# Robust space-time finite element error estimates
for parabolic distributed optimal control problems
with energy regularization

Ulrich Langer111Institute of Computational Mathematics, Johannes Kepler
University Linz, Altenberger Straße 69, 4040 Linz, Austria, Email:
<EMAIL_ADDRESS>Olaf Steinbach222Institut für Angewandte
Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria,
Email<EMAIL_ADDRESS>Huidong Yang333Institute of Science and
Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria, Email:
<EMAIL_ADDRESS>
###### Abstract

We consider space-time tracking optimal control problems for linear parabolic
initial boundary value problems that are given in the space-time cylinder
$Q=\Omega\times(0,T)$, and that are controlled by the right-hand side
$z_{\varrho}$ from the Bochner space $L^{2}(0,T;H^{-1}(\Omega))$. So it is
natural to replace the usual $L^{2}(Q)$ norm regularization by the energy
regularization in the $L^{2}(0,T;H^{-1}(\Omega))$ norm. We derive a priori
estimates for the error $\|\widetilde{u}_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}$ between the computed state
$\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ in terms of
the regularization parameter $\varrho$ and the space-time finite element mesh-
size $h$, and depending on the regularity of the desired state $\overline{u}$.
These estimates lead to the optimal choice $\varrho=h^{2}$. The approximate
state $\widetilde{u}_{\varrho h}$ is computed by means of a space-time finite
element method using piecewise linear and continuous basis functions on
completely unstructured simplicial meshes for $Q$. The theoretical results are
quantitatively illustrated by a series of numerical examples in two and three
space dimensions.

Keywords: Parabolic optimal control problems, energy regularization,
space-time finite element methods, error estimates.

2010 MSC: 49J20, 49M05, 35K20, 65M60, 65M15

##  1 Introduction

As in [8], we consider the minimization of the space-time tracking cost
functional

${\mathcal{J}}(u_{\varrho},z_{\varrho})=\frac{1}{2}\int_{0}^{T}\int_{\Omega}[u_{\varrho}(x,t)-\overline{u}(x,t)]^{2}\,dx\,dt+\frac{1}{2}\,\varrho\,\|z_{\varrho}\|^{2}_{L^{2}(0,T;H^{-1}(\Omega))}$
(1.1)

with respect to the state $u_{\varrho}$ and the control $z_{\varrho}$ subject
to the model parabolic initial boundary value problem

$\left.\begin{array}[]{rclcl}\partial_{t}u_{\varrho}(x,t)-\Delta_{x}u_{\varrho}(x,t)&=&z_{\varrho}(x,t)&&\mbox{for}\;(x,t)\in
Q:=\Omega\times(0,T),\\\\[2.84526pt]
u_{\varrho}(x,t)&=&0&&\mbox{for}\;(x,t)\in\Sigma:=\partial\Omega\times(0,T),\\\\[2.84526pt]
u_{\varrho}(x,0)&=&0&&\mbox{for}\;x\in\overline{\Sigma}_{0}:=\overline{\Omega}\times\\{0\\},\end{array}\right\\}$
(1.2)

where $\overline{u}\in L^{2}(Q)$ is the given desired state (target),
$\partial_{t}$ denotes the partial time derivative, $\Delta_{x}$ is the
spatial Laplace operator, $\Omega\subset{\mathbb{R}}^{n}$, $n=1,2,3$, is the
spatial domain that is assumed to be bounded and Lipschitz, $T>0$ is a given
time horizon, and $\varrho>0$ is a suitably chosen regularization parameter.
The standard setting of such kind of optimal control problems uses the
regularization in $L^{2}(Q)$ instead of $L^{2}(0,T;H^{-1}(\Omega))$; see,
e.g., the books [2, 6, 17], and the references given therein. The energy
regularization, as the regularization in $L^{2}(0,T;H^{-1}(\Omega))$ is also
called, permits controls $z_{\varrho}$ from the space
$L^{2}(0,T;H^{-1}(\Omega))$ that is larger than $L^{2}(Q)$, and admits more
concentrated controls. Such kind of controls that are concentrated around
hypersurfaces play an important role in electromagnetics in form of thinly
wound coils and magnets. Moreover, the space $L^{2}(0,T;H^{-1}(\Omega))$ is
the natural space for the source term in the variational formulation of the
initial boundary value problem (1.2), at least, in the Hilbert space setting;
see, e.g., [10] or [18] for solvability results. In the literature, there are
other regularization techniques aiming at specific properties of the control
such as sparsity and directional sparsity. We refer the reader to the recent
survey article [4] where a comprehensive overview of the literature on this
topic is given.

Since the state equation (1.2) in its variational form has a unique solution
$u_{\varrho}\in X:=\\{u\in Y:=L^{2}(0,T;H^{1}_{0}(\Omega)):\partial_{t}u\in
Y^{*},\;u=0\;\text{on}\;\Sigma_{0}\\}$, for every given right-hand side
$z_{\varrho}\in Y^{*}:=L^{2}(0,T;H^{-1}(\Omega))$, the corresponding optimal
control problem (1.1)-(1.2) also has a unique solution
$(u_{\varrho},z_{\varrho})\in X\times Y^{*}$ that can be computed by solving
the first-order optimality system or the reduced first-order optimality system
where the control is eliminated by the gradient equation. The unique
solvability of the state equation can also be shown by the
Banach–Nec̆as–Babus̆ka theorem as it was done in [14]. This theorem can also
be used to show well-posedness of the reduced first-order optimality system as
it was done in [8]. Now the optimal control problem (1.1)-(1.2) can be
approximately solved by discretizing the reduced optimality system. Following
[8], we discretize the reduced optimality system by means of a real space-time
finite element method working on fully unstructured, but shape regular
simplicial space-time meshes into which the space-time cylinder $Q$ is
decomposed. In [8], the authors showed a discrete inf-sup condition for the
bilinear form arising from the variational formulation of the reduced
optimality system. Once a discrete inf-sup condition is proven, one can easily
derive the corresponding estimates for the finite element discretization error
$u_{\varrho}-\widetilde{u}_{\varrho h}$ and
$p_{\varrho}-\widetilde{p}_{\varrho h}$ in the corresponding norms, where
$\widetilde{u}_{\varrho h}$ and $\widetilde{p}_{\varrho h}$ are the finite
element solutions to the reduced first-order optimality system approximating
the state $u_{\varrho}$ and the co-state (adjoint) $p_{\varrho}$,
respectively.

In this paper, we are investigating the error between the computed finite
element solution $\widetilde{u}_{\varrho h}$ and the desired state
$\overline{u}$, where we use continuous, piecewise linear finite element basis
functions. This error is obviously of primary interest since one wants to know
how well $\widetilde{u}_{\varrho h}$ approximates $\overline{u}$ in advance.
More precisely, we derive estimates for the $L^{2}(Q)$ norm of this error in
terms of $\varrho$ and $h$, and depending on the smoothness of the target
$\overline{u}$ that is assumed to belong to $H^{s}(Q)$ for some $s\in(0,2]$.
In particular, we admit discontinuous targets that are important in many
practical applications. These estimates lead to the optimal choice
$\varrho=h^{2}$ in all cases. For elliptic optimal control problems with
energy regularization, i.e., in $H^{-1}(\Omega)$, error estimates for
$\|\overline{u}-{u}_{\varrho}\|_{L^{2}(Q)}$ and
$\|\overline{u}-\widetilde{u}_{\varrho h}\|_{L^{2}(Q)}$ were recently derived
in [11] and [9], respectively. It is interesting that, in the elliptic case,
${u}_{\varrho}$ solves the singularly perturbed reaction-diffusion equation
$-\varrho\Delta u_{\varrho}+u_{\varrho}=\overline{u}$ in $\Omega$ with
homogeneous Dirichlet conditions on the boundary $\partial\Omega$, also known
as differential filter in fluid mechanics [7], whereas, in the parabolic case,
${u}_{\varrho}$ solves a similar singularly perturbed problem, but with a more
complicated space-time operator of the form $B^{*}A^{-1}B$ replacing
$-\Delta$, where $B:X\to Y^{*}$ is nothing but the state (parabolic) operator,
and $A:Y\to Y^{*}$ represents the spatial Laplacian $-\Delta_{x}$; see
Sections 2 and 3 for a more detailed discussion.

The reminder of this paper is organized as follows: Section 2 deals with the
formulation of an abstract optimal control problem, and the corresponding
error estimates between the desired state and the discrete state based on the
exact state Schur complement equation. In Section 3, we consider a model
parabolic distributed optimal control problem with energy regularization, and
derive estimates for the $L^{2}(Q)$ error between the desired state
$\overline{u}$ and the finally computed state $\widetilde{u}_{\varrho h}$ from
the perturbed state Schur complement equation for the coupled optimality
system. Several numerical tests in two and three space dimensions are
discussed in Section 4. Finally, some conclusions are drawn in Section 5, and
we also discuss some future research topics.

##  2 Abstract optimal control problems

Let $X\subset H\subset X^{*}$ and $Y\subset H\subset Y^{*}$ be Gelfand triples
of Hilbert spaces, where $X^{*},Y^{*}$ are the duals of $X,Y$ with respect to
$H$. Let $A:Y\to Y^{*}$ and $B:X\to Y^{*}$ be bounded linear operators, i.e.,

$\|Av\|_{Y^{*}}\leq c_{2}^{A}\,\|v\|_{Y}\quad\forall v\in
Y,\quad\|Bu\|_{Y^{*}}\leq c_{2}^{B}\,\|u\|_{X}\quad\forall u\in X.$ (2.1)

We assume that $A$ is self-adjoint and elliptic in $Y$, and that $B$ satisfies
an inf-sup condition, i.e., there exist positive constants $c_{1}^{A}$ and
$c_{1}^{B}$ such that

$\langle Av,v\rangle_{H}\geq c_{1}^{A}\,\|v\|_{Y}^{2}\quad\forall v\in
Y,\quad\sup\limits_{0\neq v\in Y}\frac{\langle Bu,v\rangle_{H}}{\|v\|_{Y}}\geq
c_{1}^{B}\,\|u\|_{X}\quad\forall u\in X.$ (2.2)

In addition, we assume that the dual to $B$ operator $B^{*}:Y\to X^{*}$ is
injective. Then, due to Lax–Milgram’s and Banach–Nečas–Babuška’s theorems
(see, e.g.,[5]), $A:Y\to Y^{*}$ and $B:X\to Y^{*}$ are isomorphisms.
Therefore,

$\|z\|_{Y^{*}}=\sqrt{\langle A^{-1}z,z\rangle_{H}}\quad\mbox{for}\;z\in Y^{*}$
(2.3)

defines a norm in $Y^{*}$ that is equivalent to the standard supremum norm.

We now consider the abstract minimization problem to find the minimizer
$(u_{\varrho},z_{\varrho})\in X\times Y^{*}$ of the functional

${\mathcal{J}}(u_{\varrho},z_{\varrho})=\frac{1}{2}\,\|u_{\varrho}-\overline{u}\|_{H}^{2}+\frac{1}{2}\,\varrho\,\|z_{\varrho}\|^{2}_{Y^{*}}\quad\mbox{subject
to}\;Bu_{\varrho}=z_{\varrho},$ (2.4)

when $\overline{u}\in H$ is given, and $\varrho\in{\mathbb{R}}_{+}$ is some
regularization parameter. For the time being, our particular interest is
focused on the behavior of $\|u_{\varrho}-\overline{u}\|_{H}$ as $\varrho\to
0$. The minimizer $(u_{\varrho},z_{\varrho})$ of (2.4) is determined as the
unique solution of the optimality system, see, e.g., [8],

$Bu_{\varrho}=z_{\varrho},\qquad
B^{*}p_{\varrho}=u_{\varrho}-\overline{u},\qquad p_{\varrho}+\varrho
A^{-1}z_{\varrho}=0.$ (2.5)

Eliminating the control $z_{\varrho}\in Y^{*}$ and the adjoint variable
$p_{\varrho}\in Y$ results in the operator equation to find $u_{\varrho}\in X$
such that

$\varrho
B^{*}A^{-1}Bu_{\varrho}+u_{\varrho}=\overline{u}\quad\mbox{in}\;X^{*}.$ (2.6)

Let us introduce the operator $S:=B^{*}A^{-1}B:X\to X^{*}$, for which we have
the following result:

######  Lemma 1.

There hold the inequalities

$\langle Su,u\rangle_{H}\geq
c_{1}^{S}\,\|u\|_{X}^{2}\quad\mbox{and}\quad\|Su\|_{X^{*}}\leq
c_{2}^{S}\,\|u\|_{X}\quad\mbox{for all}\;u\in X$

with constants

$c_{1}^{S}=c_{1}^{A}\left(\frac{c_{1}^{B}}{c_{2}^{A}}\right)^{2}\quad\mbox{and}\quad
c_{2}^{S}=\frac{[c_{2}^{B}]^{2}}{c_{1}^{A}}\,.$

###### Proof.

For arbitrary, but fixed $u\in X$, we define $\overline{p}=A^{-1}Bu$ to obtain

$\langle Su,u\rangle_{H}=\langle A^{-1}Bu,Bu\rangle_{H}=\langle
A\overline{p},\overline{p}\rangle_{H}\geq
c_{1}^{A}\,\|\overline{p}\|_{Y}^{2}\,.$

From the inf-sup condition (2.2) we further conclude

$c_{1}^{B}\,\|u\|_{X}\leq\sup\limits_{0\neq v\in Y}\frac{\langle
Bu,v\rangle_{H}}{\|v\|_{Y}}=\sup\limits_{0\neq v\in Y}\frac{\langle
A\overline{p},v\rangle_{H}}{\|v\|_{Y}}\leq\|A\overline{p}\|_{Y^{*}}\leq
c_{2}^{A}\,\|\overline{p}\|_{Y}.$

This gives

$\langle Su,u\rangle_{H}\geq c_{1}^{A}\,\|\overline{p}\|^{2}_{Y}\geq
c_{1}^{A}\left(\frac{c_{1}^{B}}{c_{2}^{A}}\right)^{2}\|u\|_{X}^{2}=c_{1}^{S}\,\|u\|_{X}^{2}.$

To prove the second estimate, we consider

$c_{1}^{A}\,\|\overline{p}\|_{Y}^{2}\leq\langle
A\overline{p},\overline{p}\rangle_{H}=\langle
Bu,\overline{p}\rangle_{H}\leq\|Bu\|_{Y^{*}}\|\overline{p}\|_{Y}\leq
c_{2}^{B}\,\|u\|_{X}\|\overline{p}\|_{Y},$

i.e.,

$\|\overline{p}\|_{Y}\leq\frac{c_{2}^{B}}{c_{1}^{A}}\,\|u\|_{X}\,.$

With this we finally obtain

$\displaystyle\|Su\|_{X^{*}}$ $\displaystyle=$
$\displaystyle\sup\limits_{0\neq v\in X}\frac{\langle
Su,v\rangle_{H}}{\|v\|_{X}}=\sup\limits_{0\neq v\in X}\frac{\langle
A^{-1}Bu,Bv\rangle_{H}}{\|v\|_{X}}$ $\displaystyle=$
$\displaystyle\sup\limits_{0\neq v\in
X}\frac{\langle\overline{p},Bv\rangle_{H}}{\|v\|_{X}}\leq\sup\limits_{0\neq
v\in X}\frac{\|\overline{p}\|_{Y}\|Bv\|_{Y^{*}}}{\|v\|_{X}}$
$\displaystyle\leq$ $\displaystyle
c_{2}^{B}\,\|\overline{p}\|_{Y}\leq\frac{[c_{2}^{B}]^{2}}{c_{1}^{A}}\,\|u\|_{X}=c_{2}^{S}\,\|u\|_{X}\,.$

∎

As a consequence of Lemma 1 we also have

$\langle Su,u\rangle_{H}\leq\|Su\|_{X^{*}}\|u\|_{X}\leq
c_{2}^{S}\,\|u\|_{X}^{2},$

i.e.,

$\|u\|_{S}^{2}:=\langle Su,u\rangle_{H}=\langle A^{-1}Bu,Bu\rangle_{H}$

defines an equivalent norm in $X$ satisfying the norm equivalence inequalities

$c_{1}^{S}\,\|u\|_{X}^{2}\leq\|u\|_{S}^{2}\leq
c_{2}^{S}\,\|u\|_{X}^{2}\quad\mbox{for all}\;u\in X.$ (2.7)

Now we consider the abstract operator equation to find $u_{\varrho}\in X$ such
that

$\varrho Su_{\varrho}+u_{\varrho}=\overline{u}\quad\mbox{in}\;X^{*},$ (2.8)

and its equivalent variational formulation

$\varrho\,\langle Su_{\varrho},v\rangle_{H}+\langle
u_{\varrho},v\rangle_{H}=\langle\overline{u},v\rangle_{H}\quad\mbox{for
all}\;v\in X.$ (2.9)

Since $S$ induces an equivalent norm in $X$, unique solvability of (2.9)
follows.

######  Lemma 2.

For the unique solution $u_{\varrho}\in X$ of the variational formulation
(2.9), there hold the estimates

$\|u_{\varrho}\|_{H}\leq\|\overline{u}\|_{H}\quad\mbox{and}\quad\varrho\,\|u_{\varrho}\|^{2}_{S}\leq\|\overline{u}\|^{2}_{H}.$
(2.10)

###### Proof.

For the particular choice $v=u_{\varrho}$ within the variational formulation
(2.9), we obtain

$\varrho\,\|u_{\varrho}\|_{S}^{2}+\|u_{\varrho}\|^{2}_{H}=\varrho\,\langle
Su_{\varrho},u_{\varrho}\rangle_{H}+\langle
u_{\varrho},u_{\varrho}\rangle_{H}=\langle\overline{u},u_{\varrho}\rangle_{H}\leq\|\overline{u}\|_{H}\|u_{\varrho}\|_{H},$

from which we conclude

$\|u_{\varrho}\|_{H}\leq\|\overline{u}\|_{H}$

as well as

$\varrho\,\|u_{\varrho}\|^{2}_{S}\leq\|\overline{u}\|_{H}\|u_{\varrho}\|_{H}\leq\|\overline{u}\|^{2}_{H}.$

∎

Analogously to [11, Theorem 3.2] we can state the following estimates, which
depend on the regularity of the given target $\overline{u}\in H$.

######  Lemma 3.

Let $u_{\varrho}\in X$ be the unique solution of the variational formulation
(2.9). For $\overline{u}\in H$ there holds

$\|u_{\varrho}-\overline{u}\|_{H}\leq\|\overline{u}\|_{H},$ (2.11)

while for $\overline{u}\in X$ the following estimates hold true:

$\|u_{\varrho}-\overline{u}\|_{H}\leq\varrho^{1/2}\,\|\overline{u}\|_{S},$
(2.12) $\|u_{\varrho}-\overline{u}\|_{S}\leq\|\overline{u}\|_{S}.$ (2.13)

If in addition $S\overline{u}\in H$ is satisfied for $\overline{u}\in X$,

$\|u_{\varrho}-\overline{u}\|_{H}\leq\varrho\,\|S\overline{u}\|_{H}$ (2.14)

as well as

$\|u_{\varrho}-\overline{u}\|_{S}\leq\varrho^{1/2}\,\|S\overline{u}\|_{H}$
(2.15)

follow.

###### Proof.

From the variational formulation (2.9) and for the particular test function
$v=u_{\varrho}$, we obtain

$\varrho\,\|u_{\varrho}\|_{S}^{2}=\varrho\,\langle
Su_{\varrho},u_{\varrho}\rangle_{H}=\langle\overline{u}-u_{\varrho},u_{\varrho}\rangle_{H}=\langle\overline{u}-u_{\varrho},\overline{u}\rangle_{H}-\langle\overline{u}-u_{\varrho},\overline{u}-u_{\varrho}\rangle_{H},$

which gives

$\varrho\,\|u_{\varrho}\|_{S}^{2}+\|u_{\varrho}-\overline{u}\|_{H}^{2}=\langle\overline{u}-u_{\varrho},\overline{u}\rangle_{H}\leq\|\overline{u}-u_{\varrho}\|_{H}\|\overline{u}\|_{H},$

i.e., (2.11) follows.

When assuming $\overline{u}\in X$, we can choose
$v=\overline{u}-u_{\varrho}\in X$ as test function in (2.9) to conclude

$\displaystyle\|\overline{u}-u_{\varrho}\|_{H}^{2}$ $\displaystyle=$
$\displaystyle\langle\overline{u}-u_{\varrho},\overline{u}-u_{\varrho}\rangle_{H}$
$\displaystyle=$ $\displaystyle\varrho\,\langle
Su_{\varrho},\overline{u}-u_{\varrho}\rangle_{H}$ $\displaystyle=$
$\displaystyle\varrho\,\langle
S\overline{u},\overline{u}-u_{\varrho}\rangle_{H}-\varrho\,\langle
S(\overline{u}-u_{\varrho}),\overline{u}-u_{\varrho}\rangle_{H},$

i.e.,

$\varrho\,\|\overline{u}-u_{\varrho}\|_{S}^{2}+\|\overline{u}-u_{\varrho}\|^{2}_{H}=\varrho\,\langle
S\overline{u},\overline{u}-u_{\varrho}\rangle_{H}\leq\varrho\,\|\overline{u}\|_{S}\|\overline{u}-u_{\varrho}\|_{S}\,.$

In a first step this gives (2.13),

$\|u_{\varrho}-\overline{u}\|_{S}\leq\|\overline{u}\|_{S}.$

With this we further obtain

$\|u_{\varrho}-\overline{u}\|^{2}_{H}\leq\varrho\,\|\overline{u}\|_{S}\|\overline{u}-u_{\varrho}\|_{S}\leq\varrho\,\|\overline{u}\|_{S}^{2},$

i.e., (2.12) follows.

If, for $\overline{u}\in X$, we have in addition $S\overline{u}\in H$, from
the estimate (2), we also conclude

$\varrho\,\|\overline{u}-u_{\varrho}\|_{S}^{2}+\|\overline{u}-u_{\varrho}\|^{2}_{H}=\varrho\,\langle
S\overline{u},\overline{u}-u_{\varrho}\rangle_{H}\leq\varrho\,\|S\overline{u}\|_{H}\|u_{\varrho}-\overline{u}\|_{H},$

from which (2.14) follows. Finally, the estimates

$\varrho\,\|\overline{u}-u_{\varrho}\|_{S}^{2}\leq\varrho\,\|S\overline{u}\|_{H}\|u_{\varrho}-\overline{u}\|_{H}\leq\varrho^{2}\,\|S\overline{u}\|_{H}^{2}$

imply (2.15). ∎

Based on the estimates as given in Lemma 3 and in the case of the particular
application we have in mind, we can derive more general estimates which are
based on interpolation arguments in a scale of Sobolev spaces. This will be
discussed later in more detail.

For some conforming approximation space $X_{h}\subset X$, we now consider the
Galerkin variational formulation of (2.9), i.e., find $u_{\varrho h}\in X_{h}$
such that

$\varrho\,\langle Su_{\varrho h},v_{h}\rangle_{H}+\langle u_{\varrho
h},v_{h}\rangle_{H}=\langle\overline{u},v_{h}\rangle_{H}\quad\forall\,v_{h}\in
X_{h}.$ (2.17)

Using again standard arguments, we conclude unique solvability of (2.17), and
the following Cea type a priori error estimate,

$\|u_{\varrho}-u_{\varrho h}\|_{H}\leq\inf\limits_{v_{h}\in
X_{h}}\sqrt{\varrho\,\|u_{\varrho}-v_{h}\|^{2}_{S}+\|u_{\varrho}-v_{h}\|^{2}_{H}}.$
(2.18)

As a particular application of (2.18) we obtain, when choosing $v_{h}=0$, and
using (2.10),

$\|u_{\varrho}-u_{\varrho
h}\|_{H}^{2}\leq\varrho\,\|u_{\varrho}\|^{2}_{S}+\|u_{\varrho}\|^{2}_{H}\,\leq\,2\,\|\overline{u}\|_{H}^{2}.$

Now, using (2.11), we conclude the abstract error estimate

$\|u_{\varrho
h}-\overline{u}\|_{H}\leq\|u_{\varrho}-\overline{u}\|_{H}+\|u_{\varrho}-u_{\varrho
h}\|_{H}\leq(1+\sqrt{2})\,\|\overline{u}\|_{H}.$ (2.19)

when assuming $\overline{u}\in H$ only.

##  3 Parabolic distributed optimal control problem

The parabolic optimal control problem (1.1)-(1.2) as given in the introduction
is obviously a special case of the abstract optimal control problem (2.4).
Indeed, in view of the abstract setting, we have $H:=L^{2}(Q)$,
$Y:=L^{2}(0,T;H^{1}_{0}(\Omega))$, and

$X:=\\{u\in W(0,T):u=0\;\mbox{on}\;\Sigma_{0}\\},$

with $W(0,T):=\\{u\in Y:\partial_{t}u\in Y^{*}=L^{2}(0,T;H^{-1}(\Omega))\\}$.
The related norms in $Y$, $X$, and $Y^{*}$ are given by

$\|v\|_{Y}:=\|\nabla_{x}v\|_{L^{2}(Q)},\;\|u\|_{X}:=\sqrt{\|u\|^{2}_{Y}+\|\partial_{t}u\|^{2}_{Y^{*}}},\;\text{and}\;\|\partial_{t}u\|_{Y^{*}}=\|\nabla_{x}w_{u}\|_{L^{2}(Q)},$

respectively, where $w_{u}\in Y$ is the unique solution of the variational
problem

$\langle\nabla_{x}w_{u},\nabla_{x}v\rangle_{L^{2}(Q)}=\langle\partial_{t}u,v\rangle_{Q}\quad\forall\,v\in
Y.$

For later use, we will prove the following embedding:

######  Lemma 4.

For $u\in X\cap H^{1}(Q)$ there holds

$\|u\|_{X}\leq\max\\{\sqrt{c_{F}},1\\}\,\|u\|_{H^{1}(Q)}$ (3.1)

with the constant $c_{F}>0$ from the spatial Friedrichs inequality in
$H^{1}_{0}(\Omega)$,

$\int_{\Omega}[v(x)]^{2}\,dx\,\leq\
c_{F}\int_{\Omega}|\nabla_{x}v(x)|^{2}\,dx\quad\forall\,v\in
H^{1}_{0}(\Omega).$ (3.2)

###### Proof.

Recall that we can write

$\|u\|^{2}_{X}=\|\partial_{t}u\|_{Y^{*}}^{2}+\|\nabla_{x}u\|^{2}_{L^{2}(Q)},$

and since $\partial_{t}u\in L^{2}(Q)$ for $u\in H^{1}(Q)$, we can bound
$\|\partial_{t}u\|_{Y^{*}}$ as follows:

$\|\partial_{t}u\|_{Y^{*}}=\sup\limits_{0\neq v\in
Y}\frac{\langle\partial_{t}u,v\rangle_{Q}}{\|v\|_{Y}}\leq\sup\limits_{0\neq
v\in
Y}\frac{\|\partial_{t}u\|_{L^{2}(Q)}\|v\|_{L^{2}(Q)}}{\|\nabla_{x}v\|_{L^{2}(Q)}}\leq\sqrt{c_{F}}\,\|\partial_{t}u\|_{L^{2}(Q)}.$

Here we have used the Friedrichs inequality

$\|v\|_{L^{2}(Q)}^{2}=\int_{0}^{T}\|v(t)\|^{2}_{L^{2}(\Omega)}\,dt\leq
c_{F}\int_{0}^{T}\|\nabla_{x}v(t)\|^{2}_{L^{2}(\Omega)}\,dt=c_{F}\,\|\nabla_{x}v\|^{2}_{L^{2}(Q)}$

that holds for all $v\in Y=L^{2}(0,T;H_{0}^{1}(\Omega))$ due to (3.2). Hence,
the estimates

$\|u\|_{X}^{2}\leq
c_{F}\,\|\partial_{t}u\|^{2}_{L^{2}(Q)}+\|\nabla_{x}u\|^{2}_{L^{2}(Q)}\leq\max\\{c_{F},1\\}\,\|u\|^{2}_{H^{1}(Q)}$

follow. ∎

The variational formulation of the state equation (1.2) can now be written in
the form: Find $u_{\varrho}\in X$ such that

$\int_{0}^{T}\int_{\Omega}\Big{[}\partial_{t}u_{\varrho}(x,t)\,v(x,t)+\nabla_{x}u_{\varrho}(x,t)\cdot\nabla_{x}v(x,t)\Big{]}\,dx\,dt=\int_{0}^{T}\int_{\Omega}z_{\varrho}(x,t)\,v(x,t)\,dx\,dt$

for all $v\in Y$, where the first term in the bilinear form and the right-hand
side must be understood as duality pairing between $Y^{*}$ and $Y$. This
variational formulation can be rewritten as operator equation
$Bu_{\varrho}=z_{\varrho}$ in $Y^{*}=L^{2}(0,T;H^{-1}(\Omega))$. The operator
$B:X\to Y^{*}$ is therefore defined by the variational identity

$\langle
Bu,v\rangle_{Q}=\int_{0}^{T}\int_{\Omega}\Big{[}\partial_{t}u(x,t)\,v(x,t)+\nabla_{x}u(x,t)\cdot\nabla_{x}v(x,t)\Big{]}\,dx\,dt$
(3.3)

for all $u\in X$ and $v\in Y$, while $A:Y\to Y^{*}$ is given as

$\langle
Aw,v\rangle_{Q}=\int_{0}^{T}\int_{\Omega}\nabla_{x}w(x,t)\cdot\nabla_{x}v(x,t)\,dx\,dt,\quad\forall\,w,v\in
Y.$ (3.4)

We obviously have $c_{1}^{A}=c_{2}^{A}=1$. Following [14, 15], the operator
$B:X\to Y^{*}$ is bounded,

$\langle Bu,v\rangle_{Q}\leq\sqrt{2}\,\|u\|_{X}\|v\|_{Y}\quad\forall\,u\in
X,\;v\in Y,$

and satisfies the inf-sup condition

$\frac{1}{\sqrt{2}}\,\|u\|_{X}\leq\sup\limits_{0\neq v\in Y}\frac{\langle
Bu,v\rangle_{Q}}{\|v\|_{Y}}\quad\forall\,u\in X,$

i.e., $c_{1}^{B}=1/\sqrt{2}$ and $c_{2}^{B}=\sqrt{2}$. Hence we obtain the
statements of Lemma 1 with $c_{1}^{S}=1/2$ and $c_{2}^{S}=2$. With these
definitions, the reduced first-order optimality system can be written in the
following operator form: Find $(u_{\varrho},p_{\varrho})\in X\times Y$ such
that

$\begin{pmatrix}\varrho^{-1}A&B\\\
B^{*}&-I\end{pmatrix}\begin{pmatrix}p_{\varrho}\\\
u_{\varrho}\end{pmatrix}=\begin{pmatrix}0\\\
-\overline{u}\end{pmatrix}\quad\text{in}\quad Y^{*}\times X^{*},$ (3.5)

from which the control $z_{\varrho}=-\varrho^{-1}Ap_{\varrho}$ can be
computed; cf. also (2.5) and (2.6).

For the Galerkin formulation (2.17), we introduce a conforming finite element
space $X_{h}=S_{h}^{1}(Q)\cap X\subset X$ of piecewise linear and continuous
basis functions which are defined with respect to some admissible
decomposition of the space-time domain $Q$ into shape regular simplicial
finite elements of mesh width $h$; see, e.g., [3]. Then the finite element
approximation of (2.9) reads to find $u_{\varrho h}\in X_{h}$ such that

$\varrho\,\langle B^{*}A^{-1}Bu_{\varrho h},v_{h}\rangle_{Q}+\langle
u_{\varrho
h},v_{h}\rangle_{L^{2}(Q)}=\langle\overline{u},v_{h}\rangle_{L^{2}(Q)}$ (3.6)

is satisfied for all $v_{h}\in X_{h}$.

######  Theorem 1.

Assume $\overline{u}\in[L^{2}(Q),X]_{s}\cap H^{s}(Q)$ for $s\in[0,1)$ or
$\overline{u}\in X\cap H^{s}(Q)$ for $s\in[1,2]$. For the unique solution
$u_{\varrho h}\in X_{h}$ of (3.6), the finite element error estimate

$\|u_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}\,\leq\,c\,h^{s}\,\|\overline{u}\|_{H^{s}(Q)}$
(3.7)

holds provided that $\varrho=h^{2}$.

###### Proof.

For $\overline{u}\in L^{2}(Q)$, we can write the error estimate (2.19) as

$\|u_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}\leq(1+\sqrt{2})\,\|\overline{u}\|_{L^{2}(Q)}.$

Due to $X\subset H^{1}(Q)$, we now assume $\overline{u}\in X\cap H^{1}(Q)$ for
which we can write the error estimate (2.18) as

$\displaystyle\|u_{\varrho}-u_{\varrho h}\|_{L^{2}(Q)}^{2}$
$\displaystyle\leq$ $\displaystyle\inf\limits_{v_{h}\in
X_{h}}\Big{[}\varrho\,\|u_{\varrho}-v_{h}\|_{S}^{2}+\|u_{\varrho}-v_{h}\|_{L^{2}(Q)}^{2}\Big{]}$
$\displaystyle\leq\,2\left[\varrho\,\|u_{\varrho}-\overline{u}\|_{S}^{2}+\|u_{\varrho}-\overline{u}\|^{2}_{L^{2}(Q)}+\inf\limits_{v_{h}\in
X_{h}}\Big{[}\varrho\,\|\overline{u}-v_{h}\|^{2}_{S}+\|\overline{u}-v_{h}\|^{2}_{L^{2}(Q)}\Big{]}\right]$
$\displaystyle\leq\,4\,\varrho\,\|\overline{u}\|_{S}^{2}+2\inf\limits_{v_{h}\in
X_{h}}\Big{[}\varrho\,\|\overline{u}-v_{h}\|^{2}_{S}+\|\overline{u}-v_{h}\|^{2}_{L^{2}(Q)}\Big{]}$
$\displaystyle\leq\,8\,\varrho\,\|\overline{u}\|_{X}^{2}+2\inf\limits_{v_{h}\in
X_{h}}\Big{[}2\,\varrho\,\|\overline{u}-v_{h}\|^{2}_{X}+\|\overline{u}-v_{h}\|^{2}_{L^{2}(Q)}\Big{]}$
$\displaystyle\leq\,8\,\max\\{c_{F},1\\}\,\varrho\,\|\overline{u}\|_{H^{1}(Q)}^{2}$
$\displaystyle+2\inf\limits_{v_{h}\in
X_{h}}\Big{[}2\,\max\\{c_{F},1\\}\,\varrho\,\|\overline{u}-v_{h}\|^{2}_{H^{1}(Q)}+\|\overline{u}-v_{h}\|^{2}_{L^{2}(Q)}\Big{]}$

when using (2.13) and (2.12), the upper norm equivalence inequality in (2.7)
with $c_{2}^{S}=2$, and $\|\overline{u}\|_{H^{1}(Q)}$ as upper bound of
$\|\overline{u}\|_{X}$, see (3.1). Now inserting a suitable $H^{1}$-stable
quasi-interpolation $v_{h}=P_{h}\overline{u}\in X_{h}$ of the desired state
$\overline{u}\in H^{1}(Q)$, e.g., Scott–Zhang’s interpolation [3], we
immediately obtain the estimate

$\|u_{\varrho}-u_{\varrho h}\|_{L^{2}(Q)}^{2}\leq
c\,[\varrho+h^{2}]\,\|\overline{u}\|^{2}_{H^{1}(Q)}.$

Combining this estimate with (2.12) and chosing $\varrho=h^{2}$ finally gives

$\|u_{\varrho h}-\overline{u}\|_{L^{2}(Q)}\leq
c\,h\,\|\overline{u}\|_{H^{1}(Q)}.$

Next we consider $\overline{u}\in X\cap H^{2}(Q)$ which guarantees
$S\overline{u}\in L^{2}(Q)$. Similar as above, but now using (2.14) and
(2.15), we then obtain the estimates

$\displaystyle\|u_{\varrho h}-\overline{u}\|_{L^{2}(Q)}^{2}$
$\displaystyle\leq$ $\displaystyle 2\,\|u_{\varrho
h}-u_{\varrho}\|_{L^{2}(Q)}^{2}+2\,\|u_{\varrho}-\overline{u}\|_{L^{2}(Q)}^{2}$
$\displaystyle\leq$ $\displaystyle
10\,\varrho^{2}\,\|S\overline{u}\|^{2}_{L^{2}(Q)}+4\inf\limits_{v_{h}\in
X_{h}}\Big{[}\varrho\,\|\overline{u}-v_{h}\|^{2}_{S}+\|\overline{u}-v_{h}\|^{2}_{L^{2}(Q)}\Big{]}$
$\displaystyle\leq$ $\displaystyle c\,[\varrho^{2}+\varrho
h^{2}+h^{4}]\,\|\overline{u}\|_{H^{2}(Q)}\,.$

Here we have used the estimate

$\|S\overline{u}\|_{L^{2}(Q)}\leq c\,\|\overline{u}\|_{H^{2}(Q)}$

that can be shown by Fourier analysis; cf. [15]. Chosing $\varrho=h^{2}$
yields

$\|u_{\varrho h}-\overline{u}\|_{L^{2}(Q)}\leq
c\,h^{2}\,\|\overline{u}\|_{H^{2}(Q)}\,.$

The general estimate for $s\in[0,1)$ and $s\in[1,2]$ now follows from a space
interpolation argument; see, e.g., [16]. ∎

######  Corollary 1.

Let us assume that $\overline{u}\in X\cap H^{s}(Q)$ for some $s\in[1,2]$. Then
there holds the error estimate

$\|u_{\varrho h}-\overline{u}\|_{X}\leq
c\,h^{s-1}\|\overline{u}\|_{H^{s}(Q)}.$ (3.8)

###### Proof.

Let $P_{h}\overline{u}\in X_{h}$ be again Scott–Zhang’s interpolation of
$\overline{u}\in H^{1}(Q)$. Using an inverse inequality and standard arguments
we obtain

$\displaystyle\|u_{\varrho h}-\overline{u}\|_{X}$ $\displaystyle\leq$
$\displaystyle\|u_{\varrho h}-\overline{u}\|_{H^{1}(Q)}$ $\displaystyle\leq$
$\displaystyle\|u_{\varrho
h}-P_{h}\overline{u}\|_{H^{1}(Q)}+\|P_{h}\overline{u}-\overline{u}\|_{H^{1}(Q)}$
$\displaystyle\leq$ $\displaystyle c\,h^{-1}\,\|u_{\varrho
h}-P_{h}\overline{u}\|_{L^{2}(Q)}+c\,h^{s-1}\,\|\overline{u}\|_{H^{s}(Q)}$
$\displaystyle\leq$ $\displaystyle c\,h^{-1}\,\Big{[}\|u_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}+\|\overline{u}-P_{h}\overline{u}\|_{L^{2}(Q)}\Big{]}+c\,h^{s-1}\,\|\overline{u}\|_{H^{s}(Q)}$
$\displaystyle\leq$ $\displaystyle c\,h^{s-1}\,\|\overline{u}\|_{H^{s}(Q)}\,.$

∎

Since (3.6) requires, for any given $w\in X$, the evaluation of
$Sw=B^{*}A^{-1}Bw$, we have to define a suitable computable approximation
$\widetilde{S}w$. This can be done as follows. For given $w\in X$, we
introduce $p_{w}=A^{-1}Bw\in Y$ as the unique solution of the variational
formulation

$\langle Ap_{w},q\rangle_{Q}=\langle Bw,q\rangle_{Q}\quad\forall\,q\in Y.$

Let $p_{wh}\in Y_{h}\subset Y$ be the continuous, piecewise linear space-time
finite element approximation to $p_{w}\in Y$, satisfying

$\langle Ap_{wh},q_{h}\rangle_{Q}=\langle
Bw,q_{h}\rangle_{Q}\quad\forall\,q_{h}\in Y_{h}.$ (3.9)

With this we define the approximate operator $\widetilde{S}w:=B^{*}p_{wh}$ of
$Sw=B^{*}p_{w}$. The boundedness of $B:X\to Y^{*}$ implies

$\|\widetilde{S}w\|_{X^{*}}=\|B^{*}p_{wh}\|_{X^{*}}\leq
c_{2}^{B}\,\|p_{wh}\|_{X},$

while the ellipticity of $A:Y\to Y^{*}$ gives

$c_{1}^{A}\,\|p_{wh}\|^{2}_{Y}\leq\langle Ap_{wh},p_{wh}\rangle_{Q}=\langle
Bw,p_{wh}\rangle_{Q}\leq c_{2}^{B}\,\|w\|_{X}\|p_{wh}\|_{Y},$

i.e.,

$\|p_{wh}\|_{Y}\leq\frac{c_{2}^{B}}{c_{1}^{A}}\,\|w\|_{X}\,.$

Hence, we conclude the boundedness of the approximate operator
$\widetilde{S}:X\to X^{*}$,

$\|\widetilde{S}w\|_{X^{*}}\leq
c_{2}^{\widetilde{S}}\,\|w\|_{X}\quad\forall\,w\in X,\quad
c_{2}^{\widetilde{S}}=\frac{[c_{2}^{B}]^{2}}{c_{1}^{A}}=2.$ (3.10)

Instead of (3.6), we now consider the perturbed variational formulation to
find $\widetilde{u}_{\varrho h}\in X_{h}$ such that

$\varrho\,\langle\widetilde{S}\widetilde{u}_{\varrho
h},v_{h}\rangle_{Q}+\langle\widetilde{u}_{\varrho
h},v_{h}\rangle_{L^{2}(Q)}=\langle\overline{u},v_{h}\rangle_{L^{2}(Q)}$ (3.11)

is satisfied for all $v_{h}\in X_{h}$. Unique solvability of (3.11) follows
since the stiffness matrix of $\widetilde{S}$ is positive semi-definite, while
the mass matrix, which is related to the inner product in $L^{2}(Q)$, is
positive definite.

######  Lemma 5.

Let $u_{\varrho h}\in X_{h}$ and $\widetilde{u}_{\varrho h}\in X_{h}$ be the
unique solutions of the variational formulations (3.6) and (3.11),
respectively. Assume $\overline{u}\in X\cap H^{1}(Q)$. Then, there holds the
error estimate

$\|u_{\varrho h}-\widetilde{u}_{\varrho h}\|_{L^{2}(Q)}\leq
c\,h\,\|\overline{u}\|_{H^{1}(Q)}.$

###### Proof.

The difference of the variational formulations (3.6) and (3.11) first gives
the Galerkin orthogonality

$\varrho\,\langle Su_{\varrho h}-\widetilde{S}\widetilde{u}_{\varrho
h},v_{h}\rangle_{Q}+\langle u_{\varrho h}-\widetilde{u}_{\varrho
h},v_{h}\rangle_{L^{2}(Q)}=0\quad\forall\,v_{h}\in X_{h},$

which can be written as

$\varrho\,\langle\widetilde{S}(\widetilde{u}_{\varrho h}-u_{\varrho
h}),v_{h}\rangle_{Q}+\langle\widetilde{u}_{\varrho h}-u_{\varrho
h},v_{h}\rangle_{L^{2}(Q)}=\varrho\,\langle(S-\widetilde{S})u_{\varrho
h},v_{h}\rangle_{Q}\quad\forall\,v_{h}\in X_{h}.$

In particular, chosing $v_{h}=\widetilde{u}_{\varrho h}-u_{\varrho h}\in
X_{h}$, using $\langle\widetilde{S}w,w\rangle_{Q}\geq 0$ for all $w\in X$,
applying an inverse inequality in $X_{h}$, i.e., using the dual norm for
$\|\partial_{t}v_{h}\|_{Y^{*}}$ and Friedrich’s inequality (3.2), we arrive at
the estimates

$\displaystyle\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|^{2}_{L^{2}(Q)}$
$\displaystyle\leq$ $\displaystyle\varrho\,\langle(S-\widetilde{S})u_{\varrho
h},\widetilde{u}_{\varrho h}-u_{\varrho h}\rangle_{Q}$ $\displaystyle\leq$
$\displaystyle\varrho\,\|(S-\widetilde{S})u_{\varrho
h}\|_{X^{*}}\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{X}$
$\displaystyle\leq$ $\displaystyle
c\,\varrho\,h^{-1}\,\|(S-\widetilde{S})u_{\varrho
h}\|_{X^{*}}\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)},$

i.e.,

$\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}\leq
c\,\varrho\,h^{-1}\,\|(S-\widetilde{S})u_{\varrho h}\|_{X^{*}}.$

Since $\overline{u}\in X$, we can further estimate

$\displaystyle\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}$
$\displaystyle\leq$ $\displaystyle
c\,\varrho\,h^{-1}\,\Big{[}\|(S-\widetilde{S})(u_{\varrho
h}-\overline{u})\|_{X^{*}}+\|(S-\widetilde{S})\overline{u}\|_{X^{*}}\Big{]}$
$\displaystyle\leq$ $\displaystyle c\,\varrho\,h^{-1}\,\Big{[}4\,\|u_{\varrho
h}-\overline{u}\|_{X}+\sqrt{2}\,\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\Big{]}\,,$

where we used the boundedness of $S$ and $\widetilde{S}$. We note that
$p_{\overline{u}}=A^{-1}B\overline{u}$, and $p_{\overline{u}h}\in Y_{h}$
solves (3.9) with $w=\overline{u}$. For $\overline{u}\in H^{1}(Q)$, we can use
standard arguments as well as (3.1) to bound

$\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\leq\|p_{\overline{u}}\|_{Y}=\|A^{-1}B\overline{u}\|_{Y}\leq\frac{c_{2}^{B}}{c_{1}^{A}}\,\|\overline{u}\|_{X}\leq\sqrt{2}\,\max\\{\sqrt{c_{F}},1\\}\,\|\overline{u}\|_{H^{1}(Q)},$

and using (3.8) for $s=1$ we finally obtain, using $\varrho=h^{2}$,

$\|\widetilde{u}_{\varrho h}-u_{\varrho
h}\|_{L^{2}(Q)}\,\leq\,c\,h\,\|\overline{u}\|_{H^{1}(Q)}.$

∎

######  Theorem 2.

Assume $\overline{u}\in[L^{2}(Q),X]_{s}\cap H^{s}(Q)$ for $s\in[0,1]$, and
$\varrho=h^{2}$. Then,

$\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}\leq
c\,h^{s}\,\|\overline{u}\|_{H^{s}(Q)}.$ (3.12)

###### Proof.

For $s=1$ the assertion is an immediate consequence of Theorem 1 and Lemma 5.
Now we consider (3.11) for $v_{h}=\widetilde{u}_{\varrho h}$,

$\varrho\,\langle\widetilde{S}\widetilde{u}_{\varrho h},\widetilde{u}_{\varrho
h}\rangle_{Q}+\langle\widetilde{u}_{\varrho
h}-\overline{u},\widetilde{u}_{\varrho
h}-\overline{u}\rangle_{L^{2}(Q)}=\langle\overline{u}-\widetilde{u}_{\varrho
h},\overline{u}\rangle_{L^{2}(Q)},$

from which we immediately conclude

$\|\widetilde{u}_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}\leq\|\overline{u}\|_{L^{2}(Q)}.$

The assertion then again follows by a space interpolation argument. ∎

The error estimate as given in (3.12) covers in particular the case when the
target is either discontinuous, or does not satisfy the required boundary or
initial conditions. It remains to consider the case when the target
$\overline{u}$ is smooth. As in the proof of Lemma 5, and using (3.8) for
$s=2$, we now have, recall $\varrho=h^{2}$,

$\displaystyle\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}$
$\displaystyle\leq$ $\displaystyle c\,\varrho\,h^{-1}\,\Big{[}4\,\|u_{\varrho
h}-\overline{u}\|_{X}+\sqrt{2}\,\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\Big{]}$
$\displaystyle\leq$ $\displaystyle
c_{1}\,h^{2}\,\|\overline{u}\|_{H^{2}(Q)}+c_{2}\,h\,\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\,.$

When using the approximation result as given in [14, Theorem 3.3] we have

$\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\leq
c\,h\,\|p_{\overline{u}}\|_{H^{2}(Q)},$ (3.13)

i.e., we obtain

$\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}\leq
c\,h^{2}\,\Big{[}\|\overline{u}\|_{H^{2}(Q)}+\|p_{\overline{u}}\|_{H^{2}(Q)}\Big{]}.$
(3.14)

While the error estimate (3.13) holds for any admissible decomposition of the
space-time domain $Q$ into simplicial finite elements, in addition to
$\overline{u}\in X\cap H^{2}(Q)$, we have to assume
$p_{\overline{u}}=A^{-1}B\overline{u}\in H^{2}(Q)$, i.e., $\overline{u}\in
H^{2,3}(Q)$. This additional regularity requirement in time is due to the
finite element error estimate (3.13) which does not reflect the anisotropic
behavior in space and time of the norm in $Y=L^{2}(0,T;H^{1}_{0}(\Omega))$.
However, and as already discussed in [14, Corollary 4.2], we can improve the
error estimate (3.13) under additional assumptions on the underlying space-
time finite element mesh. In fact, when considering as in [14, Section 4]
right-angled space-time finite elements, or space-time tensor product meshes,
instead of (3.13) we obtain the error estimate

$\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\leq
c\,h\,|\nabla_{x}p_{\overline{u}}|_{H^{1}(Q)},$ (3.15)

when assuming $\nabla_{x}p_{\overline{u}}\in H^{1}(Q)$ for
$p_{\overline{u}}=A^{-1}B\overline{u}$,, i.e., there are no second order time
derivatives yet. This is the reason to further conclude the bound

$\|p_{\overline{u}}-p_{\overline{u}h}\|_{Y}\leq
c\,h\,\|\overline{u}\|_{H^{2}(Q)},$

and hence,

$\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}\leq
c\,h^{2}\,\|\overline{u}\|_{H^{2}(Q)}$ (3.16)

follows, when assuming $\overline{u}\in X\cap H^{2}(Q)$. Now, interpolating
(3.12) for $s=1$ and (3.16), we conclude

$\|\widetilde{u}_{\varrho h}-u_{\varrho h}\|_{L^{2}(Q)}\leq
c\,h^{s}\,\|\overline{u}\|_{H^{s}(Q)}\quad\mbox{for}\;\overline{u}\in X\cap
H^{s}(Q),\;s\in[1,2]$ (3.17)

that together with estimate (3.7) from Theorem 1 finally gives

$\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}\leq
c\,h^{s}\,\|\overline{u}\|_{H^{s}(Q)}.$ (3.18)

While we can prove this result for some structured space-time finite element
meshes only, numerical experiments indicate that (3.18) remains true for any
admissible decomposition of the space-time domain into simplicial finite
elements.

##  4 Numerical results

In the numerical experiments, we choose the spatial domain $\Omega=(0,1)^{n}$
with $n=2$ (Subsection 4.1) and $n=3$ (Subsection 4.2), and final time $T=1$,
resulting in the $n+1$-dimensional space-time cylinder $Q=(0,1)^{n+1}$. We
follow the space-time finite element method on fully unstructured simplicial
meshes as considered in [8] for the coupled optimality system of the parabolic
distributed control problem (1.1). This finally leads to the solution of a
saddle-point system that is nothing but the discrete version of (3.5): Find
the nodal parameter vectors $\underline{p}\in\mathbb{R}^{M_{Y}}$
($M_{Y}=\text{dim}(Y_{h})$) and $\underline{u}\in\mathbb{R}^{M_{X}}$
($M_{X}=\text{dim}(X_{h})$) such that

$\begin{pmatrix}\varrho^{-1}A_{h}&B_{h}\\\
B_{h}^{\top}&-M_{h}\end{pmatrix}\begin{pmatrix}\underline{p}\\\
\underline{u}\end{pmatrix}=\begin{pmatrix}\underline{0}\\\
-\underline{f}\end{pmatrix},$ (4.1)

where the finite element matrices $A_{h}$, $B_{h}$, and $M_{h}$ correspond to
the bilinear forms (3.4) and (3.3), and to the $L_{2}(Q)$ inner product,
respectively. The matrices $A_{h}\in{\mathbb{R}}^{M_{Y}\times M_{Y}}$ and
$M_{h}\in{\mathbb{R}}^{M_{X}\times M_{X}}$ are symmetric and positive
definite, while the matrix $B_{h}\in{\mathbb{R}}^{M_{Y}\times M_{X}}$ is in
general rectangular. The load vector $\underline{f}\in{\mathbb{R}}^{N_{X}}$ is
computed from the given target $\overline{u}$ as usual. We mention that the
symmetric, but indefinite system (4.1) is equivalent to solving the related
Schur complement system

$\left(\varrho
B_{h}^{\top}A_{h}^{-1}B_{h}+M_{h}\right)\underline{u}=\underline{f}$

that corresponds to (3.11). Here, the symmetric but indefinite system (4.1) is
simply solved by the ILU(0) preconditioned GMRES method; see [8]. We stop the
GMRES iteration when the relative residual error of the preconditioned system
is reduced by a factor $10^{8}$.

###  4.1 Two space dimensions

In the first example (Example 4.1.1), we consider the smooth target

$\overline{u}(x,t)=\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi t)$ (4.2)

where we can apply the error estimate (3.14). As predicted, we observe a
second order convergence with respect to the mesh size $h$ when choosing
$\varrho=h^{2}$; see Table 1.

$h$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ | eoc
---|---|---|---
$2^{-2}$ | $2^{-4}$ |  $2.2380$e$-1$ |
$2^{-3}$ | $2^{-6}$ |  $9.0449$e$-2$ | $1.31$
$2^{-4}$ | $2^{-8}$ |  $2.6491$e$-2$ | $1.77$
$2^{-5}$ | $2^{-10}$ |  $6.9335$e$-3$ | $1.93$
$2^{-6}$ | $2^{-12}$ |  $1.7613$e$-3$ | $1.98$
$2^{-7}$ | $2^{-14}$ |  $4.4352$e$-4$ | $1.99$
$2^{-8}$ | $2^{-16}$ |  $1.0600$e$-4$ | $2.06$
$2^{-9}$ | $2^{-18}$ |  $2.6836$e$-5$ | $1.98$
Table 1: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the
case of a smooth target $\overline{u}$ given by (4.2) (Example 4.1.1).

As a second example (Example 4.1.2), we consider a piecewise linear continuous
function $\overline{u}$ being one at the mid point $(1/2,1/2,1/2)$, and zero
in all corner points of $Q=(0,1)^{3}$. In this case, we have $\overline{u}\in
X\cap H^{3/2-\varepsilon}(Q)$, $\varepsilon>0$, and we observe $1.5$ as the
order of convergence, see Table 2, which corresponds to the error estimate
(3.17).

$h$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ | eoc
---|---|---|---
$2^{-2}$ | $2^{-4}$ |  $2.0231$e$-1$ |
$2^{-3}$ | $2^{-6}$ |  $9.1319$e$-2$ | $1.15$
$2^{-4}$ | $2^{-8}$ |  $3.4303$e$-2$ | $1.41$
$2^{-5}$ | $2^{-10}$ |  $1.2428$e$-2$ | $1.46$
$2^{-6}$ | $2^{-12}$ |  $4.4443$e$-3$ | $1.48$
$2^{-7}$ | $2^{-14}$ |  $1.5797$e$-3$ | $1.49$
$2^{-8}$ | $2^{-16}$ |  $5.5868$e$-4$ | $1.50$
$2^{-9}$ | $2^{-18}$ |  $1.9786$e$-4$ | $1.50$
Table 2: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the
case of a piecewise linear continuous target $\overline{u}\in X\cap
H^{3/2-\varepsilon}(Q)$, $\varepsilon>0$ (Example 4.1.2).

As a third example (Example 4.1.3), we take a piecewise constant discontinuous
target $\overline{u}$ which is one in the inscribed cube
$(\frac{1}{4},\frac{3}{4})^{3}$, and zero elsewhere. In this case, we have
$\overline{u}\in H^{1/2-\varepsilon}(Q)$, $\varepsilon>0$. From the numerical
results given in Table 3, we observe $0.5$ for the order of convergence, as
expected from the error estimate (3.12). In this example, since the target
$\overline{u}$ is discontinuous, we may apply an adaptive refinement based on
the residual type error indicator as used in [8]. We compare the errors and
number of degrees of freedom using both uniform and adaptive refinements in
Table 4, with respect to the regularization parameter $\varrho$. We clearly
see that for each regularization parameter $\varrho$, the adaptive refinement
requires less degrees of freedom to reach a similar accuracy as for uniform
refinements. In Figure 1, we plot the state $u$, the adjoint state $p$, and
the control $z$ at time $t=0.5$, and the adaptive meshes in space-time. For
comparison of the results with different regularization terms, we refer to the
numerical results in [8].

$h$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ | eoc
---|---|---|---
$2^{-2}$ | $2^{-4}$ |  $2.8840$e$-1$ |
$2^{-3}$ | $2^{-6}$ |  $2.0871$e$-1$ | $0.47$
$2^{-4}$ | $2^{-8}$ |  $1.4793$e$-1$ | $0.50$
$2^{-5}$ | $2^{-10}$ |  $1.0473$e$-1$ | $0.50$
$2^{-6}$ | $2^{-12}$ |  $7.4108$e$-2$ | $0.50$
$2^{-7}$ | $2^{-14}$ |  $5.2425$e$-2$ | $0.50$
$2^{-8}$ | $2^{-16}$ |  $3.7079$e$-2$ | $0.50$
$2^{-9}$ | $2^{-18}$ |  $2.6219$e$-2$ | $0.50$
Table 3: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the case of a discontinuous target $\overline{u}\in H^{1/2-\varepsilon}(Q)$, $\varepsilon>0$ (EXample 4.1.3). | uniform refinement | adaptive refinement
---|---|---
$\varrho$ | $h=\varrho^{1/2}$ | #DOFs | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ | #DOFs | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$
$2^{-4}$ | $2^{-2}$ | $250$ |  $2.8840$e$-1$ | $250$ |  $2.8840$e$-1$
$2^{-6}$ | $2^{-3}$ | $1,458$ |  $2.0871$e$-1$ | $1,230$ |  $2.0873$e$-1$
$2^{-8}$ | $2^{-4}$ | $9,826$ |  $1.4793$e$-1$ | $9,948$ |  $1.3999$e$-1$
$2^{-10}$ | $2^{-5}$ | $71,874$ |  $1.0473$e$-1$ | $34,998$ |  $1.0153$e$-1$
$2^{-12}$ | $2^{-6}$ | $549,250$ |  $7.4108$e$-2$ | $230,154$ |  $7.2804$e$-2$
$2^{-14}$ | $2^{-7}$ | $4,293,378$ |  $5.2425$e$-2$ | $1,526,400$ |  $5.1838$e$-2$
$2^{-16}$ | $2^{-8}$ | $33,949,186$ |  $3.7079$e$-2$ | $6,196,200$ |  $3.6609$e$-2$
$2^{-18}$ | $2^{-9}$ | $270,011,394$ |  $2.6219$e$-2$ | $31,419,720$ |  $2.5824$e$-2$
Table 4: Comparison of the error $\|\widetilde{u}_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}$ and the number of degrees of freedoms in the
case of a discontinuous target $\overline{u}\in H^{1/2-\varepsilon}(Q)$,
$\varepsilon>0$, when using both uniform and adaptive refinements (Example
4.1.3).

Figure 1: Visualization of the state $u_{\varrho}$, the adjoint state
$p_{\varrho}$, the control $z_{\varrho}$, and the adaptive mesh on the cutting
plane at time $t=0.5$, where the total #DOFs in space-time is $3,328,617$ at
the $58$th adaptive level from $67$ levels (last line in Table 4)
corresponding to the regularization parameter $\varrho=2^{-18}$ (Example
4.1.3).

In the last example (Example 4.1.4) of this subsection, we consider the
discontinuous target

$\overline{u}^{\delta}=\overline{u}+2\sqrt{2}\delta\sin(10\pi x_{1})\sin(10\pi
x_{2})\sin(10\pi t),$ (4.3)

that contains some noise in space and time. Here, $\overline{u}$ is one in the
inscribed cube $(1/4,3/4)^{3}\subset(0,1)^{3}$ and zero else, and $\delta>0$
is the noise level. For this polluted target, we easily see that
$\|\overline{u}^{\delta}-\overline{u}\|_{L^{2}(Q)}=\delta$. To balance the two
error contributions we take $h=16\,\delta^{2}$. This ensures an almost optimal
convergence with respect to the mesh size $h$, see Table 5.

$\delta$ | $h\,\left(=16\cdot\delta^{2}\right)$ | $\varrho\,\left(=h^{2}\right)$ | $\|\widetilde{u}_{\varrho h}^{\delta}-\overline{u}\|_{L^{2}(Q)}$ | eoc
---|---|---|---|---
$2^{-3}$ | $2^{-2}$ | $2^{-4}$ |  $2.8841$e$-1$ |
$2^{-3.5}$ | $2^{-3}$ | $2^{-6}$ |  $2.0871$e$-1$ | $0.47$
$2^{-4}$ | $2^{-4}$ | $2^{-8}$ |  $1.4796$e$-1$ | $0.50$
$2^{-4.5}$ | $2^{-5}$ | $2^{-10}$ |  $1.0535$e$-1$ | $0.49$
$2^{-5}$ | $2^{-6}$ | $2^{-12}$ |  $7.6837$e$-2$ | $0.46$
$2^{-5.5}$ | $2^{-7}$ | $2^{-14}$ |  $5.5990$e$-2$ | $0.46$
Table 5: Error $\|\widetilde{u}_{\varrho
h}^{\delta}-\overline{u}\|_{L^{2}(Q)}$ in the case of a discontinuous target
$\overline{u}\in H^{1/2-\varepsilon}(Q)$ containing some noise level $\delta$
(Example 4.1.4).

###  4.2 Three space dimensions

Now we present some numerical results for the three-dimensional spatial domain
$\Omega=(0,1)^{3}$, i.e., $Q=(0,1)^{4}$.

In the first example (Example 4.2.1), we look at the smooth target

$\overline{u}(x,t)=\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3})\sin(\pi t).$
(4.4)

As predicted by the error estimate (3.14), we observe a second order
convergence with respect to the mesh size $h$ when choosing $\varrho=h^{2}$;
see Table 6 and Figure 2.

N(#DOFs) | $h=(N/2)^{-1/4}$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$
---|---|---|---
$356$ |  $2.7378$e$-1$ |  $7.4953$e$-2$ |  $2.0985$e$-1$
$630$ |  $2.3737$e$-1$ |  $5.6344$e$-2$ |  $1.7263$e$-1$
$2,986$ |  $1.6087$e$-1$ |  $2.5880$e$-2$ |  $1.2166$e$-1$
$6,930$ |  $1.3034$e$-1$ |  $1.6988$e$-2$ |  $9.3733$e$-2$
$38,114$ |  $8.5111$e$-2$ |  $7.2439$e$-3$ |  $4.7198$e$-2$
$94,146$ |  $6.7890$e$-2$ |  $4.6091$e$-3$ |  $3.2098$e$-2$
$546,562$ |  $4.3737$e$-2$ |  $1.9129$e$-3$ |  $1.4088$e$-2$
$1,400,322$ |  $3.4570$e$-2$ |  $1.1951$e$-3$ |  $8.8652$e$-3$
$8,289,026$ |  $2.2163$e$-2$ |  $4.9121$e$-4$ |  $3.7164$e$-3$
$21,657,090$ |  $1.7432$e$-2$ |  $3.0389$e$-4$ |  $2.2967$e$-3$
$129,165,826$ |  $1.1155$e$-2$ |  $1.2444$e$-4$ |  $9.5061$e$-4$
Table 6: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the
case of the smooth target $\overline{u}$ given by (4.4) (Example 4.2.1).
Figure 2: Error $\|e\|_{L^{2}(Q)}=\|\widetilde{u}_{\varrho
h}-\overline{u}\|_{L^{2}(Q)}$ in the case of a smooth desired state
$\overline{u}\in H_{0}^{1}(Q)\cap H^{2}(Q)$ in three space dimensions (Example
4.2.1).

In the second example (Example 4.2.2), we take a piecewise linear continuous
target function $\overline{u}$ being one in the mid point $(1/2,1/2,1/2,1/2)$
and zero in all corner points of $Q=(0,1)^{4}$. In this case we have
$\overline{u}\in X\cap H^{3/2-\varepsilon}$, $\varepsilon>0$, and we observe
$1.5$ as order of convergence which corresponds to the error estimate (3.17),
see Table 7 and Figure 3.

N(#DOFs) | $h=(N/2)^{-1/4}$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$
---|---|---|---
$356$ |  $2.7378$e$-1$ |  $7.4953$e$-2$ |  $2.1510$e$-1$
$630$ |  $2.3737$e$-1$ |  $5.6344$e$-2$ |  $1.7972$e$-1$
$2,986$ |  $1.6087$e$-1$ |  $2.5880$e$-2$ |  $1.3082$e$-1$
$6,930$ |  $1.3034$e$-1$ |  $1.6988$e$-2$ |  $1.0510$e$-1$
$38,114$ |  $8.5111$e$-2$ |  $7.2439$e$-3$ |  $6.1638$e$-2$
$94,146$ |  $6.7890$e$-2$ |  $4.6091$e$-3$ |  $4.5864$e$-2$
$546,562$ |  $4.3737$e$-2$ |  $1.9129$e$-3$ |  $2.4759$e$-2$
$1,400,322$ |  $3.4570$e$-2$ |  $1.1951$e$-3$ |  $1.7766$e$-2$
$8,289,026$ |  $2.2163$e$-2$ |  $4.9121$e$-4$ |  $9.2755$e$-3$
$21,657,090$ |  $1.7432$e$-2$ |  $3.0389$e$-4$ |  $6.5280$e$-3$
$129,165,826$ |  $1.1155$e$-2$ |  $1.2444$e$-4$ |  $3.3636$e$-3$
Table 7: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the
case of a piecewise linear continuous target $\overline{u}\in X\cap
H^{3/2-\varepsilon}(Q)$, $\varepsilon>0$ (Example 4.2.2). Figure 3: Error
$\|e\|_{L^{2}(Q)}=\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in
the case of a piecewise linear continuous target $\overline{u}\in X\cap
H^{3/2-\varepsilon}(Q)$, $\varepsilon>0$ (Example 4.2.2).

In the third example (Example 4.2.3), we consider a piecewise constant
discontinuous target $\overline{u}$ which is one in the inscribed cube
$(\frac{1}{4},\frac{3}{4})^{4}$, and zero else. In this case we have
$\overline{u}\in H^{1/2-\varepsilon}(Q)$, $\varepsilon>0$. From the numerical
results as given in Table 8 we observe $0.5$ for the order of convergence, as
expected from the error estimate (3.12), see also Figure 4.

N(#DOFs) | $h=(N/2)^{-1/4}$ | $\varrho\,(=h^{2})$ | $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$
---|---|---|---
$356$ |  $2.7378$e$-1$ |  $7.4953$e$-2$ |  $2.5099$e$-1$
$630$ |  $2.3737$e$-1$ |  $5.6344$e$-2$ |  $1.9143$e$-1$
$2,986$ |  $1.6087$e$-1$ |  $2.5880$e$-2$ |  $1.8823$e$-1$
$6,930$ |  $1.3034$e$-1$ |  $1.6988$e$-2$ |  $1.7500$e$-1$
$38,114$ |  $8.5111$e$-2$ |  $7.2439$e$-3$ |  $1.4710$e$-1$
$94,146$ |  $6.7890$e$-2$ |  $4.6091$e$-3$ |  $1.3313$e$-1$
$546,562$ |  $4.3737$e$-2$ |  $1.9129$e$-3$ |  $1.0558$e$-1$
$1,400,322$ |  $3.4570$e$-2$ |  $1.1951$e$-3$ |  $9.6592$e$-2$
$8,289,026$ |  $2.2163$e$-2$ |  $4.9121$e$-4$ |  $7.7744$e$-2$
$21,657,090$ |  $1.7432$e$-2$ |  $3.0389$e$-4$ |  $6.8891$e$-2$
$129,165,826$ |  $1.1155$e$-2$ |  $1.2444$e$-4$ |  $5.5284$e$-2$
Table 8: Error $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in the
case of a piecewise constant and discontinuous target $\overline{u}\in
H^{1/2-\varepsilon}(Q)$, $\varepsilon>0$ (Example 4.2.3). Figure 4: Error
$\|e\|_{L^{2}(Q)}=\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^{2}(Q)}$ in
the case of a piecewise constant and discontinuous target $\overline{u}\in
H^{1/2-\varepsilon}(Q)$, $\varepsilon>0$ (Example 4.2.3).

##  5 Conclusions and outlook

We have derived robust space-time finite element error estimates for
distributed parabolic optimal control problems with energy regularization.
More precisely, we have estimated the $L^{2}(Q)$ norm of the error between the
desired state $\overline{u}$ and the computed state $\widetilde{u}_{\varrho
h}$ depending on the regularity of the desired state $\overline{u}$. It has
been shown that the optimal convergence rate is achieved by the proper scaling
$\varrho=h^{2}$ between the regularization parameter $\varrho$ and the mesh
size $h$. The theoretical findings are confirmed by several numerical examples
in both two and three space dimensions.

The theoretical results are valid for uniform mesh refinement. However, for
discontinuous targets $\overline{u}$ and targets that don’t fulfil the
boundary or initial conditions, we can expect layers with steep gradients in
the solutions as in Example 4.2.3. In this example, we have observed that, for
a fixed $\varrho=h^{2}$, the adaptive version needs considerably less unknowns
to achieve the same accuracy as the corresponding uniformly refined grid with
the finest mesh-size $h$. Since, for adaptively refined grids, the local mesh-
sizes are very different, one can also think about a localization of the
regularization parameter $\varrho$. Another future research topic is the
construction of fast and $\varrho$ robust solvers for the symmetric and
indefinite system (4.1) that is equivalent to (3.11); see, e.g., [1, 12, 13,
19]. Finally, the consideration of constraints imposed on the control
$z_{\varrho}$ is of practical interest; see, e.g., [17].

## Acknowledgments

The authors would like to acknowledge the computing support of the
supercomputer MACH–2444https://www3.risc.jku.at/projects/mach2/ from Johannes
Kepler Universität Linz and of the high performance computing cluster
Radon1555https://www.oeaw.ac.at/ricam/hpc from Johann Radon Institute for
Computational and Applied Mathematics (RICAM) on which the numerical examples
are performed. The first and the third author were partially supported by
RICAM.

## References

  * [1] M. Benzi, G. H. Golub, and J. Liesen.  Numerical solution of saddle point problems.  Acta Numer., 14:1–137, 2005.
  * [2] A. Borzì and V. Schulz.  Computational optimization of systems governed by partial differential equations, volume 8 of Computational Science & Engineering.  Society for Industrial and Applied Mathematics (SIAM), 2011.
  * [3] S. Brenner and R. Scott.  The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics.  Springer, New York, 2008.
  * [4] E. Casas.  A review on sparse solutions in optimal control of partial differential equations.  SeMA Journal, 74:319–344, 2017.
  * [5] A. Ern and J.-L. Guermond.  Theory and Practice of Finite Elements.  Springer-Verlag, New York, 2004.
  * [6] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich.  Optimization with PDE Constraints, volume 23.  Springer-Verlag, Berlin, 2009.
  * [7] V. John.  Finite Element Methods for Incompressible Flow Problems, volume 51 of Springer Series in Computational Mathematics.  Springer, 2016.
  * [8] U. Langer, O. Steinbach, F. Tröltzsch, and H. Yang.  Space-time finite element discretization of parabolic optimal control problems with energy regularization.  SIAM J. Numer. Anal., 59(2):660–674, 2021.
  * [9] U. Langer, O. Steinbach, and H. Yang.  Robust discretization and solvers for elliptic optimal control problems with energy regularization.  Comput. Meth. Appl. Math., 22(1):97–111, 2022.
  * [10] J. L. Lions.  Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles.  Dunod Gauthier-Villars, Paris, 1968.
  * [11] M. Neumüller and O. Steinbach.  Regularization error estimates for distributed control problems in energy spaces.  Math. Methods Appl. Sci., 44:4176–4191, 2021.
  * [12] J. Schöberl and W. Zulehner.  Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems.  SIAM J. Matrix Anal. Appl., 29:752–773, 2007.
  * [13] V. Schulz and G. Wittum.  Transforming smoothers for pde constrained optimization problems.  Comput. Visual. Sci., 11(4):207–219, 2008.
  * [14] O. Steinbach.  Space-time finite element methods for parabolic problems.  Comput. Meth. Appl. Math., 15(4):551–566, 2015.
  * [15] O. Steinbach and M. Zank.  Coercive space-time finite element methods for initial boundary value problems.  Electron. Trans. Numer. Anal., 52:154–194, 2020.
  * [16] L. Tartar.  An Introduction to Sobolev Spaces and Interpolation Spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana.  Springer, Berlin, 2007.
  * [17] F. Tröltzsch.  Optimal control of partial differential equations: Theory, methods and applications, volume 112 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, Rhode Island, 2010.
  * [18] E. Zeidler.  Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators.  Springer, New York, 1990.
  * [19] W. Zulehner.  Nonstandard norms and robust estimates for saddle point problems.  SIAM J. Matrix Anal. Appl., 32(2):536–560, 2011.