}^{1/2}\nabla
u_{h}\|^{2}_{0,\Delta(F)}+\frac{C^{2}_{\mathrm{tr}}\|\boldsymbol{K}\|_{\infty}}{4\delta
h_{F}}\|{\llbracket u_{h}\rrbracket}\|^{2}_{0,F},$

where $\Delta(F)$ denotes the union of the elements $K$ with face $F$. Since
every element $K$ has $|\mathcal{F}_{K}|$ faces, the first term occurs at most
$|\mathcal{F}_{K}|$ times after the summation. So if $\delta$ is chosen such
that $\delta(1-\theta)|\mathcal{F}_{K}|<1$, the first term can be absorbed by
the first term of (43). Condition (45) ensures that the second term can be
absorbed by the second term of (43).

The boundedness (not necessarily uniform in $h$) is obvious since all bilinear
and linear forms on finite-dimensional spaces are bounded. ∎

Thus, the Lax-Milgram lemma ensures the existence of a unique solution to
(41).

######  Remark 15.

  1. 1)

Lemma 14 also holds true for certain nonsimplicial and nonconsistent
partitions, with $|\mathcal{F}_{K}|$ substituted by the maximum number of
faces of an element.

  2. 2)

More elaborate techniques can be used to show that the NIPG method for
$\eta=0$ is also stable. This method is tributed to Oden, Babuška, and Baumann
[OBB98], and is referred to as the _OBB method_.

While the above coercivity constant $\alpha=\alpha_{h}$ is uniform in $h$, but
depends on $\eta$, the constants for boundedness are allowed to be
$h$-dependent. This justifies the use of Remark 4 provided we are able to find
an appropriate normed space $(V(h),\|\cdot\|_{V(h)})$ where $a_{h}$ is
bounded.

### Convergence analysis for the full problem

Assume that $u\in V\cap H^{2}(\mathcal{T})$. Next, we justify the assumptions
of Remark 4 by first constructing a space
$V(h)\supset\mathbb{P}_{k}(\mathcal{T})+\mathrm{span}\,(u)$ such that
$\|\cdot\|_{V_{h}}$ is a norm on $V(h)$. A suitable choice clearly is

$V(h)=H^{2}(\mathcal{T}):=\\{v\in L^{2}(\Omega)\;|\;v\in
H^{2}(K)\quad\text{for all }K\in\mathcal{T}\\}.$

However, on $H^{2}(\mathcal{T})$ we cannot apply a discrete trace inequality
to control $\\{|{\boldsymbol{K}\nabla u_{h}}|\\}$ on the faces
$F\in\mathcal{F}\cup\mathcal{F}_{3}$, as in the proof of Lemma 14. Thus,
$\|\cdot\|_{V_{h}}$ has to be extended to both control this term and to
preserve the uniform boundedness of $a_{h}$ on $V(h)\times V_{h}$. This
motivates the definition

$\|w_{h}\|^{2}_{V(h)}:=\|w_{h}\|^{2}_{V_{h}}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\frac{h_{F}}{\eta}\|\\{|{\boldsymbol{K}^{1/2}\nabla
w_{h}}|\\}\|^{2}_{0,F}+\sum_{F\in\overline{\mathcal{F}}}\frac{h_{F}}{\eta}\|\boldsymbol{c}w_{h}\|^{2}_{0,F}\quad\text{for
all }w_{h}\in V(h),$ (46)

where in the last term both traces are considered. That is, this term is
evaluated twice for interfaces.

######  Lemma 16.

Under the assumptions of Lemma 13, there is a constant $\widetilde{M}_{h}>0$
bounded in $h$ such that

$a_{h}(w_{h},v_{h})\leq\widetilde{M}_{h}\|w_{h}\|_{V(h)}\|v_{h}\|_{V_{h}}\quad\text{for
all }w_{h}\in V(h),\ v_{h}\in V_{h}.$

###### Proof.

Consider the eight terms defining $a_{h}(w_{h},v_{h})$ according to (42).
Taking the assumptions into account, the first three terms, the sixth, and the
eighth term can be estimated using the Cauchy–Schwarz inequality (using only
$\|\cdot\|_{V_{h}}$). To handle the fourth and the fifth terms, introduce
$(\eta/h_{F})^{1/2}(h_{F}/\eta)^{1/2}$ before applying the Cauchy–Schwarz
inequality to control the terms to the expense of the new terms in
$\|\cdot\|_{V(h)}$. For the seventh term and $w_{h},v_{h}\in V_{h}$, we
proceed similar to the fourth term taking advantage of a discrete trace
inequality as in the proof of Lemma 14. For $w_{h}\in V(h)$, $v_{h}\in V_{h}$,
integration by parts with the second term and adding the seventh term leads to

$\sum_{K\in\mathcal{T}}(\nabla\cdot(\boldsymbol{c}w_{h}),v_{h})_{K}-\sum_{F\in\overline{\mathcal{F}}}(\boldsymbol{c},{\llbracket
w_{h}v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}_{\mathrm{upw}}(w_{h}),{\llbracket
v_{h}\rrbracket})_{F}.$

The first term can directly be controlled due to
$\nabla\cdot(\boldsymbol{c}w_{h})=\nabla\cdot\boldsymbol{c}w_{h}+\boldsymbol{c}\cdot\nabla
w_{h}$. The second and third terms eventually cancel on boundary faces.
Otherwise, they can be directly controlled. On interfaces, we use that both
terms sum up to the downwind flux
$(\boldsymbol{c}_{\textup{down}}(w_{h}),{\llbracket v_{h}\rrbracket})_{F}$
(defined analogously to (27), (28)) to deduce that these terms can also be
controlled by the last term in (46). ∎

All that is left to do for a convergence result in $\|\cdot\|_{V_{h}}$ is to
show an estimate of the type

$\|u-\Pi u\|_{V(h)}\leq Ch^{m}|u|_{m+1,\mathcal{T}}$ (47)

with an appropriate comparison function $\Pi u\in V(h)$. A good choice for
$\Pi$ is the elementwise orthogonal $L^{2}$-projection. To verify (47), we
need an auxiliary proposition that relates face integrals to element
integrals. In case of a shape-regular triangulation there is a constant $c>0$
such that $h_{F}\geq ch_{K}$ for $K$ adjacent to $F$. This, a multiplicative
trace inequality [DF15, Lemma 2.19] and Young’s inequality imply that there is
a constant $C>0$ independent of $h$ such that

$\|v\|^{2}_{V(h)}\leq
C\sum_{K\in\mathcal{T}_{h}}\left[h_{K}^{-2}\|v\|^{2}_{0,K}+|v|^{2}_{1,K}+h_{K}^{2}|v|^{2}_{2,K}\right]\quad\text{for
all }v\in V(h).$

As the critical term, we handle the second one in (43), with $C>0$ being a
generic constant, as follows:

$\displaystyle\sum_{F\in\mathcal{F}}\frac{1}{h_{F}}\|{\llbracket
v\rrbracket}\|^{2}_{0,F}$ $\displaystyle\leq
C\sum_{K\in\mathcal{T}}\frac{1}{h_{K}}\Big{[}\|\nabla
v\|_{0,K}+\frac{1}{h_{K}}\|v\|_{0,K}\Big{]}\|v\|_{0,K}$ $\displaystyle\leq
C\sum_{K\in\mathcal{T}}\frac{1}{h^{2}_{K}}\big{[}\|v\|^{2}_{0,K}+\|\nabla
v\|^{2}_{0,K}\big{]},$

and the treatment of the additional term in (46), where ${\llbracket
v\rrbracket}$ is replaced by $\\{|{\boldsymbol{K}^{1/2}\nabla v}|\\}$ and
$h^{-1}_{F}$ by $h_{F}$, runs similarly. With this estimate, we can formulate
the main convergence result in the energy norm.

######  Theorem 17 (Convergence in the energy norm).

Assume that the family of compatible partitions is shape- and contact-regular,
and $\boldsymbol{K},\boldsymbol{c}$ are piecewise continuous. Furthermore, let
the conditions (18), (44), (45), and one of the conditions (30) or (31) be
satisfied. If $k\in\mathbb{N}$ and $u\in V\cap H^{m+1}(\mathcal{T})$ with
$1\leq m\leq k$ satisfies (22), and $u_{h}\in V_{h}$ is the IPG solution, then
there is a constant $C>0$ independent of $h$ such that it holds:

$\|u-u_{h}\|_{V_{h}}\leq Ch^{m}|u|_{m+1,\mathcal{T}}\,.$

###### Proof.

According to Remark 4, 2) with $W\subset H^{m+1}(\mathcal{T})$ and Lemmata 11,
14, 16 it remains to complete the estimate

$\inf_{w_{h}\in V_{h}}\|u-w_{h}\|_{V(h)}\leq\|u-\Pi u\|_{V(h)},$

but the latter result is known from (47). ∎

This immediately implies the following (nonoptimal) $L^{2}$-convergence
result.

######  Theorem 18 (Convergence in the $L^{2}(\Omega)$-norm).

Under the assumptions of Theorem 17, the IPG solution $u_{h}\in V(h)$
converges with order at least $m$ to $u$ in the sense of $L^{2}(\Omega)$, i.
e.,

$\|u-u_{h}\|_{0}\leq Ch^{m}|u|_{m+1,\mathcal{T}}\,.$

### Convergence analysis in weaker norms

To apply the results of Theorem 5, and in particular (14), we have to consider
the adjoint problem (6) with the forms (38).

It is not difficult to show that, under the same conditions as for the
original problem (see Lemma 13), the adjoint problem possesses a unique
solution $v=v_{g}\in V$.

The discrete adjoint forms read as

$\displaystyle a_{h}^{*}(v_{h},w_{h})$
$\displaystyle:=a_{h}(w_{h},v_{h})=\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla
w_{h}-\boldsymbol{c}w_{h},\nabla v_{h})_{K}+(rw_{h},v_{h})_{K}\big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}\theta(\\{|{\boldsymbol{K}\nabla
v_{h}}|\\},{\llbracket w_{h}\rrbracket})_{F}-\Big{(}\\{|{\boldsymbol{K}\nabla
w_{h}}|\\}-\frac{\eta}{h_{F}}{\llbracket w_{h}\rrbracket},{\llbracket
v_{h}\rrbracket}\Big{)}_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}(\boldsymbol{c}_{\mathrm{upw}}(w_{h}),{\llbracket
v_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}w_{h},v_{h})_{F}$
$\displaystyle=\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla v_{h},\nabla
w_{h})_{K}-(\boldsymbol{c}\cdot\nabla
v_{h},w_{h})_{K}+(rv_{h},w_{h})_{K}\big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}\theta(\\{|{\boldsymbol{K}\nabla
v_{h}}|\\},{\llbracket w_{h}\rrbracket})_{F}-\Big{(}{\llbracket
v_{h}\rrbracket},\\{|{\boldsymbol{K}\nabla
w_{h}}|\\}-\frac{\eta}{h_{F}}{\llbracket w_{h}\rrbracket}\Big{)}_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}({\llbracket
v_{h}\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v_{h},w_{h})_{F}$
$\displaystyle=\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla v_{h},\nabla
w_{h})_{K}-(\boldsymbol{c}\cdot\nabla
v_{h},w_{h})_{K}+(rv_{h},w_{h})_{K}\big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}-(\\{|{\boldsymbol{K}\nabla
w_{h},{\llbracket v_{h}\rrbracket}}|\\})_{F}+\theta(\\{|{\boldsymbol{K}\nabla
v_{h}}|\\},{\llbracket w_{h}\rrbracket})_{F}+\frac{\eta}{h_{F}}({\llbracket
v_{h}\rrbracket},{\llbracket w_{h}\rrbracket})_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}({\llbracket
v_{h}\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v_{h},w_{h})_{F},$
$\displaystyle\ell_{h}(w_{h})$ $\displaystyle:=\tilde{\ell}(w_{h}).$

To investigate the consistency we observe that

$\displaystyle a_{h}^{*}(v,w_{h})$ $\displaystyle-\tilde{\ell}_{h}(w_{h})$
$\displaystyle=$
$\displaystyle\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla v,\nabla
w_{h})_{K}-(\boldsymbol{c}\cdot\nabla v,w_{h})_{K}+(rv,w_{h})_{K}\big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}-(\\{|{\boldsymbol{K}\nabla
w_{h},{\llbracket v\rrbracket}}|\\})_{F}+\theta(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}+\frac{\eta}{h_{F}}({\llbracket
v\rrbracket},{\llbracket w_{h}\rrbracket})_{F}\Big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}-(g,w_{h})$
$\displaystyle=$
$\displaystyle\sum_{K\in\mathcal{T}}\big{[}(\boldsymbol{K}\nabla v,\nabla
w_{h})_{K}-(\boldsymbol{c}\cdot\nabla v,w_{h})_{K}+(rv,w_{h})_{K}\big{]}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\theta(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}-(g,w_{h}).$

Here we have used that ${\llbracket v\rrbracket}_{F}=0$ on
$F\in\mathcal{F}\cup\mathcal{F}_{3}$. Next we integrate by parts the first
term and obtain

$\displaystyle a_{h}^{*}(v,w_{h})$ $\displaystyle-\tilde{\ell}_{h}(w_{h})$
$\displaystyle=$ $\displaystyle(-\nabla\cdot\left(\boldsymbol{K}\nabla
v\right)-\boldsymbol{c}\cdot\nabla
v+rv,w_{h})+\sum_{K\in\mathcal{T}}(\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{K}\nabla
v,w_{h})_{\partial K}$
$\displaystyle+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\theta(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}-(g,w_{h})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:magicf}}}{{=}}$
$\displaystyle(-\nabla\cdot\left(\boldsymbol{K}\nabla
v\right)-\boldsymbol{c}\cdot\nabla
v+rv,w_{h})+\sum_{F\in\mathcal{F}}[(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}+({\llbracket\boldsymbol{K}\nabla
v\rrbracket},\\{|{w_{h}}|\\})_{F}]$
$\displaystyle+\sum_{F\in\partial\mathcal{F}}(\boldsymbol{K}\nabla
v,{\llbracket
w_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\theta(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}-(g,w_{h})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:strongadjointde}}}{{=}}$
$\displaystyle\sum_{F\in\mathcal{F}}[(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}+({\llbracket\boldsymbol{K}\nabla
v\rrbracket},\\{|{w_{h}}|\\})_{F}]$
$\displaystyle+\sum_{F\in\partial\mathcal{F}}(\boldsymbol{K}\nabla
v,{\llbracket
w_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\theta(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}.$

The second term in the first sum vanishes due to the regularity assumption
w.r.t. the adjoint solution $\boldsymbol{K}\nabla v\in
H(\operatorname{div};\Omega)$ (analogously to (22)). For the third term, it
holds

$\sum_{F\in\partial\mathcal{F}}(\boldsymbol{K}\nabla v,{\llbracket
w_{h}\rrbracket})_{F}=\sum_{F\in\mathcal{F}_{2}\cup\mathcal{F}_{3}}(\boldsymbol{K}\nabla
v,{\llbracket w_{h}\rrbracket})_{F}$

thanks to the homogeneous boundary condition to $\boldsymbol{K}\nabla
v\cdot\boldsymbol{{\mathfrak{n}}}$ on $\Gamma_{1}$. Hence

$\displaystyle a_{h}^{*}(v,w_{h})-\tilde{\ell}_{h}(w_{h})=$
$\displaystyle(1+\theta)\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,1}}(\boldsymbol{K}\nabla v,{\llbracket
w_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,1}}({\llbracket
v\rrbracket},\boldsymbol{c}_{\mathrm{upw}}(w_{h}))_{F}$
$\displaystyle+\sum_{F\in\mathcal{F}_{2,2}}(\boldsymbol{K}\nabla v,{\llbracket
w_{h}\rrbracket})_{F}+\sum_{F\in\mathcal{F}_{2,2}}(\tilde{\alpha}v,w_{h})_{F}.$

Since $\boldsymbol{{\mathfrak{n}}}\cdot\boldsymbol{c}=\tilde{\alpha}\geq 0$ on
$\Gamma_{2,1}$ by assumption (18), 2), all the boundary terms vanish due to
the homogeneous boundary conditions on $\Gamma_{2}$, so that we arrive at the
representation

$a_{h}^{*}(v,w_{h})-\tilde{\ell}_{h}(w_{h})=(1+\theta)\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
v}|\\},{\llbracket w_{h}\rrbracket})_{F}.$ (48)

This shows that the adjoint discrete problem is consistent only for
$\theta=-1$, i. e., for the SIPG method.

The considerations from Lemma 11 hold also true for the adjoint problem in the
SIPG case, guaranteeing a consistent method with a unique solution such that
an analogous order of convergence estimate to Theorem 17 is satisfied.

Thus, the total consistency term in (10) reduces to

$\displaystyle(a-a_{h})(u,v_{g})$
$\displaystyle=-\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}\Big{[}\theta(\\{|{\boldsymbol{K}\nabla
v_{g}}|\\},{\llbracket u\rrbracket})_{F}-\Big{(}\\{|{\boldsymbol{K}\nabla
u}|\\}-\frac{\eta}{h_{F}}{\llbracket u\rrbracket},{\llbracket
v_{g}\rrbracket}\Big{)}_{F}\Big{]}$
$\displaystyle\quad-\sum_{F\in\mathcal{F}}(\boldsymbol{c}_{\mathrm{upw}}(u),{\llbracket
v_{g}\rrbracket})_{F}$

for $u\in V$ being the weak solution of (15), $v_{g}$ the solution of the
adjoint problem (6), (7). Therefore we have the following result.

######  Theorem 19.

Assume that the family of compatible partitions is shape- and contact-regular,
and $\boldsymbol{K},\boldsymbol{c}$ are piecewise continuous. Furthermore, let
the conditions (18), (44), (45), and one of the conditions (30) or (31) be
satisfied. Let $u\in V\cap H^{k+1}(\mathcal{T})$ and $u_{h}\in V_{h}$ be its
SIPG approximate. Assume that for any right-hand side $g\in H^{m}(\Omega)$,
$0\leq m\leq k-1$, the solution $v_{g}$ of the adjoint problem belongs to
$H^{m+2}(\mathcal{T})$ such that $\|v_{g}\|_{m+2,\mathcal{T}}\leq
C_{s}\|g\|_{m,\Omega}\,$ for some constant $C_{s}>0$. Furthermore let the
solutions $u$ and $v_{g}$ satisfy (22). Then, there is a constant $C>0$
independent of $h$ such that

$\|u-u_{h}\|_{-m,\Omega}\leq Ch^{k+m+1}|u|_{k+1}\mathcal{T}.$

###### Proof.

Under the given regularity and boundary conditions the above error term
satisfies

$(a-a_{h})(u,v_{g})=0.$

∎

######  Remark 20.

According to (48), for other methods with $\theta\neq-1$, the discretization
of the adjoint problem is no longer consistent to the adjoint problem, i. e.,
according to Theorem 5, the second term

$a_{h}(u-u_{h},v_{g})-\tilde{\ell}_{h}(u-u_{h})=(1+\theta)\sum_{F\in\mathcal{F}\cup\mathcal{F}_{3}}(\\{|{\boldsymbol{K}\nabla
v_{g}}|\\},{\llbracket u-u_{h}\rrbracket})_{F}$

has to be estimated for sufficiently smooth solutions $u,v_{g}$.

##  6 Conclusion

We presented a unified approach to the analysis of FEM for boundary value
problems with linear diffusion-convection-reaction equations and boundary
conditions of mixed type, where neither conformity nor consistency properties
are assumed. Being elementary in nature, it clarifies and quantifies the
interplay between stability, approximation errors, and consistency errors –
the theme guiding PDE numerical analysis from its very beginning. As an
example, we formulated and investigated two different stabilized
discretizations and obtained stability and optimal error estimates in energy-
type norms and, as a consequence of our generalization of the Aubin-Nitsche
technique, optimal error estimates in weaker norms. We expect the described
framework to provide guidelines to set up and analyze further new stable and
convergent schemes.

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