Document [original]

Methods for the Temporal Approximation of

Nonlinear, Nonautonomous Evolution

Equations

vorgelegt von

M. Sc.

Monika Eisenmann

ORCID: 0000-0001-7428-9546

von der Fakult¨at II – Mathematik und Naturwissenschaften

der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktorin der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Martin Skutella

Gutachter: Prof. Dr. Etienne Emmrich

Gutachter: Dr. Raphael Kruse

Gutachterin: Prof. Dr. Mechthild Thalhammer

Tag der wissenschaftlichen Aussprache: 15.10.2019

Berlin 2019

Abstract

Differential equations are an important building block for modeling processes in physics,

biology, and social sciences. Usually, their exact solution is not known explicitly though.

Therefore, numerical schemes to approximate the solution are of great importance. In

this thesis, we consider the temporal approximation of nonlinear, nonautonomous evolution

equations on a finite time horizon. We present two independent approaches that can be

used to find a temporal approximation of the solution.

As the solution of a nonlinear equation typically lacks global higher-order regularity, it

cannot be expected to obtain higher-order convergence rates. Thus, we only concentrate on

schemes that are formally of first order.

In the first part of the thesis, we consider the question of how nonsmooth temporal data

can be handled. A common method for the approximation of the integral of an irregular

function is a Monte Carlo type quadrature rule. We take on this idea and use a similar

approach to approximate the solution to a nonautonomous evolution equation. If the data

is evaluated at the points of a randomly shifted grid, we can prove the convergence of the

backward Euler scheme. Moreover, we prove explicit error estimates. Here, we introduce a

second set of randomized points, where the data is evaluated, and make additional assump-

tions on the data and the solution.

Secondly, we approximate the solution via an operator splitting based scheme. We work

with both an implicit-explicit splitting and a product type splitting. First, we decompose

the operator into a monotone and a bounded part. The implicit-explicit splitting is used to

obtain one implicit equation that contains the monotone part. The bounded part is solved

in an explicit fashion. This way, we only solve as many implicit equations as necessary.

Further, we use a product type splitting on the monotone part. Even though this leads to

more problems, they are potentially easier to solve individually. For this splitting scheme,

we follow a similar approach as in the first part of the thesis. After proving the convergence

of the scheme, we provide error bounds under additional assumptions on both the data and

the solution.

In order to provide an interesting field of application, we show that the schemes can be

applied for the temporal approximation of certain nonlinear, parabolic problems.

Zusammenfassung

Differentialgleichungen bilden einen wichtigen Bestandteil f¨ur die Modellierung von Pro-

zessen in der Physik, Biologie und Sozialwissenschaft. Allerdings l¨asst sich ihre L¨osung nur in

seltenen F¨allen analytisch bestimmen. Aus diesem Grund ist eine numerische N¨aherung der

L¨osung von großer Wichtigkeit. In dieser Arbeit betrachten wir die Zeitdiskretisierung von

nichtlinearen, nichtautonomen Evolutionsgleichungen auf einem endlichen Zeitintervall. Wir

pr¨asentieren zwei unabh¨angig voneinander anwendbare L¨osungsverfahren f¨ur die zeitliche

Approximation der L¨osung.

Da die L¨osung einer nichtlinearen Gleichung h¨aufig irregul¨ar ist, k¨onnen keine beson-

ders hohen Konvergenzraten erwartet werden. Aus diesem Grund konzentrieren wir uns

ausschließlich auf Verfahren, die formal eine Konvergenzordnung von eins aufweisen.

Im ersten Teil der Arbeit besch¨aftigen wir uns mit der Frage, wie zeitlich irregul¨are

Daten behandelt werden k¨onnen. F¨ur die Approximation des Integrals einer nichtglatten

Funktion ist ein Monte-Carlo-Algorithmus h¨aufig eine gute Wahl. Wir verfolgen hier einen

¨ahnlichen Ansatz, um die L¨osung einer nichtautonomen Evolutionsgleichung zu approx-

imieren. Wir zeigen die Konvergenz des impliziten Euler Verfahrens unter der Verwendung

eines zuf¨allig verschobenes Zeitgitters. Weiterhin k¨onnen unter zus¨atzlichen Voraussetzun-

gen an die Daten und die L¨osung explizite Fehlerschranken angegeben werden. Um diese zu

zeigen, wenden wir eine weitere Randomisierung an.

Der zweite Teil der Arbeit enth¨alt ein Approximationsverfahren, das ein Operatorsplit-

ting nutzt. Wir verwenden sowohl ein implizit-explizites Splitting als auch ein Produktsplit-

ting. Hierbei zerlegen wir den Operator zun¨achst in einen monotonen und einen beschr¨ank-

ten Anteil. Das implizit-explizit Splitting wird genutzt, um eine implizite Gleichung zu

erhalten, die den monotonen Anteil enth¨alt. Der beschr¨ankte Anteil kann in einer expliziten

Gleichung gel¨ost werden. Auf diese Weise entstehen nur so viele implizite Gleichungen, wie

tats¨achlich notwendig sind. Weiterhin verwenden wir das Produktsplitting, um die implizite

Gleichung weiter aufzuteilen. Hierbei erhalten wir zwar mehr Gleichungen, diese sind aber

m¨oglicherweise leichter zu l¨osen. F¨ur das Splittingverfahren gehen wir ¨ahnlich vor wie im

ersten Teil der Arbeit. Nachdem die Konvergenz des Verfahrens gezeigt ist, wenden wir uns

auch hier expliziten Fehlerabsch¨atzungen zu, die unter zus¨atzlichen Voraussetzungen an die

Daten und die L¨osung m¨oglich sind.

Schlußendlich pr¨asentieren wir f¨ur beide Verfahren ein Anwendungsbeispiel aus dem

Bereich der nichtlinearen, parabolischen Differentialgleichungen.

iii

Acknowledgments

I would like to thank Etienne Emmrich for the chance to be part of this working group, for

always trusting me to follow my interests, and the valuable input throughout the years. I

am grateful to Raphael Kruse and Eskil Hansen for all the projects we worked on together.

I learned much from this and always enjoyed the joint work. I want to thank Raphael for

always taking the time to listen to my questions and giving me a lot of encouragement.

Further, I want to thank Eskil for inviting me to Lund for a visit and for the help to come

back later this year.

I want to thank my other co-authors Misi Kov´acs, Stig Larsson, and Volker Mehrmann

for the fruitful cooperation and hope there will be more projects to come. In particular,

I thank Stig and Misi for inviting me to Gothenburg. Moreover, I would like to thank

Mechthild Thalhammer for refereeing this thesis.

I am very grateful for having been a part of the differential equations working group

and I want to thank everybody who was part of this group over the last years. I enjoyed

the mathematical discussions and even more our non-mathematical ones. This made my

time here more memorable and much more fun. The whole thesis would not have been

possible without the support from all of you and I would specifically like to thank Andr´e,

Aras, Dana, Henrik, Lukas, Melanie, Nilasis, and Patrick for the proofreading. Without

their helpful remarks, this thesis would certainly be in a worse state. Further, I want to

thank all of the people mentioned above as well as Christian, Robert, and Rico for helpful

answers to my questions over the years. Moreover, I am grateful for Alex’s help with the

administration for the whole time I have been here.

Last but not least, I want to thank all of my friends and family who have supported me

and made the last years all the more enjoyable.

Contents

Introduction ix

1 Solvability and Regularity 1

1.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Regularity of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Randomized Schemes 9

2.1 Convergence on a Randomly Shifted Grid . . . . . . . . . . . . . . . . . . . . 11

2.2 Explicit Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Example: A Problem of p-Laplacian Type . . . . . . . . . . . . . . . . . . . . 40

3 Operator Splitting 47

3.1 Convergence of the Splitting Scheme . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 An Explicit Error Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Example: A Nonlinear Parabolic Problem . . . . . . . . . . . . . . . . . . . . 83

A Appendix 91

A.1 UsefulInequalities ................................. 91

A.2 BochnerIntegral .................................. 92

A.3 StochasticBackground............................... 94

Bibliography 97

vii

viii CONTENTS

Introduction

This work intends to present results on the approximation of nonlinear, nonautonomous

evolution equations on a finite time horizon. This type of equation appears when modeling

complex processes in physics, biology, and social sciences. Yet the solution can rarely be

written down explicitly. Therefore, it is important to find ways to approximate a solution

efficiently. In this thesis, we will present two independent approaches that can be helpful for

the temporal approximation of such equations. For the presented schemes, our aim is always

twofold: We begin to prove the convergence of a proposed scheme, while we do not make any

additional regularity assumptions on the solution. This verifies that our approaches work in

general settings. For practical uses, a certain classification of the size of the error becomes

important to rule out an arbitrarily slow convergence of the method. Thus, our second aim

is to show certain error bounds if the exact solution uis more regular. This quantifies the

error at least under additional assumptions.

We only concentrate on methods with a convergence order of at most one. Higher-order

schemes usually only lead to a better convergence rate if the solution is sufficiently smooth.

For general nonlinear problems, the solution usually lacks global higher-order spatial and

temporal regularity. Thus, we only concentrate on simpler schemes.

In the first part of this work, we consider a randomized scheme for the approximation of

the solution to an evolution equation. Precisely, let T∈(0,∞) as well as a Banach space V

and a Hilbert space Hbe given such that Vd

,→H∼

=H∗d

,→V∗. We consider the problem

(u0(t) + A(t)u(t) = f(t) in V∗,for almost all t∈(0, T ),

u(0) = u0in H.

Here, {A(t)}t∈[0,T ]is a family of operators such that A(t): V→V∗for every t∈[0, T ].

Further, A(t), t∈[0, T ], is a monotone, coercive, radially continuous operator, of at most

polynomial growth. The function f: [0, T]→V∗is integrable and u0∈H. Our starting

point for the approximation is rather simple: We approximate the solution of the evolution

equation with the well-known backward Euler scheme. To this end, for N∈N, let the

equidistant temporal grid 0 = t0< t1<· · · < tN=Tbe given with tn=nk,n∈ {0, . . . , N},

and the step size k=T

N. In order to find an approximation Un≈u(tn), a recursion of the

type

Un−Un−1

k+AnUn=fn, n ∈ {1, . . . , N},(1)

with U0=u0can be solved. Here, (An)n∈{1,...,N}and (fn)n∈{1,...,N}are approximations

for the operator and the right-hand side.

The question we want to address is what kind of approximations (An)n∈{1,...,N}and

(fn)n∈{1,...,N}should be used. While standard point evaluations for merely integrable data

xINTRODUCTION

are not well-defined, a suitable choice is

An=1

kZtn

tn−1

A(t) dt, fn=1

kZtn

tn−1

f(t) dt, n ∈ {1, . . . , N}.

In order to obtain such values in practice, quadrature rules are applied. As these rules

mostly depend on point evaluations, irregular data remains problematic. For functions with

very low regularity, a useful approach to approximate their integral is given by Monte Carlo

integration techniques. Instead of using this approach to obtain such integrals, we include it

directly to our scheme. This can be done via randomized point evaluations and measuring

only the expectation of the error.

Here, we propose two different ways to randomize the evaluation points. First, we

consider a temporal grid, which is randomly shifted. We evaluate the data at these points.

Under general assumptions on the data and no additional regularity condition imposed on the

solution u, we can show that piecewise polynomial prolongations of the values (Un)n∈{1,...,N}

converge to the solution in a certain probabilistic sense.

Secondly, we always choose a randomized point between two randomly shifted grid points.

When the data is evaluated at these points, we can provide certain error bounds if the

solution is more regular and the data fulfills some additional assumptions. More precisely,

we assume that the solution uis an element of a fractional Sobolev space and we suppose

that the operator A(t), t∈[0, T ], fulfills a stronger monotonicity condition and is Lipschitz

continuous on bounded sets. Then we can provide error estimates, where we prove that the

expectation of the error is sufficiently small.

A second, independent method to improve the computation of a solution, is to decompose

the operator and consider an operator splitting. Here, we allow a monotone part A(t): V→

V∗,t∈[0, T ], as before and a Lipschitz continuous part B(t): H→H,t∈[0, T]. Then we

consider

(u0(t) + A(t)u(t) + B(t)u(t) = f(t) in V∗,for almost all t∈(0, T ),

u(0) = u0in H.

We again have the same starting point and want to solve the recursion

Un−Un−1

k+AnUn+BnUn=fn, n ∈ {1, . . . , N},(2)

with U0=uk

0. This time our aim is different and we assume that suitable approximations

0, (An)n∈{1,...,N}, (Bn)n∈{1,...,N}, and (fn)n∈{1,...,N}are known. We concentrate on finding

modifications for one single backward Euler step

(I+kAn+kBn)Un=kfn+Un−1, n ∈ {1, . . . , N},

that make this step potentially easier to solve. When approximating the solution to an

operator equation that is governed by a monotone operator, it is convenient to use a back-

ward Euler scheme. Compared to the forward Euler scheme, this has much better stability

properties. The downside of the backward Euler method is that it becomes necessary to

solve an implicit equation in each step. In our setting, we assume that the operator B(t),

t∈[0, T ], is bounded on the pivot space H. Here, the better stability properties of the

implicit scheme are not present. Thus, we exchange BnUnin (2) by BnUn−1. We then

work with the implicit-explicit structure given by

(I+kAn)Un=kfn+Un

0with Un

0= (I−kBn)Un−1

for n∈ {1, . . . , N}. This has the advantage that not more implicit equations appear than

really necessary. As the implicit equation containing Ancan be complex to solve, we further

introduce M∈Nvalues An

mand fn

m,m∈ {1, . . . , M}, such that An=PM

m=1 An

mand

fn=PM

m=1 fn

m. Then the backward Euler step containing Anis equivalent to

I+k

m=1

mUn=k

m=1

m+Un

An application of the well-known product splitting leads to the system of equations given

I+kAn

mUn

m=kfn

m+Un

m−1, m ∈ {1, . . . , M}.

Finally, we obtain a system of the type

0= (I−kBn)Un−1

and

(I+kAn

m)Un

m=kfn

m+Un

m−1, m ∈ {1, . . . , M},

with Un=Un

Mfor n∈ {1, . . . , N}and U0=uk

0. This method is not intended to lead

to an increased convergence rate. But we will see that the additional error caused by the

splitting scheme does not affect the magnitude of the error. Compared to the standard

backward Euler scheme, this approach leads to more subproblems. These can potentially

be easier to solve such that the total computational time may decrease. A suitable choice

of the decomposition for A(t) and f(t), t∈[0, T ], can even lead to a problem that is easier

to parallelize. In modern hardware structures, parallelization can be a powerful tool to

accelerate the algorithm.

We follow a similar intention as for the randomized scheme and prove the convergence

of piecewise polynomial prolongations of the values (Un)n∈{1,...,N}. Under the additional

regularity condition that uis H¨older continuous and that the operator A(t), t∈[0, T], fulfills

a stronger monotonicity condition and a bounded Lipschitz condition, we can prove explicit

error bounds.

This thesis consists of extensions of the following works, which were developed over the

last years.

(i) M. Eisenmann and E. Hansen. [34]

Convergence analysis of domain decomposition based time integrators for degenerate

parabolic equations. Numer. Math., 140(4):913–938, 2018.

(ii) M. Eisenmann and E. Hansen. [35]

A variational approach to splitting schemes, with applications to domain decomposi-

tion integrators. ArXiv Preprint, arXiv:1902.10023, 2019.

(iii) M. Eisenmann, M. Kov´acs, R. Kruse, and S. Larsson. [37]

On a randomized backward Euler method for nonlinear evolution equations with time-

irregular coefficients. Found. Comput. Math., Jan 2019 (Online First).

(iv) M. Eisenmann and R. Kruse. [38]

Two quadrature rules for stochastic Itˆo-integrals with fractional Sobolev regularity.

Commun. Math. Sci., 16(8):2125–2146, 2018.

xii INTRODUCTION

In the separate chapters, we explain in more detail how the content of this thesis is related

to the works mentioned above. Furthermore, two additional papers appeared within the last

years. Their content is not included directly in this thesis.

(v) M. Eisenmann, E. Emmrich, and V. Mehrmann. [33]

Convergence of the backward Euler scheme for the operator-valued Riccati differential

equation with semi-definite data. Evol. Equ. Control Theory, 8(2):315–342, 2019.

(vi) M. Eisenmann, M. Kov´acs, R. Kruse, and S. Larsson. [36]

Error estimates of the backward Euler-Maruyama method for multi-valued stochastic

differential equations. ArXiv Preprint, arXiv:1906.11538, 2019.

This monograph mainly consists of variations for the well-known Rothe method, which is

used in [99]. We prove convergence results of a semidiscrete scheme in a variational setting.

Similar approaches can be found in [35, 37, 40, 41, 42, 44].

When it comes to discretizing evolution equations, different strategies can be used. It

is possible to use a semi-discretization either via a temporal discretization or a spatial

discretization. In order to obtain a full discretization, these two strategies can be combined.

This leads, in particular, to an implementable method. A more basic introduction to the

numerical approximation of linear evolution equations can be found in [80, 111]. There, the

solutions of linear parabolic equations are approximated using a fully discretized scheme

involving a Galerkin approximation.

We only concentrate on a temporal discretization in this work and use the concept of

variational solutions. A full discretization can be obtained in a somewhat natural way. One

of the main aspects of the concept of variational solutions is to look at the problem in a

certain tested way. A Galerkin scheme can easily be integrated through a finite-dimensional

space of test functions. This also includes the finite element method.

Two important classes of methods to approximate the solution of a differential equation

are Runge–Kutta and multistep methods. For a more basic introduction, we refer the

reader to [105]. Applications of Runge–Kutta methods to evolution equations can be found

in [44, 53, 60, 87, 95] and multistep methods in [40, 41, 58, 81]. The backward Euler method

is one of the most simple prototypes of these classes of algorithms. Randomized point

evaluations for both the backward and the forward Euler scheme have been considered in

[25, 37, 71, 75] for different types of problem classes.

An introduction to operator splittings can be found in [70]. There exist many different

schemes that are based on operator splittings. Similar ones to the product splitting can

be found in [34, 35, 106]. Approaches with an implicit-explicit splitting are discussed in

[2, 5, 17, 24, 64]. The main difference of an operator splitting based scheme compared to

Runge–Kutta or multistep methods is that instead of evaluating the data at different points,

we decompose the data in several parts but evaluate them at the same point. Various types

of useful decompositions can be used. For many differential equations, there exists an

intuitive choice for a splitting given by different structures within one equation. Often,

these different structures can be handled easier by themselves. For example, a linear main

part and nonlinear perturbation can be split, see [65, 107, 108]. It is also possible to split

different partial derivatives or to decompose the domain, see [34, 35, 61, 62].

In our work, we only assume that the solution of the problem fulfills an additional

regularity condition that involves a fractional derivative when proving error bounds. Similar

error bounds for linear problems can be found in [15, 69]. The analysis becomes more

involved as soon as a nonlinearity is part of the equation. A first generalization is to consider

a linear main part and allow for some nonlinear perturbation. A numerical analysis for such

xiii

semilinear problems can be found in [3, 63, 79, 88, 92]. When the equation is only linear

with respect to the highest appearing derivative it is called quasilinear. Such problems have

been considered in [4, 53, 87]. The numerical treatment of a fully nonlinear equation has

been studied in [52, 95]. Evolution equations containing a maximal monotone main part are

treated in [58, 59, 60, 64, 66, 101].

This monograph is build up as follows. At the end of the introduction, a collection of

the used notation can be found. In Chapter 1, we begin with a short recollection of both the

solvability of evolution equations and appropriate regularity results. We give some examples

of equations, where the solution is more regular. These examples contain settings that fit

the conditions imposed for our explicit error bounds. The two approaches discussed above

are explained in detail in Chapter 2 and Chapter 3. In Chapter 2, we consider randomized

schemes to approximate the solution of an evolution equation. Here, we begin to prove

the convergence of the scheme when the temporal grid is randomly shifted. In order to

obtain more information about the magnitude of the error, we prove error bounds for the

scheme. The chapter ends with an example of a parabolic problem of p-Laplacian type. The

following Chapter 3 is build up similarly. We begin to prove the convergence of a method

that is based on an operator splitting scheme. We also prove that under some additional

assumptions, certain error bounds can be provided. Again, we show that the abstract

theory has applications for nonlinear parabolic problems. At the end of the monograph, we

collect some auxiliary results in the appendix. Here, useful inequalities can be found. This

is followed by a brief introduction to spaces of Bochner integrable functions on a general

measure space and some results from stochastic analysis that are needed for the randomized

schemes.

Notation

Let D ⊂ Rd,d≥1, be a bounded domain. We denote the boundary of Dby ∂Dand its

closure by D. The space of uniformly continuous functions v:D → Ris denoted by C(D),

while the space of continuously differentiable functions on Dis denoted by C1(D). On D

we also consider the space

C1(D) = nv∈C(D) : ∂ivexists and ∂iv∈C(D), i ∈ {1, . . . , d}o,

where ∂iv:= ∂v

∂xifor i∈ {1, . . . , d}. The space C∞

c(D) contains all functions v:D → Rthat

are infinitely many times differentiable and have a compact support in D. Furthermore, for

a function v∈C1(D), we write ∇v=∂1v, . . . , ∂dvTfor its gradient while the divergence

of a function v= (v1, . . . , vd)T∈C1(D)dis denoted by ∇ · v=Pd

i=1 ∂ivi. For T∈(0,∞)

and a function v: (0, T )× D → R, we write ∂tvfor the partial derivative with respect to the

temporal parameter.

For p∈[1,∞] and `∈N, we write Lp(D)`for the space of Lebesgue measurable functions

v:D → R`such that

kvkLp(D)`=







ZD

|v|pdx

p, p ∈[1,∞),

ess sup

|v|, p =∞

is finite. If `= 1, we just write Lp(D). Here, ess supDis the essential supremum over D.

For more details and properties, see [1, Chapter 2]. The Sobolev space W1,p(D) consists of

xiv INTRODUCTION

all functions v∈Lp(D) such that every weak partial derivative ∂iv,i∈ {1, . . . , d}, exists

and is an element of Lp(D). This space is equipped with the norm

kvkW1,p(D)=(kvkp

Lp(D)+k∇vkp

Lp(D)d

p, p ∈[1,∞),

kvkL∞(D)+k∇vkL∞(D)d, p =∞

for v∈W1,p(D). If all the mixed partial derivatives up to order j∈Nexist and are

elements of Lp(D), we denote the space of such functions by Wj,p(D). Additionally, for

p∈[1,∞) we consider the space W1,p

0(D), which is the closure of C∞

c(D) with respect to

the norm of W1,p(D). Due to Poincar´e’s inequality (cf. [19, Corollary 9.19]) the seminorm

kvkW1,p

0(D)=k∇vkLp(D)d,v∈W1,p

0(D), is a full norm on this space. In the case p= 2, we

also write Hj(D) = Wj,2(D), j∈N, and H1

0(D) = W1,2

0(D). More details and properties

for Sobolev spaces can be found in [1, 19, 82]. For fractional differentiability exponents, we

use the same notation and refer the reader to [26, Chapter 4] for the precise definition.

In the following, let (X, k·kX) be a real Banach space. We write X∗for its dual space

that is equipped with the induced norm given by

kfkX∗= sup

x∈X,

kxkX≤1

hf, xiX∗×X, f ∈X∗,

where hf, xiX∗×X=f(x) stands for the duality pairing. For a finite value T∈(0,∞), the

space of uniformly continuous functions v: [0, T]→Xis denoted by C([0, T ]; X). A norm

on this space is given by

kvkC([0,T ];X)= max

t∈[0,T ]kv(t)kX

for v∈C([0, T ]; X). We denote the H¨older seminorm to the exponent α∈(0,1] of a function

v: [0, T ]→Xby

|v|C0,α([0,T ];X)= sup

s,t∈[0,T ],

s6=t

kv(s)−v(t)kX

|s−t|α.

We then call the space

C0,α([0, T]; X) = {v∈C([0, T]; X) : |v|C0,α([0,T ];X)<∞}

the H¨older space with exponent αand use the norm

kvkC0,α([0,T ];X)=kvkC([0,T ];X)+|v|C0,α([0,T ];X)

for v∈C0,α([0, T]; X). For α= 1 this is the space of Lipschitz continuous functions with

values in X. For more details on continuous X-valued functions, see [39, Abschnitt 7.1] and

for H¨older continuous functions, see [6, Section 3.7].

For p∈[1,∞] the space of Bochner integrable functions on [0, T ] with values in Xis

denoted by Lp(0, T ;X). This space is equipped with the norm

kvkLp(0,T ;X)=









ZT

kv(t)kp

Xdt

p, p ∈[1,∞),

ess sup

t∈[0,T ]

kv(t)kX, p =∞

for v∈Lp(0, T ;X). For more details, see [39, Abschnitt 7.1] or [96, Section 4.2]. In

Appendix A.2 there is a short explanation and collection of results for Bochner integrable

functions that are defined on a general measure space.

For p∈[1,∞), the space W1,p(0, T ;X) contains the functions in Lp(0, T;X) that possess

a weak derivative in Lp(0, T ;X). A norm for this space is given by

kvkW1,p(0,T ;X)=kvkp

Lp(0,T ;X)+kv0kp

Lp(0,T ;X)

compare [99, Section 7.1]. For α∈(0,1), p∈[1,∞), and v: [0, T ]→Xwe consider the

Sobolev–Slobodecki˘ı seminorm

|v|Wα,p(0,T ;X)=ZT

0ZT

kv(s)−v(t)kp

|s−t|αp+1 dsdt

Then the Sobolev–Slobodecki˘ı space is given by

Wα,p(0, T;X) = {v∈Lp(0, T;X) : |v|Wα,p(0,T ;X)<∞},

which is endowed with the norm

kvkWα,p(0,T ;X)=kvkp

Lp(0,T ;X)+|v|p

Wα,p(0,T ;X)

for v∈Wα,p(0, T;X). A full introduction and properties can be found in [26, Chapter 4] or

[27]. In [102], there is a wide range of embedding theorems for such function spaces.

The spaces V,H, and V∗are called a Gelfand triple if (V, k·kV) is a separable, reflexive

Banach space that is continuously and densely embedded into the separable Hilbert space

(H, (·,·)H,k·kH). The space His identified with its dual and we consider

,→H∼

=H∗d

,→V∗.

For f∈V∗and v∈Vwe write hf, viV∗×V=f(v), which is the continuous extension of the

inner product of H. On such spaces, we introduce

Wp(0, T ) = {v∈Lp(0, T;V) : v0exists and v0∈Lq(0, T;V∗)},

where p∈(1,∞), q=p

p−1, and v0denotes the weak derivative of v. In the case p= 2, we

write W(0, T ) = W2(0, T). The space Wp(0, T ) is equipped with the norm

kvkWp(0,T )=kvkLp(0,T ;V)+kv0kLq(0,T ;V∗)

for v∈ Wp(0, T ) and is continuously embedded into C([0, T ]; H). For more details, see [39,

Abschnitt 8.1 and 8.4], [99, Section 7.2], and [96, Section 4.2].

xvi INTRODUCTION

Chapter 1

Solvability and Properties of the

Solutions to Evolution Equations

Before we come to the numerical analysis of evolution equations, we state a general setting

with known existence results, where the solution is also unique. This is in mind, we explain

some settings where the solution fulfills additional regularity conditions.

1.1 Existence and Uniqueness

In this thesis, we work with variational solutions of evolution equations. We only give a

brief introduction to this concept. For more details, we refer the reader to the following

monographs [39, 46, 49, 99, 117, 118]. Another widely spread solution concept is the theory

of mild solutions. Sometimes, we refer to results in the literature, where this notion of

solution is used. For more details, see [16, 89, 118].

In the following, let (H, (·,·)H,k·kH) be a real, separable Hilbert space and let (V, k·kV)

be a real, separable, reflexive Banach space, which is continuously and densely embedded

into H. Identifying Hwith its dual, we obtain the Gelfand triple

,→H∼

=H∗d

,→V∗.

For a finite end time T∈(0,∞), we consider a family {A(t)}t∈[0,T ]of operators such that

A(t): V→V∗for every t∈[0, T ]. We assume that the mapping Av : [0, T ]→V∗given by

t7→ A(t)vis Bochner measurable for every v∈V. Further, we suppose that A(t) is radially

continuous for every t∈[0, T], i.e., the mapping s7→ hA(t)(v+sw), wiV∗×Vis continuous

on [0,1] for every v, w ∈V. For κ∈[0,∞), we assume that A(t) + κI is monotone, i.e., the

inequality

hA(t)v−A(t)w, v −wiV∗×V+κkv−wk2

H≥0

is fulfilled for all v, w ∈V. For a fixed p∈(1,∞), we assume that the operator A(t) fulfills

a growth condition and A(t) + κI fulfills a semi-coercivity condition such that there exist

β, λ ∈[0,∞) and µ∈(0,∞) with

kA(t)vkV∗≤β1 + kvkp−1

V,hA(t)v, viV∗×V+κkvk2

H+λ≥µ|v|p

for every t∈[0, T ] and v∈V. Here, |·|Vis a seminorm on Vsuch that there exists

cV∈(0,∞) with k·kV≤cVk·kH+|·|V. Note that since Vis continuously embedded

2CHAPTER 1. SOLVABILITY AND REGULARITY

into H, it is always possible to choose |·|V=k·kV. In this case, the operator A(t) + κI,

t∈[0, T ], is coercive. As pointed out in [41], it is possible to rewrite the semi-coercivity

condition as a coercivity condition of the type

h(A(t)+(κ+µ)I)v, viV∗×V≥µ|v|p

V+kvk2

H−λ≥µ|v|˜p

V+kvk˜p

H−1−λ

≥21−˜pµ|v|V+kvkH˜p−µ−λ≥21−˜pµc−˜p

Vkvk˜p

V−µ−λ(1.1)

for v∈Vwith ˜p= min{2, p}. In Chapter 3, we will use the semi-coercivity condition as

this slightly improves some convergence results compared to using (1.1).

For a source term f∈Lq(0, T ;V∗) + L1(0, T ;H), q=p

p−1, and u0∈H, we consider the

initial value problem

(u0+Au =fin Lq(0, T ;V∗) + L1(0, T;H),

u(0) = u0in H.

We are looking for a solution uthat is an element of the space

1(0, T ) = {v∈Lp(0, T;V) : v0exists and v0∈Lq(0, T;V∗) + L1(0, T;H)},

where v0denotes the weak derivative of v. This space is continuously embedded into

C([0, T ]; H), compare [109, Chapter III, Section 1.5]. Thus, the initial condition is well-

defined.

The existence of a solution to the initial value problem has been proved in [99, Theo-

rem 8.9], [39, Satz 8.4.2], and [85, Section 2.7] if Vis compactly embedded into H. A further

existence results for κ= 0 can be found in [118, Chapter 30]. In Chapter 2, we also only

consider this particular case. It is not a real restriction for p∈[2,∞) as it is possible to use

a transformation trick, compare [117, Remark 23.25] or [49, Folgerung on page 211]. For

t∈[0, T ], we transform A(t) and f(t) to ˜

A(t) = e−κtA(t)eκt +κI and ˜

f(t) = e−κtf(t) and

find a solution to the problem

(˜u0+˜

A˜u=˜

fin Lq(0, T ;V∗) + L1(0, T;H),

˜u(0) = u0in H. (1.2)

The family {˜

A(t)}t∈[0,T ]of operators ˜

A(t): V→V∗,t∈[0, T], fulfills all the condition

imposed above for κ= 0. The solution of the original problem is given by u(t) = eκt ˜u(t) in

Hfor t∈[0, T ]. Usually, it is a requirement to assume that Vis compactly embedded into

Hwhen allowing for κ∈(0,∞). With this transformation, we do not have to make this

assumption.

Furthermore, the solutions of both the original and the transformed problem are unique,

compare [99, Theorem 8.31]. Thus, from an analytical point of view, it makes sense to use

the transformed problem. In applications, it might be preferable to work with the original

problem without any transformation. In Chapter 3, we allow for arbitrary κ∈[0,∞) in

terms of an additionally appearing family {B(t)}t∈[0,T ]of operators such that B(t): H→H.

This kind of operator cannot easily be included in Chapter 2 due to a missing compactness

result, which we will point out later.

1.2 Regularity of the Solution

For the results in Sections 2.2 and 3.2, we need some additional regularity of the solution.

Precisely, we need that the solution is in a certain Sobolev–Slobodecki˘ı space or in a H¨older

1.2. REGULARITY OF THE SOLUTION 3

space with values in Vdepending on the exact statement. These particular conditions are

not available under general assumptions on the data. We will focus briefly on some settings

where we can obtain this higher-order regularity. For a general overview, we refer the reader

to [89] and [7, Chapter III]. Some further results for a fractional derivative can be found in

[78, Chapter IV, Section 5] and [84, Chapter 4, Section 5].

For general nonlinear problems, it is difficult to state suitable regularity results that fit

our assumptions. Here, it becomes necessary to look at the specific problem more closely

to obtain any appropriate results if possible. In the following, we concentrate on known

regularity results for linear problems. For certain semilinear problems, bootstrap arguments

can be applied to recover the regularity of the linear problem, compare [110, Section 3] or

Example 1.2.4 below. Some further regularity results for more specific nonlinear problems

can be found in [28, 51].

In the following, we assume that Vis a real, separable Hilbert space, which is continuously

and densely embedded into the real, separable Hilbert space Hsuch that we again obtain a

Gelfand triple Vd

,→H∼

=H∗d

,→V∗. We assume that the family {A(t)}t∈[0,T ]of operators

A(t): V→V∗,t∈[0, T], fulfills the conditions stated in the previous section for p= 2 and

that A(t): V→V∗is linear for every t∈[0, T ]. For f∈L2(0, T ;V∗), we consider the linear

equation

(u0+Au =fin L2(0, T ;V∗),

u(0) = u0in H. (1.3)

For such a linear problem, compatibility conditions lead to a more regular solution, see [39,

Abschnitt 8.5], [46, Chapter 7, Theorem 6], [48, Chapter 10, Section 6–7], and [114, §27].

To this end, we assume that the temporal derivative of fexists and it fulfills f0∈

L2(0, T ;V∗). We also need an additional assumption for the family of operators {A(t)}t∈[0,T ].

Here, we assume that the classical derivative of t7→ hA(t)v, wiV∗×Vexists on [0, T ], is

measurable, and there exists β0∈[0,∞) such that |d

dthA(t)v, wiV∗×V| ≤ β0kvkVkwkV

for every v, w ∈Vand t∈[0, T ]. This derivative can be used to define the operator

A0(t): V→V∗,t∈[0, T], by hA0(t)v, wiV∗×V=d

dthA(t)v, wiV∗×Vfor v, w ∈V. The

operator A0(t), t∈[0, T ], is linear and bounded independently of t. Further, if the initial

conditions u(0) = u0in Vand u0

0:= f(0) −A(0)u0in Hare fulfilled, it follows that

u, u0∈L2(0, T ;V) and u00 ∈L2(0, T;V∗). This shows, in particular, that u∈W1,2(0, T ;V)

and u0∈W1,2(0, T ;V∗). Using embedding theorems, we obtain that

u∈W1,2(0, T ;V),→Wα,2(0, T;V),→C0,α−1

2([0, T ]; V),

u0∈W1,2(0, T ;V∗),→Wα,2(0, T;V∗),→C0,α−1

2([0, T ]; V∗),(1.4)

for every α∈(0,1), cf. [102, Corollary 26]. This includes the regularity conditions stated

in Theorem 2.2.7 and 3.2.3 in the case p= 2. For a nonlinear, yet autonomous, operator, a

similar idea has been considered in [99, Theorem 8.18].

A further approach to prove higher-order regularity of the solution to the linear problem

(1.3) is the concept of maximal Lp-regularity. The initial value problem (1.3) is said to have

maximal Lp-regularity in Hfor some p∈[2,∞) if for every f∈Lp(0, T;H) the unique

solution u∈ W(0, T ) fulfills that u0∈Lp(0, T ;H) and Au ∈Lp(0, T ;H). Suitable regularity

results can be obtained by embedding theorems as we will see in one of the examples below.

A survey about this concept can be found in [93]. Sufficient conditions to obtain maximal

Lp-regularity are stated in [9, 10, 11, 32, 47, 55].

Another approach is to only look for local regularity. Due to the parabolic smoothing

property, it is possible to prove certain regularity results away from the initial data. If

4CHAPTER 1. SOLVABILITY AND REGULARITY

we assume that f∈L2(0, T ;V∗) fulfills t7→ tf0(t)∈L2(0, T;V∗) and A0(t): V→V∗,

t∈[0, T ], is linear and bounded independently of t, then the solution of (1.3) fulfills that

t7→ tu0(t)∈ W(0, T ). See [39, Satz 8.5.3] and [111, Chapter 3] for more details. For a fully

nonlinear problem, a regularity result of this type can be found in [95, Lemma 3]. These local

results are not enough for our analysis and are more useful in a non-smooth data analysis

as has been done in [79, 88, 111]. Still, they indicate that after a certain time the methods

should work well if the error from the beginning has not become too large. Moreover, under

the assumption that the solution is bounded, it is possible to prove local H¨older regularity

for nonlinear problems, compare [28, Chapter III].

In the following, we present a few different examples and discuss the regularity of the

solution.

Example 1.2.1. For a finite end time T∈(0,∞) and a bounded Lipschitz domain D ⊂ Rd,

d∈N, we look at the problem











∂tu(t, x)−∇·a(t, x)∇u(t, x)+b(t, x, u(t, x)) = f(t, x),(t, x)∈(0, T )× D,

u(t, x) = 0,(t, x)∈(0, T )×∂D,

u(0, x) = u0(x), x ∈ D.

We assume that a= (aij)i,j∈{1,...,d}: [0, T ]×D → Rd,d is an element of L∞((0, T)×D;Rd,d)

and there exists µ∈(0,∞) with

a(t, x)z·z=

i,j=1

aij(t, x)zizj≥µ|z|2(1.5)

for all t∈[0, T ], almost all x∈ D, and all z∈Rd. We assume that b: [0, T]×D × R→R

fulfills b(·,·, z)∈L∞((0, T )× D) for every z∈Rand there exist κ, ρ ∈[0,∞) such that

|b(t, x, z)−b(t, x, ˜z)| ≤ κ|z−˜z|,|b(t, x, 0)| ≤ ρ(1.6)

for all t∈[0, T ], almost all x∈ D, and all z, ˜z∈R.

We choose the Hilbert spaces V=H1

0(D) and H=L2(D) equipped with the norms given

in the introduction. We obtain the Gelfand triple Vd

,→H∼

=H∗d

,→V∗, where we identify H

with its dual space. We introduce the operators A(t): V→V∗and B(t): H→H,t∈[0, T ],

which are given by

hA(t)v, wiV∗×V=ZD

a(t, ·)∇v· ∇wdx, v, w ∈V, (1.7)

(B(t)v, w)H=ZD

b(t, ·, v)wdx, v, w ∈H. (1.8)

We assume that for f: [0, T ]×D → Rthe abstract function [f(t)](x) = f(t, x), (t, x)∈

(0, T )× D, is an element of L2(0, T;V∗). For u0∈H, we obtain the variational formulation

of the problem

(u0+Au +Bu =fin L2(0, T ;V∗),

u(0) = u0in H. (1.9)

In the following, we verify that this equation fits into the setting introduced in the previous

section. The proof for this is quite basic. As all the examples mentioned in this section have

the same underlying structure, we add it for the sake of completeness.

1.2. REGULARITY OF THE SOLUTION 5

First, we prove that t7→ B(t)vis measurable for every v∈H. The measurability of

t7→ A(t)vfor v∈Vcan be argued analogously. By assumption, t7→ b(t, x, z) is measurable

for almost every x∈ D and every z∈R. Thus, there exists a sequence (bi)i∈Nof functions

bi: [0, T ]×D × R→R,i∈N, that are simple with respect to the first argument such

that bi(t, x, z)→b(t, x, z) as i→ ∞ and |bi(t, x, z)| ≤ |b(t, x, z)|,i∈N, for almost

every (t, x)∈(0, T )× D and every z∈Rd. For t∈[0, T ], we defined the simple operator

Bi(t): H→Hgiven by

(Bi(t)v, w)H=ZD

bi(t, ·, v)wdx, v, w ∈H.

Applying the conditions from (1.6), it follows that

b(t, ·, v)−bi(t, ·, v)w≤2|b(t, ·, v)||w|

≤2|b(t, ·, v)−b(t, ·,0)||w|+ 2|b(t, ·,0)||w|

≤2κ|v||w|+ 2ρ|w|(1.10)

for every v, w ∈H, for almost every t∈(0, T), and almost everywhere in D. As (1.10) is

an integrable function on D, we can apply Lebesgue’s dominated convergence theorem to

obtain that

lim

i→∞(B(t)v−Bi(t)v, w)H=ZD

lim

i→∞ b(t, ·, v)−bi(t, ·, v)wdx= 0

for every v, w ∈Hand almost every t∈(0, T ). This implies that t7→ B(t)v,v∈H, is

weakly measurable. As His also separable, the mapping is Bochner measurable.

It is easy to see that v7→ A(t)vis linear and inserting the definition of A(t) implies

hA(t)v, wiV∗×V=ZD

a(t, ·)∇v· ∇wdx≤ kakL∞((0,T )×D;Rd,d)kvkVkwkV(1.11)

for every v, w ∈Vand t∈[0, T ]. Hence, this shows that v7→ A(t)vis continuous because

kA(t)kL(V,V ∗)≤ kakL∞((0,T )×D;Rd,d)holds true for every t∈[0, T], where k·kL(V,V ∗)denotes

the induced operator norm. Further, the Lipschitz continuity of bin the third argument

shows

(B(t)v1−B(t)v2, w)H=ZDb(t, ·, v1)−b(t, ·, v2)wdx≤κkv1−v2kHkwkH

for every v1, v2, w ∈Hand t∈[0, T ]. Therefore, we see that B(t): H→His Lipschitz

continuous, as kB(t)v1−B(t)v2kH≤κkv1−v2kHis fulfilled for every v1, v2∈Hand

t∈[0, T ]. This shows that v7→ A(t)v+B(t)vis continuous and A(t) + B(t) radially

continuous for every t∈[0, T ].

Next, we prove that A(t)+B(t)+κI,t∈[0, T ], is monotone. To this end, we first notice

that

hA(t)v, viV∗×V=ZD

a(t, ·)∇v· ∇vdx≥µZD

|∇v|2dx=µkvk2

is fulfilled for every v∈Vand t∈[0, T ] due to (1.5). Thus, we see that

hA(t)v1−A(t)v2+B(t)v1−B(t)v2, v1−v2iV∗×V+κkv1−v2k2

=ZD

a(t, ·)∇v1− ∇v2·∇v1− ∇v2dx

+ZDb(t, ·, v1)−b(t, ·, v2)(v1−v2) dx+κkv1−v2k2

≥µkv1−v2k2

6CHAPTER 1. SOLVABILITY AND REGULARITY

holds for every v1, v2∈Vand t∈[0, T ].

Using (1.6), it follows that

(B(t)0, w)H=ZD

b(t, ·,0)wdx≤ρ|D|1

2kwkH(1.12)

and together with the Lipschitz continuity of B(t)

kB(t)vkH≤ kB(t)v−B(t)0kH+kB(t)0kH≤κkvkH+ρ|D|1

2(1.13)

for every v, w ∈Hand t∈[0, T ], where |D| denotes the Lebesgue measure of D. Thus, we

see that A(t) + B(t)+(κ+ 1)Iis coercive since

hA(t)v+B(t)v, viV∗×V+ (κ+ 1)kvk2

H≥µkvk2

V− kB(t)vkHkvkH+ (κ+ 1)kvk2

≥µkvk2

V−κkvk2

H−ρ|D|1

2kvkH+ (κ+ 1)kvk2

≥µkvk2

V−ρ2

4|D|

is fulfilled for every v∈Vand t∈[0, T ], where we applied the weighted Young inequality.

Combining (1.11) and (1.13), the operator A(t)+B(t), t∈[0, T ], is bounded in the sense

that

kA(t)v+B(t)vkV∗≤ kakL∞((0,T )×D;Rd,d)kvkV+c1κkvkH+ρ|D|1

2,

holds. Here, c1∈(0,∞) denotes the embedding constant of Hinto V∗.

In [39, Satz 8.3.5], it is shown that there exists a unique u∈ W(0, T ) that solves (1.9).

Example 1.2.2. We consider the same setting as in Example 1.2.1 with b≡0. Additionally,

we assume that for every i, j ∈ {1, . . . , d}the function t7→ aij(t, x) is absolutely continuous

for almost every x∈ D,aij (0,·)∈W1,∞(D), and ∂taij ∈L∞((0, T )× D). The operator

A0(t): V→V∗fulfills

hA0(t)v, wiV∗×V=ZD

∂ta(t, ·)∇v· ∇wdx≤ k∂takL∞((0,T )×D;Rd,d)kvkVkwkV

for every v, w ∈Vand t∈[0, T ]. This implies that A0(t): V→V∗,t∈[0, T], is linear and

bounded independently of t. Moreover, we choose f∈W1,2(0, T;V∗) with f(0) ∈Hand

u0∈H2(D)∩V. Then it follows that u0

0:= f(0) + ∇ · a(0,·)∇u0∈H. We can apply [39,

Satz 8.5.1], where regularity is obtained through compatibility conditions of the data. Then

we obtain that u∈W1,2(0, T ;V) and u0∈W1,2(0, T;V∗). Using the embedding theorem

from [102, Corollary 26], it follows that

u∈W1,2(0, T ;V),→Wα,2(0, T;V),→C0,α−1

2([0, T ]; V),

u0∈W1,2(0, T ;V∗),→Wα,2(0, T;V∗),→C0,α−1

2([0, T ]; V∗).

for every α∈(0,1).

Example 1.2.3. We consider the same setting as in Example 1.2.1 with b≡0. We apply

[29, Corollary 7.1] for α∈(0,1

2), f∈L2(0, T ;H2α−1(D)), u0∈H2α

0(D). Further, we assume

that for ε∈0,1

2the coefficients aij ,i, j ∈ {1, . . . , d}, are in Wα+ε, 1

α(0, T ;L∞(D)) and

therefore continuous. Then it follows that u∈Wα,2(0, T ;V)∩W1,2(0, T;H2α−1(D)).

1.2. REGULARITY OF THE SOLUTION 7

Additionally, we can find a corresponding result for the derivative u0. To this end, we

use that in [29, Theorem 6.2, Corollary 7.1] it is noted that A∈Wα+ε, 1

α(0, T ;L(V, V ∗)).

Together with a suitable result for the Nemytski˘ı operator of Afrom [29, Lemma 5.3] and an

extension result from [27, Theorem 5.4], it follows that Au ∈Wα,2(0, T ;V∗). For a function

f∈Wα,2(0, T ;V∗), this also shows that u0∈Wα,2(0, T ;V∗).

Example 1.2.4. Again, we consider the same setting as in Example 1.2.1. At first, we

set b≡0 but show later that we can recover the regularity for more general b. In the

following, we denote the Friedrichs extension of an operator A(t) by AF(t): dom(AF(t)) →

H,AF(t)u=A(t)u, where dom(AF(t)) = H2(D)∩H1

0(D) for every t∈[0, T ]. We assume

that there exists p∈(2,∞) such that afulfills the additional condition

|aij(t, x)−aij (s, x)| ≤ ω(|t−s|)

for every i, j ∈ {1, . . . , d},s, t ∈[0, T ], as well as almost every x∈ D. Here, ω: [0, T ]→

[0,∞) is a non-decreasing function that fulfills

ω(t)

dt < ∞and ZT

0ω(t)

tp

dt < ∞.

Then we can apply [55, Theorem 2] and find that (1.9) has maximal Lp-regularity for all

u0within the interpolation space (H, dom(AF(0)))1−1

p,p. As a proper explanation of the

concept of interpolation spaces is out of place in this section, we only refer the reader to [90,

Chapter 1] and [112, Chapter 1] for further details.

This means that for every f∈Lp(0, T ;H) the solution uis an element of W1,p(0, T ;H)

and Au ∈Lp(0, T ;H). Analogously to [37, Theorem 7.2], it follows that u∈C0,α([0, T ]; V)

for every α∈[0,1

2−1

p−ε) and an arbitrary ε∈(0,1

2−1

p). Note that using the first

embedding result from [8, Theorem 5.2] in the proof of [37, Theorem 7.2] instead, it also

follows that u∈Wα+1

p,2(0, T ;V). Regularity results for u0can be obtained analogously as

in the previous example after possibly asking for some additional regularity for the data.

Using a similar idea as in [37, Theorem 7.8] and [92, Theorem 2.10], we can allow for a

nontrivial function band can still recover the same regularity for the solution. To this end,

let u∈C([0, T ]; H) be the unique solution of (1.9). The function g=f−Bu fulfills

kgkLp(0,T ;H)≤ kfkLp(0,T ;H)+kBu −B0kLp(0,T ;H)+kB0kLp(0,T ;H)

≤ kfkLp(0,T ;H)+κkukLp(0,T ;H)+ZT

kB(t)0kp

Hdt

≤ kfkLp(0,T ;H)+κkukLp(0,T ;H)+T1

pρ|D|1

where we use the Lipschitz continuity of B(t), t∈[0, T], and (1.12). Thus, we have g∈

Lp(0, T ;H) and it follows that the solution vof

(v0+Av =gin L2(0, T ;V∗),

v(0) = u0in H(1.14)

is an element of C0,α([0, T ]; V) and Wα+1

p,2(0, T ;V) as we have seen above.

Now, we can use a bootstrap argument to show that the solution uof (1.9) has the same

regularity. Both (1.9) and (1.14) have a unique solution. Inserting uin (1.14), we see that

ualso solves this problem. Thus, uand vcoincide and fulfill the same regularity condition.

8CHAPTER 1. SOLVABILITY AND REGULARITY

Chapter 2

Randomized Schemes for Nonlinear,

Nonautonomous Evolution Equations

In this chapter, we introduce randomized schemes that can be used to approximate the so-

lution of a nonlinear, nonautonomous evolution equation on a finite time interval. Precisely,

for T∈(0,∞), we consider

(u0(t) + A(t)u(t) = f(t) in V∗,for almost all t∈(0, T ),

u(0) = u0in H(2.1)

for a Gelfand triple Vd

,→H∼

=H∗d

,→V∗as well as a family {A(t)}t∈[0,T ]of monotone

operators A(t): V→V∗,t∈[0, T ], a source term f: [0, T ]→V∗, and an initial value u0∈

H. The approach presented here mainly offers advantages for a temporal approximation.

Hence, we only consider a discretization in time. This leads to a semidiscrete problem. We

begin to follow a standard approach given by the backward Euler scheme. For N∈N,

we consider an equidistant partition 0 = t0<· · · < tN=Twith k=T

Nand tn=nk,

n∈ {0, . . . , N}, of the interval [0, T ] to find an approximation Unof u(tn), n∈ {1, . . . , N}.

To this end, we solve the recursion

Un−Un−1

k+AnUn=fnin V∗, n ∈ {1, . . . , N},

for U0=u0, where (An)n∈{1,...,N}and (fn)n∈{1,...,N}are approximations of the data. A

common choice of such values is of the form

An=1

kZtn

tn−1

A(t) dt, fn=1

kZtn

tn−1

f(t) dt, n ∈ {1, . . . , N}.

These values prove themselves to be very suitable if they are known. In practice though,

they are not necessarily available and additional approximation techniques are needed. An

easy to get, yet not always well-defined, alternative of merely integrable data would be

An=A(tn),fn=f(tn), n ∈ {1, . . . , N}.

Due to the low regularity, point evaluations will in general not offer a suitable approximation.

As a pre-designed grid and even the amount of points in a sequence of such grids is only

a null set, it is possible to find functions that ”fool” the scheme. This can be done by

10 CHAPTER 2. RANDOMIZED SCHEMES

redefining the data on the grid points. In order to bypass this problem, we will work with

two different types of randomization. The first is ideal for proving convergence in a setting

without any further regularity assumptions on the solution. For a complete probability space

(Ωθ,Fθ,Pθ) and a uniformly distributed random variable θ: Ωθ→[0,1], we consider the

randomly shifted grid

0 = tθ

0< tθ

1<· · · < tθ

N=T−k(1 −θ) with tθ

n=tn−1+kθ, n ∈ {1, . . . , N}.

Note that we write θas an index to the probability space. This is not supposed to show

any dependence on θbut only means that this is the probability space that θis defined

on. Later, we also introduce a further family of random variables on a different probability

space. This second probability space is also indexed so there will be no mix up between the

two spaces. For

An=A(tθ

n),fn=f(tθ

n), n ∈ {1, . . . , N},

we prove in Section 2.1 that the piecewise polynomial prolongations of (Un)n∈{1,...,N}con-

verge to upointwise strongly in L2(Ωθ;H). Furthermore, depending on the monotonicity

condition imposed on A(t), t∈[0, T], we obtain weak or strong convergence in the space

Lp(0, T ;Lp(Ωθ;V)), where the value pdepends on A(t), t∈[0, T ]. Measuring the expecta-

tion of the error on a randomly shifted grid, offers a way to handle data that is non-smooth

with respect to the temporal input. This convergence result can be obtained with fairly

general assumptions on the data and no additional regularity requirements on the solution.

If higher regularity of the solution might be available though, we prove in Section 2.2 that a

simple modification of the scheme leads to explicit error bounds with an order that depends

on the Sobolev–Slobodecki˘ı regularity of the solution. Here, a second randomization can

be used to exploit the additional regularity. To this end, for a second complete probabil-

ity space (Ωτ,Fτ,Pτ), let (τn)n∈{1,...,N}be a family of independent, uniformly distributed

random variables with τn: Ωτ→[0,1]. For

ξn=tθ

n−1+ (tθ

n−tθ

n−1)τn, n ∈ {1, . . . , N},(2.2)

we use

An=A(ξn),fn=f(ξn), n ∈ {1, . . . , N},

to prove that there exists C∈(0,∞), which depends on uand u0, such that

max

n∈{1,...,N}Eku(tθ

n)−Unk2

H+

n=1

E(tθ

n−tθ

n−1)ku(tθ

n)−Unkp

V≤Ckαp

p−1(2.3)

is fulfilled for every N∈N. Here, pdepends on A(t), t∈[0, T ], and α∈(0,1) denotes the

differentiability exponent of the Sobolev–Slobodecki˘ı spaces that contain uand u0. We write

Efor the expectation on the product probability space (Ωθ×Ωτ,Fθ⊗ Fτ,Pθ⊗Pτ). Even

though, we have to make additional regularity assumptions on the solution, we will see that

they are rather low. In order to demonstrate the relevance of these theoretical convergence

results and error bounds, we show how they can be applied to a nonlinear parabolic problem.

This includes, in particular, the well-known parabolic p-Laplacian equation. Note that the

porous media equation in a very weak formulation as considered in [34, 45] also fits in the

abstract framework.

The approach of using randomized schemes has already been studied for certain classes

of problems in the literature. Within the context of Monte Carlo algorithms, these methods

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 11

are well-known for the approximation of integrals. See [94] for an introduction. The concept

of stratified sampling was first introduced in [56, 57]. There, a subdivision of the domain of

integration is made and one random value within each subset is chosen. This approach is

useful to make sure that the random variables are distributed more evenly compared to a

standard Monte Carlo algorithm. In [103, 104], a randomization was used to approximate

the solution of an ordinary differential equation for the first time. There are many works

thereafter that continue this approach. The use of explicit randomized schemes was further

studied in [25, 71, 75]. When compared to deterministic schemes, randomized schemes can

often handle low regularity assumptions on the data better while their complexity does not

increase. In the theory of information-based complexity, upper and lower bounds of classes

of randomized schemes are studied, see [68, 72]. Similar to ours, yet explicit, schemes for

stochastic ordinary differential equations have been considered in [77, 97, 98] or for stochastic

partial differential equations in [76]. In [18], a randomized grid was used for this type of

problem. Note that a randomization of the grid is only helpful if the regularity is measured

in an appropriate way. As pointed out in [50], there are problem classes where an additional

randomization does not yield any advantages compared to a deterministic time grid. Data

from a Sobolev–Slobodecki˘ı space seems to be well suited for this kind of scheme.

The central idea of the work in the following chapter is based on both [37] and [38]. In [38],

a randomized grid was used to prove the convergence of a quadrature rule to approximate a

stochastic Itˆo-integral. It was shown that the rate of convergence depends on the Sobolev–

Slobodecki˘ı regularity of the integrand instead of the H¨older regularity. In [67], it is even

pointed out that the obtained rate is optimal. Therefore, better error estimates can be

proved if the differentiability exponent of the Sobolev–Slobodecki˘ı space is higher than of

the H¨older space. In [37], a randomization of the type ˜

ξn=tn−1+kτn,n∈ {1, . . . , N},

with tn−1and τnas introduced above, was used to approximate the solution of a problem

like (2.1) with a Lipschitz continuous operator A(t), t∈[0, T ]. There, the solution is

assumed to be H¨older continuous to prove a result like (2.3) with an error bound whose

order depends on the H¨older regularity of the solution. The results of this chapter are a

combination of these works. More precisely, in Theorem 2.1.11 and Theorem 2.1.12, we prove

the convergence of the piecewise polynomial prolongations of the solution to the semidiscrete

problem to the exact solution of the evolution equation (2.1). Here, the data is evaluated

on a randomly shifted grid. This expands the theory from [37] to a more general problem

class and shows that even without any additional regularity assumptions made on the exact

solution a randomized scheme leads to a useful numerical approximation. In Theorem 2.2.6

and 2.2.7, we evaluate the data at the points (ξn)n∈{1,...,N}explained in (2.2) and obtain

error bounds that depend on the Sobolev–Slobodecki˘ı regularity of the exact solution. This

lowers the regularity assumptions from [37] and allows for operators A(t), t∈[0, T], which

fulfill a bounded Lipschitz condition instead of a global Lipschitz condition.

This chapter is organized as follows. In Section 2.1, we begin to state the precise assump-

tions made. Then we prove the convergence of the scheme with evaluations on a randomly

shifted grid. This is followed by a setting where explicit error estimates are proved in Sec-

tion 2.2. For these bounds, we need to make additional assumptions on the solution and the

data. The chapter is concluded with an example of a p-Laplacian type problem. Here, we

show that the abstract theory can be applied to such a problem.

2.1 Convergence on a Randomly Shifted Grid

In this section, it is our overall goal to prove the convergence of the backward Euler scheme on

a randomly shifted grid. We allow for a fairly general setting without any further regularity

12 CHAPTER 2. RANDOMIZED SCHEMES

assumption on the solution. First, we introduce a nonlinear, nonautonomous operator A(t),

t∈[0, T ], and a source term f.

Assumption 2.1.1. Let T∈(0,∞),p∈[2,∞)be given. Let (H, (·,·)H,k · kH)be a real,

separable Hilbert space and (V, k·kV)be a real, separable, reflexive Banach space, which

is continuously and densely embedded into H. Let {A(t)}t∈[0,T ]be a family of operators

A(t): V→V∗such that the following conditions are fulfilled:

(1) The mapping Av : [0, T ]→V∗given by t7→ A(t)vis measurable for every v∈V.

(2) The operator A(t): V→V∗,t∈[0, T ], is radially continuous, i.e., the mapping

s7→ hA(t)(v+sw), wiV∗×Vis continuous on [0,1] for every v, w ∈V.

(3) The operator A(t): V→V∗,t∈[0, T ], is monotone, i.e.,

hA(t)v−A(t)w, v −wiV∗×V≥0

is fulfilled for every v, w ∈V.

(4) The operator A(t): V→V∗,t∈[0, T ], is uniformly bounded such that there exists

β∈[0,∞), which does not depend on t, with

kA(t)vkV∗≤β1 + kvkp−1

V

for every v∈V.

(5) The operator A(t): V→V∗,t∈[0, T ], fulfills a coercivity condition in the sense that

there exist µ∈(0,∞)and λ∈[0,∞), which do not depend on t, such that

hA(t)v, viV∗×V≥µkvkp

V−λ

for every v∈V.

In the following, we always identify Hwith its dual space and consider the Gelfand triple

,→H∼

=H∗d

,→V∗.

In applications, it can be of advantage to generalize the conditions (3) and (5) of the previous

assumption in such a way that there exists κ∈[0,∞) such that A(t) + κI,t∈[0, T ], fulfills

these conditions. Here, I:V→V∗denotes the identity mapping. Then it is necessary

to suppose that Vis compactly embedded into H. Due to the randomization, we cannot

use this compact embedding in a straight forward way. If the underlying problem contains

strictly positive κ, we can still use the transformation from (1.2) to obtain data that fulfills

our requirements.

Furthermore, it is also possible to consider the case p∈(1,2) in Assumption 2.1.1,

compare also [40, 41, 42]. Here, it will become necessary to work with slightly different

function spaces. In the second section of this chapter, we impose a stronger monotonicity

condition on the operator A(t), t∈[0, T ]. It is possible to show, that there exists no operator

that fulfills this condition for p∈(1,2). Thus, we concentrate on p∈[2,∞) for simplicity

and to be more consistent throughout this chapter.

We consider a source term f∈Lq(0, T ;V∗) with q=p

p−1, where pis the same as

in Assumption 2.1.1. Note that it is not trivial to allow for a more general source term

f∈Lq(0, T ;V∗) + L1(0, T;H). We will explain this in more detail at a later point. In

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 13

Section 1.1, we have seen that the evolution equation (2.1) is uniquely solvable under the

imposed conditions on A(t), t∈[0, T ], and ffor an initial value u0∈H.

It will also be important to consider the operator A(t), t∈[0, T], as a mapping on the

space of Bochner integrable functions Lp(0, T ;V) into its dual space. To this end, we collect

some properties of the Nemytski˘ı operator in the following lemma. We omit the proof as it

can be found in [39, Lemma 8.4.4] or [118, Section 30].

Lemma 2.1.2. Let Assumption 2.1.1 be fulfilled. Then v7→ Av with (Av)(t) = A(t)v(t)

maps Lp(0, T ;V)into Lq(0, T;V∗), where q=p

p−1. Then the operator is radially continuous,

i.e., the mapping s7→ hA(v+sw), wiLq(0,T ;V∗)×Lp(0,T ;V)is continuous on [0,1] for all v, w ∈

Lp(0, T ;V). Further, Afulfills a monotonicity, a boundedness, and a coercivity condition

such that

hAv −Aw, v −wiLq(0,T ;V∗)×Lp(0,T ;V)≥0,

kAvkLq(0,T ;V∗)≤βT1

q+kvkp−1

Lp(0,T ;V),

hAv, viLq(0,T ;V∗)×Lp(0,T ;V)+λT ≥µkvkp

Lp(0,T ;V)

hold true for all v, w ∈Lp(0, T ;V).

For the temporal discretization of (2.1), we introduce a randomly shifted grid to bypass

classical point evaluations at the points of a predetermined temporal grid.

Assumption 2.1.3. Let T∈(0,∞)and N∈Nbe given. Consider the equidistant partition

0 = t0<· · · < tN=Twith k=T

Nand tn=nk,n∈ {0, . . . , N}. Further, let (Ωθ,Fθ,Pθ)be

a complete probability space such that L1(Ωθ)is separable. Let θ: Ωθ→[0,1] be a uniformly

distributed random variable. The randomly shifted grid is denoted by 0 = tθ

0< tθ

1<· · · <

tθ

N=T−k(1 −θ)with tθ

n=tn−1+kθ for n∈ {1, . . . , N}.

The expectation on the probability space (Ωθ,Fθ,Pθ) is denoted by Eθ. It is necessary

to assume that L1(Ωθ) is a separable space to argue that some of the Bochner spaces, which

will appear further below, are separable. For applications, this assumption is unproblematic.

We only need one uniformly distributed random variable θ: Ωθ→[0,1] in this section.

Thus, we could simply take Ωθ= [0,1] equipped with the Lebesgue σ-algebra Fθand the

Lebesgue-measure Pθand choose θ(ω) = ωfor ω∈[0,1].

Under Assumptions 2.1.1 and 2.1.3 as well as f∈Lq(0, T ;V∗), we can now consider the

recursion

(Un+kA(tθ

n)Un=kf(tθ

n) + Un−1almost surely in V∗, n ∈ {1, . . . , N},

U0=u0in H, (2.4)

which is the classical backward Euler scheme, but on a randomly shifted grid. In the follow-

ing, we will prove that (2.4) admits a unique solution (Un)n∈{1,...,N}. For n∈ {1, . . . , N}

the mapping Un: Ωθ→Vis Fθ-measurable and its expectation fulfills an a priori bound.

We begin by proving a general auxiliary result to show the Fθ-measurability of a solution to

such an implicit equation. A similar result can be found in [54, Lemma 3.8]. The structure

of the proof is comparable to [31, Proposition 1] and [37, Lemma 4.3]. We adapt it to fit

our setting in an infinite-dimensional space.

Lemma 2.1.4. Let (Ω,F,P)be a complete probability space and let (V, k·kV)be a real,

separable Banach space. Further, let h:Ω×V→V∗fulfill the following conditions for some

N ⊂ Ωwith P(N) = 0:

14 CHAPTER 2. RANDOMIZED SCHEMES

(1) The mapping u7→ hh(ω, u), viV∗×Vis continuous for every v∈Vand ω∈Ω\ N .

(2) The mapping ω7→ h(ω, u)is F-measurable for every u∈V.

(3) For every ω∈Ω\N , there exists a unique element U(ω)∈Vsuch that h(ω, U(ω)) = 0.

Consider the mapping U: Ω →V,ω7→ U(ω), where U(ω)is the unique element in V

described in (3) for ω∈Ω\ N and U(ω) = 0 for ω∈ N . Then Uis F-measurable.

Proof. For ε > 0 and an arbitrary v∈V, we introduce the multivalued function Uv

ε: Ω →

P(V) by

ε(ω) = u∈V:hh(ω, u), viV∗×V∈Iε,

where P(V) denotes the power set of Vand Iε= (−ε, ε). Note that when considering

measurability conditions for set-valued mappings, it is usually necessary to work with an

image that is a closed set, compare [14, Section 8.1]. Since we do not use any specific results

imposed on such mappings, it is unproblematic to define it like this.

Let Cbe an open set within the Borel σ-algebra B(V) of V. It is our first intention to

prove that for Cthe set

(Uv

ε)−1(C) = {ω∈Ω : there exists u∈Csuch that u∈Uv

ε(ω)}

={ω∈Ω : there exists u∈Csuch that hh(ω, u), viV∗×V∈Iε}

is an element of F. Note that if Conly consists of a single element, it follows that

(Uv

ε)−1({u}) = {ω∈Ω : u∈Uv

ε(ω)}=hh(·, u), viV∗×V−1Iε

is an element of Fsince ω7→ hh(ω, u), viV∗×Vis measurable. If Ccontains more than one

element, we can still write

(Uv

ε)−1(C) = [

u∈Chh(·, u), viV∗×V−1Iε.

For a countable set C, it is easy to see that (Uv

ε)−1(C)∈Fas it is a countable union of

sets in F. By assumption, the space Vis separable. Thus, there exists a countable, dense

subset Qof Vsuch that

(Uv

ε)−1(C∩Q) = [

u∈C∩Qhh(·, u), viV∗×V−1Iε

is an element of F. Using the continuity of u7→ hh(ω, u), viV∗×Vfor every ω∈Ω\ N , we

will justify that (Uv

ε)−1(C∩Q)=(Uv

ε)−1(C). As C∩Q⊂Cholds true, (Uv

ε)−1(C∩Q)⊆

(Uv

ε)−1(C) follows directly. Thus, it only remains to prove (Uv

ε)−1(C)⊆(Uv

ε)−1(C∩Q).

Here, we consider two cases. In the first case, we assume that (Uv

ε)−1(C)⊆ N. Then the

completeness of the probability space yields (Uv

ε)−1(C)∈ F. Else, for ω∈(Uv

ε)−1(C)\ N,

there exists u1∈Csuch that u1∈Uv

ε(ω). As u7→ hh(ω, u), viV∗×Vis continuous,

D=Uv

ε(ω) = hh(ω, ·), viV∗×V−1Iε

is an open set in Vand u1∈C∩D. Since both Cand Dare open, their intersection is

open and, as we have just seen, it is nonempty. Thus, there exists u2∈C∩D∩Qsuch

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 15

that hh(ω, u2), viV∗×V∈Iε. Altogether, this implies ω∈(Uv

ε)−1(C∩Q) and therefore, in

particular, (Uv

ε)−1(C)=(Uv

ε)−1(C∩Q).

For every ω∈Ω\ N there exists a unique element U(ω)∈Vsuch that h(ω, U(ω)) = 0

in V∗. Hence, there exists at least one element u∈Vsuch that u∈Uv

ε(ω) for every v∈Q

with kvkV≤1. For such an element u, it follows that

kh(ω, u)kV∗= sup

v∈V,

kvkV≤1

|hh(ω, u), viV∗×V|= sup

v∈Q,

kvkV≤1

|hh(ω, u), viV∗×V| ≤ ε,

as every element of V∗is continuous. This in mind, we obtain that the following intersection

fulfills

v∈Q,

kvkV≤1

ε(ω) = \

v∈Q,

kvkV≤1u∈V:hh(ω, u), viV∗×V∈Iε=u∈V:h(ω, u)∈Bε,V ∗(0),

where Bε,V ∗(0) denotes the open ball in V∗with radius εand center 0 ∈V∗. Therefore, we

can write for the unique element U(ω)∈Vsuch that h(ω, U(ω)) = 0 in V∗that

U(ω)∈\

i∈N\

v∈Q,

kvkV≤1

i−1(ω)⊆\

i∈Nu∈V:h(ω, u)∈Bi−1,V ∗(0)={U(ω)}.

For an arbitrary open set C∈ B(V), we then obtain

U−1(C) = {ω∈Ω : there exists u∈Csuch that h(ω, u) = 0}

i∈N\

v∈Q,

kvkV≤1

{ω∈Ω : there exists u∈Csuch that hh(ω, u), viV∗×V∈Ii−1}

i∈N\

v∈Q,

kvkV≤1

(Uv

i−1)−1(C) = \

i∈N\

v∈Q,

kvkV≤1

(Uv

i−1)−1(C∩Q)∈F,

since we have a countable intersection of measurable sets. This proves the measurability of

ω7→ U(ω).

Lemma 2.1.5. Let Assumptions 2.1.1 and 2.1.3 be fulfilled and for q=p

p−1, let f∈

Lq(0, T ;V∗)be given. Then there exists a unique solution (Un)n∈{1,...,N}to the recursion

(2.4). The mapping Un: Ωθ→V,ω7→ Un(ω)is Fθ-measurable for every n∈ {1, . . . , N}.

Proof. The set

N={ω∈Ωθ: there exists n∈ {1, . . . , N}such that kf(tθ

n(ω))kV∗=∞} (2.5)

is a null set in Ωθdue to the integrability of f. In the following, let n∈ {1, . . . , N}and

ω∈Ωθ\ N be arbitrary but fixed. Assuming that Un−1(ω) exists, we apply the Browder–

Minty Theorem, see [99, Theorem 2.14]. To this end, we consider the equation

I+kA(tθ

n(ω))U=kf(tθ

n(ω)) + Un−1(ω) in V∗.(2.6)

The operator I+kA(tθ

n(ω)) is radially continuous, as both Iand A(tθ

n(ω)) are radially

continuous (cf. Assumption 2.1.1 (2)). Moreover, Iis strictly monotone and A(tθ

n(ω)) is

16 CHAPTER 2. RANDOMIZED SCHEMES

monotone (cf. Assumption 2.1.1 (3)), so the operator I+kA(tθ

n(ω)) is also strictly monotone.

For arbitrary v∈V, we obtain

hI+kA(tθ

n(ω))v, viV∗×V≥ kvk2

H+k(µkvkp

V−λ)→ ∞ as kvkV→ ∞,

due to Assumption 2.1.1 (5). Thus, I+kA(tθ

n(ω)) is radially continuous, strictly monotone

and coercive. So there exists a unique element U=Un(ω)∈Vsuch that (2.6) is fulfilled.

It remains to prove the Fθ-measurability of the mapping ω7→ Un(ω). This can be done

by applying Lemma 2.1.4 to the function

hn: Ωθ×V→V∗, hn(ω, U) = I+kA(tθ

n(ω))U−kf(tθ

n(ω)) −Un−1(ω)

for n∈ {1, . . . , N}. By Assumption 2.1.1 (2) and (3), the operator A(t), t∈[0, T], is radially

continuous and monotone. Thus, it is also hemicontinuous (cf. [49, Kapitel III, Lemma 1.3]),

i.e., U7→ hhn(ω, U), viV∗×Vis continuous for every ω∈Ωθ\ N and v∈V. Moreover, the

mapping ω7→ hn(ω, v) is Fθ-measurable for every v∈Vdue to Assumption 2.1.1 (1) and

the measurability of fand tθ

n. By the argumentation above, there exists a unique element

Un(ω)∈Vthat is the root of hn(ω, ·). Thus, it follows that Un: Ωθ→V,ω7→ Un(ω) is

Fθ-measurable.

Now that the existence of a unique Fθ-measurable family (Un)n∈{1,...,N}is proven, we

need a priori bounds for the solution. First, we state a result that shows that the appearing

terms containing fare bounded.

Lemma 2.1.6. Let Assumption 2.1.3 be fulfilled and let (X, k · kX)be a real Banach space.

For f∈Lq(0, T ;X),q∈[1,∞),

n=1

Eθkf(tθ

n)kq

X=kfkq

Lq(0,T ;X)

is fulfilled.

Proof. For n∈ {1, . . . , N}, we use a substitution as in (A.2) and can write

Eθkf(tθ

n)kq

X=Z1

kf(tn−1+ks)kq

Xds=1

kZtn

tn−1

kf(s)kq

Xds.

Thus, it follows that

n=1

Eθkf(tθ

n)kq

X=

n=1 Ztn

tn−1

kf(s)kq

Xds=ZT

kf(s)kq

Xds.

The following lemma is a central part of our argumentation. These bounds in mind,

we can prove the boundedness of sequences of prolongations of the values obtained in

our semidiscrete scheme (2.4). The structure of the proof to these bounds consists of

standard techniques. However, in our setting, it is not trivial to include a source term

f∈Lq(0, T ;V∗) + L1(0, T;H), as in [109, Chapter III, Section 1.5], or a semi-coercivity

condition as in [99, Theorem 8.9]. The main difficulty consists of the additional expectation

we have to include to the bounds. Whereas it is easy to see that kwk2

H<∞for w∈Him-

plies that kwkp

H<∞for every p∈[2,∞), it is not possible to conclude that EθkWk2

H<∞

also implies EθkWkp

H<∞for a random variable W: Ωθ→H. The techniques proposed

in [21, Lemma 3.1] could offer a possibility to allow for these more general assumptions.

This remains a question for future work.

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 17

Lemma 2.1.7. Let Assumptions 2.1.1 and 2.1.3 be fulfilled. Further, let f∈Lq(0, T ;V∗),

q=p

p−1, and u0∈Hbe given. Then there exists K∈(0,∞)such that for all k=T

N∈N, the unique solution (Un)n∈{1,...,N}of (2.4) fulfills

max

n∈{1,...,N}EθkUnk2

H+

n=1

EθkUn−Un−1k2

H+k

n=1

EθkUnkp

V≤K(2.7)

and

k1−q

i=1

EθkUi−Ui−1kq

V∗=k

i=1

Eθh



Ui−Ui−1

k



V∗i≤K. (2.8)

Proof. In the following, let i∈ {1, . . . , N}be fixed. Furthermore, for the set Ndefined in

(2.5), we consider the following calculations for ω∈Ωθ\ N without explicitly stating ωin

each step. For a single Euler step (2.4), it holds true that

Ui−Ui−1+kA(tθ

i)Ui=kf(tθ

i) in V∗.

Testing this equation with Ui, we obtain

(Ui−Ui−1,Ui)H+khA(tθ

i)Ui,UiiV∗×V=khf(tθ

i),UiiV∗×V.(2.9)

Recalling the identity from Lemma A.1.4 and applying it to (2.9), it follows

2kUik2

H− kUi−1k2

H+kUi−Ui−1k2

H+khA(tθ

i)Ui,UiiV∗×V

=khf(tθ

i),UiiV∗×V.

(2.10)

The weighted Young inequality applied to the right-hand side of (2.10) shows

khf(tθ

i),UiiV∗×V≤kc1kf(tθ

i)kq

V∗+kµ

2kUikp

where c1=(pµ)1−q

q21−q. Inserting this bound and the coercivity condition from Assump-

tion 2.1.1 (5) in (2.10), it follows that

2kUik2

H− kUi−1k2

H+kUi−Ui−1k2

H+kµ

2kUikp

V≤kλ +kc1kf(tθ

i)kq

V∗.

Multiplying both sides with the factor two and summing up this inequality from i= 1 to

n∈ {1, . . . , N}, yields

kUnk2

H+

i=1

kUi−Ui−1k2

H+kµ

i=1

kUikp

≤ kU0k2

H+ 2tnλ+ 2kc1

i=1

kf(tθ

i)kq

V∗.

(2.11)

Taking the expectation, it follows that

EθkUnk2

H+

i=1

EθkUi−Ui−1k2

H+kµ

i=1

EθkUikp

V

≤ ku0k2

H+ 2Tλ + 2kc1

i=1

Eθkf(tθ

i)kq

V∗=ku0k2

H+ 2Tλ + 2c1kfkq

Lq(0,T ;V∗),

18 CHAPTER 2. RANDOMIZED SCHEMES

due to Lemma 2.1.6. It remains to prove the second a priori bound (2.8). To this end, let

v∈Vbe arbitrary but fixed. Then we test (2.4) with vto obtain

Un−Un−1

k, vH=hf(tθ

n)−A(tθ

n)Un, viV∗×V≤ kf(tθ

n)−A(tθ

n)UnkV∗kvkV,

n∈ {1, . . . , N}. Together with the boundedness condition from Assumption 2.1.1 (4) this

implies



Un−Un−1

k



V∗

≤ kf(tθ

n)kV∗+kA(tθ

n)UnkV∗≤ kf(tθ

n)kV∗+β1 + kUnkp−1

V.

Taking the q-th power and the expectation as well as summing up the inequality from n= 1

to Nshows that

n=1

Eθh



Un−Un−1

k



V∗i≤

n=1

Eθkf(tθ

n)kV∗+β1 + kUnkp−1

Vq.

We then multiply by kand take the 1

q-th power again in order to use the triangle inequality

and obtain the desired bound

k

n=1

Eθh



Un−Un−1

k



V∗i

≤k

n=1

Eθkf(tθ

n)kq

V∗

q+k

n=1

βq

q+βk

n=1

EθkUnkp

V

=kfkLq(0,T ;V∗)+T1

qβ+βk

n=1

EθkUnkp

V

This is bounded independently of the step size kdue to Lemma 2.1.6 and the first a priori

bound (2.7).

For the time discrete solution (Un)n∈{1,...,N}to (2.4) corresponding to the shifted grid

stated in Assumption 2.1.3, we construct piecewise polynomial prolongations defined on the

entire interval [0, T]. To this end, we introduce the piecewise constant prolongations for

t∈(tn−1, tn], n∈ {1, . . . , N},

Uk(t) = Un, Ak(t) = A(tθ

n), fk(t) = f(tθ

n) (2.12)

as well as the piecewise affine-linear function

Uk(t) = Un−1+t−tn−1

k(Un−Un−1) (2.13)

on Ωθ\N , where Nis defined in (2.5). For t= 0, we set ¯

Uk(0) = Uk(0) = U0,Ak(0) = A(tθ

1),

and fk(0) = f(tθ

1). Then we have

((Uk)0(t) + Ak(t)¯

Uk(t) = fk(t) in Lq(Ωθ;V∗), t ∈(0, T),

Uk(0) = u0in H, (2.14)

where (Uk)0denotes the weak derivative of Uk. Here, the weak derivative coincides with

the classical derivative, where the latter exists. Note that due to the a priori bounds of

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 19

Lemma 2.1.7, the boundedness condition for A(t), t∈[0, T ], from Assumption 2.1.1 (4),

and the fact that f∈Lq(0, T ;V∗) the equation is indeed fulfilled in Lq(Ωθ;V∗).

In the following, we always consider step sizes k=T

N`, where (N`)`∈Nis a sequence of

natural numbers such that N`→ ∞ as `→ ∞. We abbreviate the corresponding sequence

(¯

N`)`∈Nby ( ¯

Uk)k>0and analogously for the other functions introduced above.

This in mind, we can prove the convergence of ¯

Ukand Ukto the exact solution pointwise

strongly in L2(Ωθ;H). Further, we can prove that ¯

Ukconverges to uin Lp(0, T ;Lp(Ωθ;V)).

The Bochner spaces L2(Ωθ;H) and Lp(Ωθ;V) appear because of the additional dependence

of the solution (Un)n∈{1,...,N}to values of the space Ωθ. More information on a Bochner

space defined on a general measure space can be found in Appendix A.2. There, a collection

of the properties and important results can be found. This information available, a space

Lp(0, T ;Lp(Ωθ;V)) or alike has a similar structure. Since (0, T ) equipped with the Lebesgue

σ-algebra and the Lebesgue measure is a finite measure space, the properties of Lp(Ωθ;V)

can be transferred to Lp(0, T ;Lp(Ωθ;V)).

Lemma 2.1.8. Let Assumptions 2.1.1 and 2.1.3 be fulfilled and let f∈Lq(0, T;V∗),q=

p−1, and u0∈Hbe given. Further, let (N`)`∈Nbe a sequence of natural numbers with

N`→ ∞ as `→ ∞, let k=T

N`be the corresponding step sizes, and let the sequences of

piecewise constant and piecewise linear prolongations be given as in (2.12) and (2.13). Then

there exists a subsequence of step sizes, again denoted by k, such that

Uk* U in Lp(0, T ;Lp(Ωθ;V)),

Uk∗

* U, Uk∗

* U in L∞(0, T ;L2(Ωθ;H)),

(Uk)0* U0in Lq(0, T ;Lq(Ωθ;V∗))

as k→0. The function Uis an element of Lp(0, T ;Lp(Ωθ;V)) ∩L∞(0, T;L2(Ωθ;H)) and

U0is the weak temporal derivative of U, which is an element of Lq(0, T ;Lq(Ωθ;V∗)).

In particular, this shows that Uis an element of the space

Ωθ(0, T ) = {W∈Lp(0, T;Lp(Ωθ;V)) :

W0exists and W0∈Lq(0, T ;Lq(Ωθ;V∗))},(2.15)

which is continuously embedded into C([0, T ]; L2(Ωθ;H)).

Proof of Lemma 2.1.8. For simplicity, we do not denote every subsequence differently within

this proof and we drop the index `. Due to the a priori bound (2.7) from Lemma 2.1.7, the

sequence ( ¯

Uk)k>0of piecewise constant prolongations is bounded in both Lp(0, T;Lp(Ωθ;V))

and L∞(0, T ;L2(Ωθ;H)). Further, the sequence (Uk)k>0of piecewise linear prolongations

is bounded in the space L∞(0, T ;L2(Ωθ;H)). Using the second a priori bound (2.8) from

Lemma 2.1.7, it follows that the sequence ((Uk)0)k>0is bounded in Lq(0, T ;Lq(Ωθ;V∗)).

As Lp(0, T ;Lp(Ωθ;V)) is a reflexive Banach space, there exists a subsequence of ( ¯

Uk)k>0

and an element U∈Lp(0, T ;Lp(Ωθ;V)) such that ¯

Uk* U in Lp(0, T;Lp(Ωθ;V)) as k→0.

An analogous argumentation yields that there exists a subsequence of (Uk)0k>0and W∈

Lq(0, T ;Lq(Ωθ;V∗)) such that (Uk)0* W in Lq(0, T ;Lq(Ωθ;V∗)) as k→0.

As stated in Assumption 2.1.1, the space L1(Ωθ) is separable. Applying the fact that H

is separable as well as [96, Proposition 2.3.24, Proposition 4.2.22], it follows that L2(Ωθ;H)

is separable. Thus, the space L∞(0, T;L2(Ωθ;H)) is the dual space of the separable Banach

space L1(0, T ;L2(Ωθ;H)), compare Appendix A.2. Thus, we can extract weakly∗converging

subsequences of ( ¯

Uk)k>0and (Uk)k>0. Due to the uniqueness of the limit of weak and

20 CHAPTER 2. RANDOMIZED SCHEMES

weak∗convergent sequences, it follows that ¯

Uk∗

* U in L∞(0, T;L2(Ωθ;H)) as k→0

and therefore U∈Lp(0, T ;Lp(Ωθ;V)) ∩L∞(0, T ;L2(Ωθ;H)). Furthermore, there exists an

element ˜

U∈L∞(0, T ;L2(Ωθ;H)) such that Uk∗

*˜

Uin L∞(0, T ;L2(Ωθ;H)) as k→0.

In order to prove that the two limits Uand ˜

Ucoincide, we consider

Eθk¯

Uk(t)−Uk(t)k2

Hdt=

n=1 Ztn

tn−1

Eθh



Un−Un−1−t−tn−1

kUn−Un−1



Hidt

n=1

EθkUn−Un−1k2

HZtn

tn−1

(tn−t)2dt

n=1

EθkUn−Un−1k2

H→0 as k→0,

where we also used the a priori bound (2.7) from Lemma 2.1.7. This shows that U=˜

in L2(0, T ;L2(Ωθ;H)). The spaces L∞(0, T ;L2(Ωθ;H)) and Lp(0, T;Lp(Ωθ;V)) are contin-

uously embedded into L2(0, T;L2(Ωθ;H)) and Uis an element of Lp(0, T ;Lp(Ωθ;V)) ∩

L∞(0, T ;L2(Ωθ;H)) and ˜

Uof L∞(0, T ;L2(Ωθ;H)). This shows that U=˜

Uin both

Lp(0, T ;Lp(Ωθ;V)) and L∞(0, T;L2(Ωθ;H)) as the embedding is always injective.

It remains to prove that Wis the weak derivative of Uwith respect to the temporal

input. For arbitrary v∈Lp(Ωθ;V) and ϕ∈C∞

c(0, T ), it follows that

EθhZT

0(Uk)0(t), vHϕ(t) dti=EθhZT

(Uk(t), v)Hϕ0(t) dti

since (Uk)0is the weak derivative of Ukalmost surely. Thus, an application of Fubini’s

theorem then yields

EθhW(t), viV∗×Vϕ(t) dt= lim

k→0ZT

Eθ(Uk)0(t), vHϕ(t) dt

=−lim

k→0ZT

Eθ(Uk(t), v)Hϕ0(t) dt

=−ZT

Eθ(U(t), v)Hϕ0(t) dt.

Applying [49, Kapitel IV, Lemma 1.7], it follows that W=U0in Lq(0, T ;Lq(Ωθ;V∗)) and,

in particular, that U∈ Wp

Ωθ(0, T ).

Lemma 2.1.9. Let Assumption 2.1.3 be fulfilled, let (X, k · kX)be a real Banach space, q∈

[1,∞), and v∈Lq(0, T ;X). Then the piecewise constant function vk: [0, T ]→Lq(Ωθ;X)

given by vk(0) = v(tθ

1)and vk(t) = v(tθ

n)in Lq(Ωθ;X)for t∈(tn−1, tn],n∈ {1, . . . , N},

fulfills

vk→vin Lq(0, T ;Lq(Ωθ;X))

as k→0.

Proof. As the space C([0, T ]; X) is dense in Lq(0, T;X), for every ε > 0, there exists a

function vε∈C([0, T ]; X) such that

kvε−vkLq(0,T ;X)<ε

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 21

In particular, the function vεis uniformly continuous and there exists k0>0 such that

kvε(s)−vε(t)kX<ε

2T1

is fulfilled for all s, t ∈[0, T ] with |s−t| ≤ kand k≤k0. In order to prove the assertion, we

notice that

kvk−vkq

Lq(0,T ;Lq(Ωθ;X)) =

n=1 Ztn

tn−1

Eθkv(tθ

n)−v(t)kq

Xdt.

Then for n∈ {1, . . . , N}and almost every t∈(tn−1, tn], a substitution as in (A.2) yields

Eθkv(tθ

n)−v(t)kq

X=Z1

kv(tn−1+ks)−v(t)kq

Xds=1

kZtn

tn−1

kv(s)−v(t)kq

Xds,

such that for k≤k0

1

n=1 Ztn

tn−1Ztn

tn−1

kv(s)−v(t)kq

Xdsdt

≤1

n=1 Ztn

tn−1Ztn

tn−1

kv(s)−vε(s)kq

Xdtds

+1

n=1 Ztn

tn−1Ztn

tn−1

kvε(s)−vε(t)kq

Xdtds

+1

n=1 Ztn

tn−1Ztn

tn−1

kvε(t)−v(t)kq

Xdtds

≤2ZT

kv(s)−vε(s)kq

Xds

q+1

n=1 Ztn

tn−1Ztn

tn−1

εq

2qTdtds

q< ε.

Altogether, we have proved that

kvk−vkLq(0,T ;Lq(Ωθ;X)) < ε,

which verifies the statement of the lemma as ε > 0 can be chosen arbitrarily.

The next lemma contains a comparable result for the operator Ak. Note that in con-

trast to deterministic methods that use values 1

kRtn

tn−1A(t) dtinstead of A(tθ

n) for n∈

{1, . . . , N}, we prove that Akvk→Av in Lq(0, T;Lq(Ωθ;V∗)) rather than Akv→Av in

Lq(0, T ;Lq(Ωθ;V∗)) as k→0 for v∈Lp(0, T;V), where vkis a piecewise constant function

similar to the one introduced in the previous lemma. In the proof, Akvkleads to a function

t7→ A(t)v(t) whereas Akvleads to a function (s, t)7→ A(s)v(t), which is harder to compare

to the claimed limit.

Lemma 2.1.10. Let Assumptions 2.1.1 and 2.1.3 be fulfilled and let an arbitrary v∈

Lp(0, T ;V)be given. Then for Akdefined in (2.12) and the piecewise constant func-

tion vk: [0, T ]→Lp(Ωθ;V)given by vk(0) = v(tθ

1)and vk(t) = v(tθ

n)in Lp(Ωθ;V)for

t∈(tn−1, tn],n∈ {1, . . . , N}, it follows that

Akvk→Av in Lq(0, T ;Lq(Ωθ;V∗))

as k→0.

22 CHAPTER 2. RANDOMIZED SCHEMES

Proof. In order to estimate Akvk−Av in the norm of Lq(0, T;Lq(Ωθ;V∗)), we use a substi-

tution as in (A.2) to obtain

kAkvk−Avkq

Lq(0,T ;Lq(Ωθ;V∗))

n=1 Ztn

tn−1

EθkA(tθ

n)v(tθ

n)−A(t)v(t)kq

V∗dt

n=1 Ztn

tn−1Z1

kA(tn−1+ks)v(tn−1+ks)−A(t)v(t)kq

V∗dsdt

n=1 Ztn

tn−1Ztn

tn−1

kA(s)v(s)−A(t)v(t)kq

V∗dsdt.

Using the function h(s) := A(s)v(s), we can follow an analogous argumentation as in the

proof of Lemma 2.1.9.

The previous lemmas show that the single summands in (2.14) are converging as k→0.

The remaining question is how these limits relate to equation (2.1). And indeed, combining

the results, shows that the limit Ufrom Lemma 2.1.8 is the solution of the initial value

problem (2.1).

Theorem 2.1.11. Let Assumptions 2.1.1 and 2.1.3 be fulfilled and let f∈Lq(0, T ;V∗),

q=p

p−1, as well as u0∈Hbe given. Further, let (N`)`∈Nbe a sequence of natural numbers

with N`→ ∞ as `→ ∞ and let k=T

N`be the corresponding step sizes. Then the sequences

of piecewise constant and piecewise linear prolongations as given in (2.12) and (2.13) fulfill

Uk* u in Lp(0, T ;Lp(Ωθ;V)),

Uk∗

* u, Uk∗

* u in L∞(0, T ;L2(Ωθ;H)),

(Uk)0* u0in Lq(0, T ;Lq(Ωθ;V∗)),

Ak¯

Uk* Au in Lq(0, T ;Lq(Ωθ;V∗))

as k→0, where uis the solution to (2.1) and u0its weak derivative. Furthermore, it holds

true that ¯

Uk(t)* u(t)and Uk(t)* u(t)in L2(Ωθ;H)as k→0for all t∈[0, T ].

Note that it is also possible to prove that Uk* u in Lp(0, T ;Lp(Ωθ;V)) as k→0 if we

choose Uk(0) = uk

0in V, where uk

0→u0in Has k→0 and k1

pkuk

0kVk>0is a bounded

sequence. Such a sequence always exists because Vis densely embedded into H. A short

construction of such a sequence can also be found in Chapter 3. If in this case a scheme

without randomization is used, the sequence of piecewise linear prolongations is bounded

in the space Wp(0, T ). Due to the Lions–Aubin lemma, this space is compactly embedded

into L2(0, T ;H) if Vis compactly embedded into H. Such a compact embedding is useful

when H-valued perturbations are considered. In our setting, it is not possible to apply the

Lions–Aubin lemma because of the additional Ωθdependence. We could still obtain that

the sequence (Uk)k>0is bounded in the space

Ωθ(0, T ) = {W∈Lp(0, T;Lp(Ωθ;V)) : W0exists and W0∈Lq(0, T;Lq(Ωθ;V∗))}

if the initial value is approximated with values in V. But now Lp(Ωθ;V) does not have to

be compactly embedded into L2(Ωθ;H) even if Vis compactly embedded into H. Thus, if

we want to consider equations with an H-valued perturbations, a transformation as in (1.2)

has to be used. Even without the compact embedding from the Lions–Aubin lemma, we

will see in Theorem 2.1.12 below that certain strong convergence results can be obtained.

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 23

Proof of Theorem 2.1.11. For simplicity, we again do not denote the subsequences differently

within this proof and we drop the index `. In the following, let U∈ Wp

Ωθ(0, T ) be the

limit of the sequences of piecewise constant and piecewise linear prolongations obtained

in Lemma 2.1.8. An application of the a priori bound (2.7) from Lemma 2.1.7 and the

boundedness condition from Assumption 2.1.1 (4) yields

kAk¯

UkkLq(0,T ;Lq(Ωθ;V∗)) =k

n=1

EθkA(tθ

n)Unkq

V∗

≤k

n=1

Eθβq1 + kUnkp−1

Vq

q≤βT 1

q+βK 1

As Lq(0, T ;Lq(Ωθ;V∗)) is a reflexive Banach space, we can extract a weakly converging

subsequence such that

Ak¯

Uk* b in Lq(0, T ;Lq(Ωθ;V∗)) as k→0

for b∈Lq(0, T ;Lq(Ωθ;V∗)). Next, we identify f−bwith the weak derivative of U. To this

end, for arbitrary v∈Lp(Ωθ;V) and ϕ∈C∞

c(0, T ) we see that

EθhU0(t), viV∗×Vϕ(t) dt= lim

k→0ZT

Eθh(Uk)0(t), viV∗×Vϕ(t) dt

= lim

k→0ZT

Eθhfk(t)−Ak(t)¯

Uk(t), viV∗×Vϕ(t) dt

=ZT

Eθhf(t)−b(t), viV∗×Vϕ(t) dt,

where we also used fk→fand (Uk)0* U0in Lq(0, T ;Lq(Ωθ;V∗)) as k→0, compare

Lemma 2.1.8 and Lemma 2.1.9. Thus, b=f−U0in Lq(0, T ;Lq(Ωθ;V∗)) is fulfilled.

Due to the a priori bound (2.7) from Lemma 2.1.7, for t∈[0, T ] we can again extract a

weakly converging subsequence of (Uk(t))k>0such that

Uk(t)*˜

U(t) in L2(Ωθ;H) as k→0

with ˜

U(t)∈L2(Ωθ;H). Assuming that t∈[0, T ], ϕ∈C1([0, T]), and v∈Lp(Ωθ;V), we can

see

Eθ(U(t), v)Hϕ(t)−Eθ(U(0), v)Hϕ(0) −Zt

Eθ(U(s), v)Hϕ0(s) ds

=Zt

EθhU0(s), viV∗×Vϕ(s) ds

= lim

k→0Zt

Eθh(Uk)0(s), viV∗×Vϕ(s) ds

= lim

k→0Eθ(Uk(t), v)Hϕ(t)−Eθ(Uk(0), v)Hϕ(0) −Zt

Eθ(Uk(s), v)Hϕ0(s) ds

=Eθ(˜

U(t), v)Hϕ(t)−Eθ(u0, v)Hϕ(0) −Zt

Eθ(U(s), v)Hϕ0(s) ds.

In the single steps, we use (Uk)0* U0in Lq(0, T ;Lq(Ωθ;V∗)) as well as Uk∗

* U in

L∞(0, T ;L2(Ωθ;H)), compare Lemma 2.1.8. As for every k > 0 the equality Uk(0) = U0=

24 CHAPTER 2. RANDOMIZED SCHEMES

u0in Hholds true, this implies U(t) = ˜

U(t) in L2(Ωθ;H). For the piecewise constant

prolongation ¯

Uk, we see that

Eθk¯

Uk(t)−Uk(t)kq

V∗=Eθh



Un−Un−1−t−tn−1

kUn−Un−1



V∗i

=tn−t

kq

EθkUn−Un−1kq

V∗

≤

i=1

EθkUi−Ui−1kq

V∗≤kq−1K

for all t∈(tn−1, tn], n∈ {1, . . . , N}, where we applied the a priori bound (2.8) from

Lemma 2.1.7. Thus, it follows that Eθk¯

Uk(t)−Uk(t)kq

V∗→0 as k→0 for every t∈[0, T ].

This implies that the limits of ( ¯

Uk(t))k>0and (Uk(t))k>0coincide in Lq(Ωθ;V∗). Since the

sequence ( ¯

Uk(t))k>0,t∈[0, T ], is bounded in L2(Ωθ;H) due to the a priori bound (2.7)

from Lemma 2.1.7, we can extract a weakly converging subsequence. The L2(Ωθ;H)-valued

limit of ( ¯

Uk(t))k>0then coincides with the weak limit U(t) of (Uk(t))k>0in Lq(Ωθ;V∗) for

every t∈[0, T ]. Since L2(Ωθ;H) is continuously embedded into Lq(Ωθ;V∗), it follows that

(¯

Uk(t))k>0converges weakly to U(t) in L2(Ωθ;H) for every t∈[0, T ].

It remains to verify that b=AU in Lq(0, T ;Lq(Ωθ;V∗)). Testing the differential equa-

tion in (2.14) with ¯

Uk∈Lp(0, T ;Lp(Ωθ;V)) and integrating from 0 to t∈(tn−1, tn],

n∈ {1, . . . , N}, we can write that

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

=Zt

Eθhfk(s)−(Uk)0(s),¯

Uk(s)iV∗×Vds

=Ztn

Eθhfk(s)−(Uk)0(s),¯

Uk(s)iV∗×Vds

−Ztn

Eθhfk(s)−(Uk)0(s),¯

Uk(s)iV∗×Vds.

Next, we insert the structure of the prolongations as well as the identity from Lemma A.1.4

to obtain that

Ztn

Eθh(Uk)0(s),¯

Uk(s)iV∗×Vds

i=1 Zti

ti−1

Eθ(Ui−Ui−1,Ui)Hds

i=1

Eθ(Ui−Ui−1,Ui)H

i=1 EθkUik2

H−EθkUi−1k2

H+EθkUi−Ui−1k2

H

≥1

2EθkUnk2

H−1

2kU0k2

H=1

2Eθk¯

Uk(t)k2

H−1

2ku0k2

Recall that fk→fin Lq(0, T ;Lq(Ωθ;V∗)) and ¯

Uk* U in Lp(0, T ;Lp(Ωθ;V)) as k→0. As

for every t∈(tn−1, tn], n∈ {1, . . . , N}, we have that ¯

Uk(tn) = ¯

Uk(t)* U(t) in L2(Ωθ;H)

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 25

as k→0, the lower semi-continuity of the norm then implies that

lim sup

k→0Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

= lim sup

k→0Ztn

Eθhfk(s)−(Uk)0(s),¯

Uk(s)iV∗×Vds

−Ztn

Eθhfk(s)−(Uk)0(s),¯

Uk(s)iV∗×Vds

=Zt

Eθhf(s), U(s)iV∗×Vds−lim inf

k→0Ztn

Eθh(Uk)0(s),¯

Uk(s)iV∗×Vds

≤Zt

Eθhf(s), U(s)iV∗×Vds−lim inf

k→0

2Eθk¯

Uk(t)k2

H− ku0k2

H

≤Zt

Eθhf(s), U(s)iV∗×Vds−1

2EθkU(t)k2

H− ku0k2

H.

Here, we also used that (hfk−(Uk)0,¯

UkiV∗×V)k>0is bounded uniformly with respect to k

in L1((0, T )×Ωθ). As Uis an element of the space Wp

Ωθ(0, T ), which is defined in (2.15), it

follows that

2EθkU(t)k2

H− ku0k2

H=1

2Zt

dsEθkU(s)k2

Hds=Zt

EθhU0(s), U(s)iV∗×Vds,

for every t∈[0, T ]. This implies that

lim sup

k→0Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

≤Zt

Eθhf(s)−U0(s), U(s)iV∗×Vds=Zt

Eθhb(s), U(s)iV∗×Vds.

(2.16)

In order to prove that AU =bin Lq(0, T ;Lq(Ωθ;V∗)), it is problematic to apply the Minty

monotonicity trick directly. Without further information, the limit Uis a random variable

that might have a dependence on Ωθ. This makes it difficult to apply Lemma 2.1.10 as the

function vin the statement has to be independent of Ωθ. But we can use the fact that we

already know that the initial value problem (2.1) has a unique solution u∈ Wp(0, T ), which

is constant with respect to Ωθ. We then define the piecewise constant function uk: [0, T ]→

Lp(Ωθ;V) given by uk(0) = u(tθ

1) and uk(t) = u(tθ

n) in Lp(Ωθ;V) for t∈(tn−1, tn], n∈

{1, . . . , N}. Because of the monotonicity condition from Assumption 2.1.1 (3), we see that

EθhAk(s)¯

Uk(s)−Ak(s)uk(s),¯

Uk(s)−uk(s)iV∗×Vds≥0.

Rearranging the terms yields the inequality

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

≥Zt

EθhAk(s)¯

Uk(s), uk(s)iV∗×Vds+Zt

EθhAk(s)uk(s),¯

Uk(s)−uk(s)iV∗×Vds.

26 CHAPTER 2. RANDOMIZED SCHEMES

Using the definition of bas well as (2.16), Lemma 2.1.9 and Lemma 2.1.10, we obtain that

Eθhb(s), U(s)iV∗×Vds

≥lim sup

k→0Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

≥lim

k→0Zt

EθhAk(s)¯

Uk(s), uk(s)iV∗×Vds

+ lim

k→0Zt

EθhAk(s)uk(s),¯

Uk(s)−uk(s)iV∗×Vds

=Zt

Eθhb(s), u(s)iV∗×Vds+Zt

EθhA(s)u(s), U(s)−u(s)iV∗×Vds.

This implies

Eθhb(s)−A(s)u(s), U(s)−u(s)iV∗×Vds≥0.(2.17)

This in mind and employing that Uand uare elements of Wp

Ωθ(0, T ), we see that

dskU(s)−u(s)k2

L2(Ωθ;H)=EθhU0(s)−u0(s), U(s)−u(s)iV∗×V

=Eθhf(s)−b(s)−f(s) + A(s)u(s), U(s)−u(s)iV∗×V

=−Eθhb(s)−A(s)u(s), U(s)−u(s)iV∗×V

for almost every s∈(0, T ). Integrating this equality from 0 to t∈[0, T ] and applying (2.17),

shows that

2kU(t)−u(t)k2

L2(Ωθ;H)−1

2kU(0) −u(0)k2

L2(Ωθ;H)

=−Zt

Eθhb(s)−A(s)u(s), U(s)−u(s)iV∗×Vds≤0.

Since we have already seen that U(0) and u(0) coincide in L2(Ωθ;H), it follows that U(t) =

u(t) in L2(Ωθ;H) for all t∈[0, T ]. This shows, in particular, that Uis constant in Ωθ.

The last step is to prove that Au =bin Lq(0, T ;Lq(Ωθ;V∗)). This also proves that bis

constant on Ωθ. As we have seen that U=uin L∞(0, T ;H) and both U0and u0exist it

follows that U0=u0in Lq(0, T ;Lq(Ωθ;V∗)) and because u0is constant on Ωθthe same is true

for U0. Also we have seen that b=U0−fin Lq(0, T ;Lq(Ωθ;V∗)). Since U0−f=u0−f=Au

in Lq(0, T ;V∗), it follows that b=Au in Lq(0, T ;V∗).

So far, we have only proved that every converging subsequence of ( ¯

Uk)k>0converges

to uweakly in Lp(0, T ;Lp(Ωθ;V)). An application of the subsequence principle, see [116,

Proposition 10.13] or [49, Kapitel I, Lemma 5.4], yields that the original sequence converges

weakly to uin Lp(0, T ;Lp(Ωθ;V)). Analogously, it is possible to prove that the other

assertions of this theorem hold true for the original sequence.

The previous theorem verifies that the sequences of prolongations converge to the solution

of (2.1) in the weak sense. We can strengthen the result from this theorem and show that

we obtain a strong pointwise convergence in L2(Ωθ;H). If A(t), t∈[0, T ], fulfills a stronger

monotonicity assumption, we can even show a strong convergence result for the piecewise

constant prolongation in Lp(0, T ;Lp(Ωθ;V)).

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 27

Theorem 2.1.12. Let Assumptions 2.1.1 and 2.1.3 be fulfilled and let f∈Lq(0, T;V∗),

q=p

p−1, as well as u0∈Hbe given. Further, let (N`)`∈Nbe a sequence of natural numbers

with N`→ ∞ as `→ ∞ and let k=T

N`be the corresponding step sizes. Then the sequences

of piecewise constant and piecewise linear prolongations given in (2.12) and (2.13) fulfill

that

Uk(t)→u(t), Uk(t)→u(t)in L2(Ωθ;H)as k→0

for every t∈[0, T ], where uis the solution to the initial value problem (2.1). Furthermore,

under the additional assumption that A(t),t∈[0, T ], fulfills a p-monotonicity condition such

that there exists η∈(0,∞), which does not depend on t∈[0, T], with

hA(t)v−A(t)w, v −wiV∗×V≥ηkv−wkp

V(2.18)

for every v, w ∈V, the sequence (¯

Uk)k>0satisfies that

Uk→uin Lp(0, T ;Lp(Ωθ;V)) as k→0.

Proof. For simplicity, we drop the index `within this proof. The main idea of this proof is to

use the weak convergence results proved in Theorem 2.1.11 to deduce the strong convergence

in the same space. We combine the monotonicity conditions from Assumption 2.1.1 (3)

and from (2.18). We notice that the case η= 0 in (2.18) is exactly the monotonicity

condition from Assumption 3.1.2 (3). Thus, we include η= 0 to (2.18). We point out

the additional result for η∈(0,∞) later in the proof. We consider the piecewise constant

function uk: [0, T ]→Lp(Ωθ;V) given by uk(0) = u(tθ

1) and uk(t) = u(tθ

n) in Lp(Ωθ;V) for

t∈(tn−1, tn], n∈ {1, . . . , N}. Using the p-monotonicity condition from (2.18), yields

Eθkuk(t)−¯

Uk(t)k2

H+ 2ηZt

Eθkuk(s)−¯

Uk(s)kp

Vds

≤Eθkuk(t)−¯

Uk(t)k2

H+ 2 Zt

EθhAk(s)uk(s)−Ak(s)¯

Uk(s), uk(s)−¯

Uk(s)iV∗×Vds

=Eθkuk(t)k2

H+ 2 Zt

EθhAk(s)uk(s), uk(s)iV∗×Vds

−2Eθ(uk(t),¯

Uk(t))H−2Zt

EθhAk(s)uk(s),¯

Uk(s)iV∗×Vds

−2Zt

EθhAk(s)¯

Uk(s), uk(s)iV∗×Vds

+Eθk¯

Uk(t)k2

H+ 2 Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

=: Γk

1(t)+Γk

2(t)+Γk

3(t),

for every t∈[0, T ], where

Γk

1(t) = Eθkuk(t)k2

H+ 2 Zt

EθhAk(s)uk(s), uk(s)iV∗×Vds,

Γk

2(t) = −2Eθ(uk(t),¯

Uk(t))H−2Zt

EθhAk(s)uk(s),¯

Uk(s)iV∗×Vds

−2Zt

EθhAk(s)¯

Uk(s), uk(s)iV∗×Vds,

Γk

3(t) = Eθk¯

Uk(t)k2

H+ 2 Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds.

28 CHAPTER 2. RANDOMIZED SCHEMES

As uis an element of Lp(0, T ;V), we can apply Lemma 2.1.10 to obtain that Akuk→Au

in Lq(0, T ;Lq(Ωθ;V∗)) as k→0. We can apply Lemma 2.1.9 to obtain that uk→u

in Lp(0, T ;Lp(Ωθ;V)) as k→0. The solution uto (2.1) is, in particular, an element of

C([0, T ]; H). For t∈[0, T ], we always choose n∈ {1, . . . , N}such t∈(tn−1, tn] or n= 1 if

t= 0. Then it follows that

ku(t)−uk(t)kL2(Ωθ;H)=ku(t)−u(tθ

n)kL2(Ωθ;H)→0k→0.

This means that uk(t)→u(t) in L2(Ωθ;H) as k→0 for every t∈[0, T]. Thus, it follows

that

lim

k→0Γk

1(t) = ku(t)k2

L2(Ωθ;H)+ 2 Zt

hA(s)u(s), u(s)iV∗×Vds

=ku(t)k2

H+ 2 Zt

hA(s)u(s), u(s)iV∗×Vds.

Recall that in Theorem 2.1.11 it was proved that ¯

Uk(t)* u(t) in L2(Ωθ;H) for every

t∈[0, T ], ¯

Uk* u in Lp(0, T ;Lp(Ωθ;V)) and Ak¯

Uk* Au in Lq(0, T ;Lq(Ωθ;V∗)) as k→0.

Together with the convergence results mentioned for Γk

1, this yields

lim

k→0Γk

2(t) = −2(u(t), u(t))H−2Zt

hA(s)u(s), u(s)iV∗×Vds−2Zt

hA(s)u(s), u(s)iV∗×Vds

=−2ku(t)k2

H−4Zt

hA(s)u(s), u(s)iV∗×Vds.

Handling Γk

3needs a little more attention. For every t∈(tn−1, tn], n∈ {1, . . . , N}, we can

write

Γk

3(t) = Eθk¯

Uk(t)k2

H+ 2 Zt

EθhAk(s)¯

Uk(s),¯

Uk(s)iV∗×Vds

=EθkUnk2

H+ 2 Ztn

EθhAk(s)¯

Uk(s)−fk(s),¯

Uk(s)iV∗×Vds

+ 2 Zt

Eθhfk(s),¯

Uk(s)iV∗×Vds

−2Ztn

EθhAk(s)¯

Uk(s)−fk(s),¯

Uk(s)iV∗×Vds.

Inserting equation (2.14), it follows that

Ztn

EθhAk(s)¯

Uk(s)−fk(s),¯

Uk(s)iV∗×Vds=−Ztn

Eθh(Uk)0(s),¯

Uk(s)iV∗×Vds.

Applying the specific structure of the piecewise constant and piecewise linear prolongation

from (2.12) and (2.13), the integral containing the weak derivative of Ukcan be estimated

−2Ztn

Eθh(Uk)0(s),¯

Uk(s)iV∗×Vds=−2

i=1

Eθ(Ui−Ui−1,Ui)H

≤ −

i=1 EθkUik2

H−EθkUi−1k2

H

=−EθkUnk2

H+EθkU0k2

H.

2.1. CONVERGENCE ON A RANDOMLY SHIFTED GRID 29

Here, we used the telescopic structure of the sum as well as the identity from Lemma A.1.4.

A combination of the previous arguments then gives us the bound

Γk

3(t)≤ ku0k2

H+ 2 Zt

Eθhfk(s),¯

Uk(s)iV∗×Vds

−2Ztn

EθhAk(s)¯

Uk(s)−fk(s),¯

Uk(s)iV∗×Vds

(2.19)

for every t∈(tn−1, tn], n∈ {1, . . . , N}. In Lemma 2.1.9, we proved that fk→fin

Lq(0, T ;Lq(Ωθ;V∗)) as k→0. Further, in Theorem 2.1.11, we showed that ¯

Uk* u in

Lp(0, T ;Lp(Ωθ;V)) as k→0. Thus, it follows that the first integral on the right-hand side

of (2.19) converges to Rt

0hf(s), u(s)iV∗×Vds. For the second integral, we notice that

hAk(s)¯

Uk(s)−fk(s),¯

Uk(s)iV∗×V

≤ kAk(s)¯

Uk(s)kV∗k¯

Uk(s)kV+kfk(s)kV∗k¯

Uk(s)kV

≤βk¯

Uk(s)kV+k¯

Uk(s)kp

V+1

qkfk(s)kq

V∗+1

pk¯

Uk(s)kp

V=: g(s)

for almost all s∈(0, T ). The function gis bounded by a function in L1((0, T)×Ωθ), compare

Lemma 2.1.6 and the a priori bound (2.7) from Lemma 2.1.7. Thus, the second integral in

(2.19) tends to zero as |tn−t| → 0. Thus, it follows that

lim sup

k→0

Γk

3(t)≤ ku0k2

H+ 2 Zt

hf(s), u(s)iV∗×Vds.

Now, it remains to combine all these results to find that

lim sup

k→0Eθkuk(t)−¯

Uk(t)k2

H+ 2ηZt

Eθkuk(s)−¯

Uk(s)kp

Vds

≤ ku(t)k2

H+ 2 Zt

hA(s)u(s), u(s)iV∗×Vds

−2ku(t)k2

H−4Zt

hA(s)u(s), u(s)iV∗×Vds

+ku0k2

H+ 2 Zt

hf(s), u(s)iV∗×Vds

=ku0k2

H− ku(t)k2

H+ 2 Zt

hf(s)−A(s)u(s), u(s)iV∗×Vds

=ku0k2

H− ku(t)k2

H+ 2 Zt

hu0(s), u(s)iV∗×Vds.

Since u∈ Wp(0, T ), we can apply a partial integration rule to obtain

ku0k2

H− ku(t)k2

H+ 2 Zt

hu0(s), u(s)iV∗×Vds=ku0k2

H− ku(t)k2

H+Zt

dsku(s)k2

Hds= 0.

As uk(t)→u(t) in L2(Ωθ;H) as k→0 for every t∈[0, T ], an application of the triangular

inequality yields that ¯

Uk(t)→u(t) in L2(Ωθ;H) as k→0 for every t∈[0, T ]. In the case

of η∈(0,∞), a similar argumentation yields that ¯

Uk→uin Lp(0, T ;Lp(Ωθ;V)) as k→0

as the same thing is true for (uk)k>0.

30 CHAPTER 2. RANDOMIZED SCHEMES

It remains to prove that Uk(t)→u(t) in L2(Ωθ;H) as k→0 for every t∈[0, T]. This

is mainly due to the fact that the limit of both the sequence of the piecewise constant and

piecewise linear prolongations coincide in a suitable sense, see also [43] for some further re-

sults. Recall the definition of ¯

Ukand Ukfrom (2.12) and (2.13), respectively. An application

of the triangle inequality then yields that

kUk(t)−u(t)kL2(Ωθ;H)

≤



tn−t

k¯

Uk(t−k)−u(t)



L2(Ωθ;H)+



t−tn−1

k¯

Uk(t)−u(t)



L2(Ωθ;H)

≤ k ¯

Uk(t−k)−u(t)kL2(Ωθ;H)+k¯

Uk(t)−u(t)kL2(Ωθ;H)

≤ k ¯

Uk(t−k)−u(t−k)kL2(Ωθ;H)+ku(t−k)−u(t)kL2(Ωθ;H)+k¯

Uk(t)−u(t)kL2(Ωθ;H)

for every t∈(tn−1, tn], n∈ {1, . . . , N}. Using that u∈ Wp(0, T ),→C([0, T]; H) and

Uk(t)→u(t) in L2(Ωθ;H) as k→0 for every t∈[0, T ], it also follows that Uk(t)→u(t) in

L2(Ωθ;H) as k→0.

2.2 Explicit Error Estimates

Having proved convergence under no additional regularity assumptions, we now consider

a second type of randomization. This is appropriate to prove explicit error estimates of

the scheme. Here, the size of the error depends on the regularity of the exact solution.

In the previous section, we used a randomized grid. Now, we still use a randomized grid

but evaluate the data at a randomized point in between the randomly shifted grid points.

Precisely, the random points are given in the following assumption.

Assumption 2.2.1. Let T∈(0,∞)and N∈Nbe given. Consider the equidistant partition

0 = t0<· · · < tN=Twith k=T

Nand tn=nk,n∈ {0, . . . , N}. For a complete probability

space (Ωθ,Fθ,Pθ)and a uniformly distributed random variable θ: Ωθ→[0,1], the randomly

shifted grid is denoted by 0 = tθ

0< tθ

1<· · · < tθ

N=T−k(1 −θ)with tθ

n=tn−1+kθ for

n∈ {1, . . . , N}. The step size is denoted by kn=tθ

n−tθ

n−1for n∈ {1, . . . , N}.

Let (Ωτ,Fτ,Pτ)be a second complete probability space and let (τn)n∈{1,...,N}be a family

of independent, uniformly distributed random variables such that τn: Ωτ→[0,1]. On the

product probability space (Ω,F,P) = (Ωθ×Ωτ,Fθ⊗ Fτ,Pθ⊗Pτ), let ξn: Ω →[0,1] be

given by ξn=tθ

n−1+knτnfor n∈ {1, . . . , N}.

The expectations on the probability spaces (Ωτ,Fτ,Pτ) and (Ω,F,P) are denoted by

Eτand E, respectively. Note that the grid is not equidistant since k1=θk but kn=kfor

n∈ {2, . . . , N}. We still have kn≤kfor every n∈ {1, . . . , N}. This specific structure of

randomization will be used to show our desired error bounds. We evaluate the data at the

randomized points ξn,n∈ {1, . . . , N}, and compare (Un)n∈{1,...,N}with u(tθ

n)n∈{1,...,N}.

This randomization is a mixture of the one considered in [37] for the approximation of

nonautonomous evolution equations and the one in [38] for the quadrature of stochastic

Itˆo-integrals. Similar to [38], we can now weaken the regularity assumption on the solution.

Instead of asking for a H¨older continuous solution as has been done in [37], we now assume

that it is an element of a Sobolev–Slobodecki˘ı space. Since the exponent for the fractional

derivative in a Sobolev–Slobodecki˘ı space can be larger than the exponent in a H¨older space,

our setting can fill a gap between rates of convergence that are seen in numerical examples

and rates that are theoretically derived.

In order to prove error estimates of schemes for nonlinear problems, it is possible to

use a linear approximation of the operator given by its derivative. In [86] or [95], the fully

2.2. EXPLICIT ERROR ESTIMATES 31

nonlinear equation u0(t) = F(t, u(t)) for t∈(0, T ) with an initial condition is linearized

along the exact solution u. If the partial derivative A(t) = ∂uF(t, u) exists, we can work

with it instead of the fully nonlinear problem. In the following, we want to work with an

approach that does not rely on such an approximation via a partial derivative. But we follow

an approach that uses the fact that A(t), t∈[0, T], is Lipschitz continuous on a bounded set.

Our starting point is to generalize the approach from [37], where error estimates are proved

for a globally Lipschitz continuous operator. This is extended to a generalized Lipschitz

condition, see (2.23) below. Now, we can also consider operators A(t), t∈[0, T ], that do

not have to be of linear growth. An example, which fits in our framework, is the classical

p-Laplacian in a variational formulation. Note that for p= 2, the framework is nearly the

same as in [37]. In [41, 42, 44] error estimates are provided without the additional bounded

Lipschitz condition for the operator A(t), t∈[0, T]. One advantage of our approach is

that we do not have to impose any temporal regularity on fand A. This could be of

advantage if u0=f−Au fulfills a stronger regularity condition than the functions fand Au

separately. The case where a function u0is more regular than fand Au separately probably

does not contain many relevant problems. Thus, it would also be interesting to consider the

techniques from [41, 42, 44] in the context of randomized schemes.

With the random point (ξn)n∈{1,...,N}from Assumption 2.2.1, we consider the scheme

(Un+knA(ξn)Un=knf(ξn) + Un−1almost surely in V∗, n ∈ {1, . . . , N},

U0=u0in H, (2.20)

for an initial value u0∈Hand a source term f∈Lq(0, T ;V∗).

In Assumption 2.2.1, we introduced N+1 independent random variables. In this section,

it becomes necessary to consider two filtrations (Fτ

n)n∈{0,...,N}⊂ Fτand (Fn)n∈{0,...,N}⊂

F=Fθ⊗ Fτ. The first is given by

Fτ

0:= σN ∈ Fτ:Pτ(N)=0

Fτ

n:= σστi:i∈ {1, . . . , n}∪ Fτ

0, n ∈ {1, . . . , N},(2.21)

where σdenotes the generated σ-algebra, compare (A.4). Further, we consider

Fn:= Fθ⊗ Fτ

n, n ∈ {0, . . . , N}.(2.22)

In particular, it is clear that Fτ

n⊂ Fτ

mand Fn⊂ Fmfor n≤m. Note that, for every

n∈ {1, . . . , N}, the mapping ξn: Ω →[0,1] is Fn-measurable as a composition of measur-

able functions. Also, ξn(ωθ,·): Ωτ→[tθ

n−1(ωθ), tθ

n(ωθ)] is a uniformly distributed random

variable, which is Fτ

n-measurable for every n∈ {1, . . . , N}and ωθ∈Ωθ. We begin by prov-

ing that (2.20) is uniquely solvable and its solution is adapted to the filtrations introduced

above.

Lemma 2.2.2. Let Assumptions 2.1.1 and 2.2.1 be fulfilled and let f∈Lq(0, T ;V∗)as well

as u0∈Hbe given. For a step size k=T

N, there exists a unique solution (Un)n∈{1,...,N}

to the recursion (2.20). For every n∈ {1, . . . , N}, the mapping Un: Ω →V,ω7→ Un(ω)

is Fn-measurable, while Un(ωθ,·): Ωτ→V,ωτ7→ Un(ωθ, ωτ)is Fτ

n-measurable for almost

every ωθ∈Ωθ.

Proof. The existence of (Un)n∈{1...,N}can be proved analogously as in Lemma 2.1.5.

In order to prove the measurability conditions, we again use Lemma 2.1.4. For n∈

{1, . . . , N}, we consider the mappings

hn: Ω ×V→V∗, hn(ω, U) = I+kn(ω)A(ξn(ω))U−kn(ω)f(ξn(ω)) −Un−1(ω)

32 CHAPTER 2. RANDOMIZED SCHEMES

as well as

hτ

n: Ωτ×V→V∗, hn(ωτ,U) = I+kn(ωθ)A(ξn(ωθ, ωτ))U

−kn(ωθ)f(ξn(ωθ, ωτ)) −Un−1(ωθ, ωτ)

for almost every ωθ∈Ωθ. Note that for hn, we consider kn: Ω →[0,1] given by kn(ωθ, ωτ) =

kn(ωθ). This mapping is also measurable with respect to Fn. As in the proof of Lemma 2.1.5,

we obtain that ω7→ Un(ω) is Fn-measurable and ωτ7→ Un(ωθ, ωτ) is Fτ

n-measurable for

almost every ωθ∈Ωθ.

Now that the existence of a solution to (2.20) is covered, let it be mentioned that the a

priori bound from Lemma 2.1.7 holds true for this scheme if Eθis replaced by the expectation

Eon the probability space (Ω,F,P). It remains to make sure that we have a bound for

the terms containing f. To this end, we also mention a counterpart to Lemma 2.1.6 for the

second randomization.

Lemma 2.2.3. Let Assumption 2.2.1 be fulfilled and let (X, k · kX)be a real Banach space.

Then for f∈Lq(0, T ;X),q∈[1,∞), the bound

n=1

Eknkf(ξn)kq

X≤2kfkq

Lq(0,T ;X)

is fulfilled.

Proof. For the first summand, we use a substitution as in (A.2) and k1=tθ

1=kθ to obtain

that

Eθk1Eτkf(ξ1)kq

X=Z1

ks Z1

kf(kst)kq

Xdtds

kZt1

sZ1

kf(st)kq

Xdtds

kZt1

0Zs

kf(t)kq

Xdtds≤Zt1

kf(t)kq

Xdt.

For n∈ {2, . . . , N}, we can argue similarly but notice that the step size knis equal to the

maximal step size k. Then we see that

EθknEτkf(ξn)kq

X=kZ1

0Z1

kf(tn−2+sk +kt)kq

Xdtds

=Ztn−1

tn−2Z1

kf(s+kt)kq

Xdtds

kZtn−1

tn−2Zs+k

kf(t)kq

Xdtds≤Ztn

tn−2

kf(t)kq

Xdt.

Thus, altogether this proves the bound

n=1

Eknkf(ξn)kq

X≤Zt1

kf(t)kq

Xdt+

n=2 Ztn

tn−2

kf(t)kq

Xdt≤2ZT

kf(t)kq

Xdt.

2.2. EXPLICIT ERROR ESTIMATES 33

In order to use the higher-order regularity of a function in the error estimates, we consider

the following two lemmas.

Lemma 2.2.4. Let Assumption 2.2.1 be fulfilled and let (X, k · kX)be a real Banach space.

For α∈(0,1) and q∈[1,∞), let v∈Wα,q(0, T ;X)be given. Then the bound

n=1

Eknkv(ξn)−v(tθ

n)kq

X≤kqα|v|q

Wα,q(0,T ;X)

is fulfilled.

Proof. Since we do not work on an equidistant grid, there are two cases that need to be

considered separately. The distance between 0 and tθ

1is given by the random value k1=θk.

After a substitution as in (A.2), we see for the first summand using ξ1=tθ

0+k1τ1=kθτ1

Eθk1Eτkv(ξ1)−v(tθ

1)kq

X=Z1

ks Z1

kv(kst)−v(ks)kq

Xdtds

kZt1

sZ1

kv(st)−v(s)kq

Xdtds

kZt1

0Zs

kv(t)−v(s)kq

Xdtds

≤kqα Zt1

0Zs

kv(t)−v(s)kq

|t−s|qα+1 dtds.

For n∈ {2, . . . , N}, we use that the distance between tθ

nand tθ

n−1is always given by k.

Thus, all further terms can be estimated by

EθknEτkv(ξn)−v(tθ

n)kq

X

=kZ1

0Z1

kv(tn−2+ks +kt)−v(tn−1+ks)kq

Xdtds

=Ztn

tn−1Z1

kv(s+kt −k)−v(s)kq

Xdtds

kZtn

tn−1Zs

s−k

kv(t)−v(s)kq

Xdtds

≤kqα Ztn

tn−1Zs

s−k

kv(t)−v(s)kq

|t−s|qα+1 dtds.

Combining these estimates, yields

n=1

Eknkv(ξn)−v(tθ

n)kq

X

≤kqα Zt1

0Zs

kv(t)−v(s)kq

|t−s|qα+1 dtds+kqα ZT

t1Zs

s−k

kv(t)−v(s)kq

|t−s|qα+1 dtds

≤kqα ZT

0ZT

kv(t)−v(s)kq

|t−s|qα+1 dtds=kqα|v|q

Wα,q(0,T ;X).

34 CHAPTER 2. RANDOMIZED SCHEMES

Lemma 2.2.5. Let Assumption 2.2.1 be fulfilled and let (X, k · kX)be a real Banach space.

For α∈(0,1) and q∈[1,∞), let v∈Wα,q(0, T ;X)be given. Then the bound

n=1

EhZtθ

tθ

n−1

kv(t)−v(ξn)kq

Xdti≤2kqα|v|q

Wα,q(0,T ;X)

is fulfilled.

Proof. The proof is very similar to the proof of Lemma 2.2.4 but an additional integral

appears in the estimates. Again, we consider the first summand separately. A substitution

as in (A.2) and using the equality ξ1=tθ

0+k1τ1=kθτ1yields

EθhEτhZtθ

kv(t)−v(ξ1)kq

Xdtii=Z1

0Z1

0Zks

kv(t)−v(ksr)kq

Xdtdrds

kZt1

0Z1

0Zs

kv(t)−v(sr)kq

Xdtdrds

kZt1

sZs

0Zs

kv(t)−v(r)kq

Xdtdrds

≤1

kZt1

sqα Zs

0Zs

kv(t)−v(r)kq

|t−r|qα+1 dtdrds

≤kqα Zt1

0Zt1

kv(t)−v(r)kq

|t−r|qα+1 dtdr.

Similarly, for n∈ {2, . . . , N}, we see that

EθhEτhZtθ

tθ

n−1

kv(t)−v(ξn)kq

Xdtii=Z1

0Z1

0Ztn−1+ks

tn−2+ks

kv(t)−v(tn−2+ks +kr)kq

Xdtdrds

kZtn−1

tn−2Z1

0Zs+k

kv(t)−v(s+kr)kq

Xdtdrds

k2Ztn−1

tn−2Zs+k

sZs+k

kv(t)−v(r)kq

Xdtdrds

≤kqα−1Ztn−1

tn−2Zs+k

sZs+k

kv(t)−v(r)kq

|t−r|qα+1 dtdrds

≤kqα Ztn

tn−2Ztn

tn−2

kv(t)−v(r)kq

|t−r|qα+1 dtdr.

A combination of the estimates then shows

n=1

EhZtθ

tθ

n−1

kv(t)−v(ξn)kq

Xdti

≤kqα Zt1

0Zt1

kv(t)−v(r)kq

|t−r|qα+1 dtdr+kqα

n=2 Ztn

tn−2Ztn

tn−2

kv(t)−v(r)kq

|t−r|qα+1 dtdr

≤2kqα ZT

0ZT

kv(t)−v(r)kq

|t−r|qα+1 dtdr= 2kqα|v|q

Wα,q(0,T ;X).

2.2. EXPLICIT ERROR ESTIMATES 35

The previous lemma can also be proved with the help of Lemma 2.2.4. Using the trian-

gular inequality yields

N

n=1

EhZtθ

tθ

n−1

kv(t)−v(ξn)kq

Xdti

≤N

n=1

EhZtθ

tθ

n−1

kv(t)−v(tθ

n)kq

Xdti

q+N

n=1

Eknkv(tθ

n)−v(ξn)kq

X

= 2N

n=1

Eknkv(tθ

n)−v(ξn)kq

X

q≤2kα|v|Wα,q(0,T ;X).

For large q, this leads to a worse constant though.

Now, we are well prepared to prove the two main statements of this section. Both show

error bounds for the expectation of the distance between the exact solution at a shifted

grid point and the numerical approximation. The magnitude of the error depends on the

regularity of the exact solution u. Here, we measure the temporal regularity of uand u0

within a space of Sobolev–Slobodecki˘ı functions. In the first theorem, we assume that the

temporal derivative of the exact solution is Hvalued. In the second theorem, we show that

this can be weakened to values in V∗. To obtain the same error bound, we require more

temporal regularity for the V∗-valued result though.

Further, we state two different bounded Lipschitz conditions in each theorem. In the

first one, the operator is Lipschitz continuous on bounded sets in H. Alternatively, it is also

possible to ask for a Lipschitz condition on a bounded set in V. As every bounded set in Vis

also bounded in H, but not necessarily the other way around, the second condition is more

general. For the second Lipschitz condition, we additionally need that the solution uis an

element of L∞(0, T ;V). Note that this assumption is fulfilled directly if the differentiability

exponent of the Sobolev–Slobodecki˘ı space is large enough (cf. [102, Corollary 32]). In

this section, we only make the assumption that the specific regularity is fulfilled without

any further explanation. Some information for additional regularity and more concrete

examples that fit this setting can be found in Section 1.2.

Theorem 2.2.6. Let Assumptions 2.1.1 and 2.2.1 be fulfilled and let f∈Lq(0, T ;V∗),

q=p

p−1, as well as the initial value u0∈Hbe given. Let the operator A(t),t∈[0, T ],

fulfill a bounded Lipschitz condition in the sense that for every R∈(0,∞)there exists

L(R)∈[0,∞)such that

kA(t)v−A(t)wkV∗≤L(R)kv−wkV(2.23)

is fulfilled for all t∈[0, T]and v, w ∈Vwith kvkH,kwkH≤R. If the exact solution uis

an element of L∞(0, T ;V), then it is sufficient that (2.23) is fulfilled for all v, w ∈Vwith

kvkV,kwkV≤R. Furthermore, let A(t),t∈[0, T], satisfy a p-monotonicity condition such

that there exists η∈(0,∞)with

hA(t)v−A(t)w, v −wiV∗×V≥ηkv−wkp

V(2.24)

for all v, w ∈Vand t∈[0, T ].

Let the exact solution ube an element of Wα,q(0, T ;V)for α∈(0,1). Further, let the

temporal derivative u0of the exact solution be an element of Wγ,2(0, T ;H)for γ∈0,1

2.

Then there exists C∈(0,∞)such that for every maximal step size k=T

N,N∈N, the error

36 CHAPTER 2. RANDOMIZED SCHEMES

estimate

max

n∈{1,...,N}Eku(tθ

n)−Unk2

H+

n=1

Eknku(tθ

i)−Unkp

V≤C(k2γ+1 +kqα)

is fulfilled for the solution (Un)n∈{1,...,N}to (2.20).

Due to the boundedness condition from Assumption 2.1.1 (4), for the relevant exam-

ples that fulfill (2.23), we get L(R) = L0(max kvkH,kwkH)1 + kvkp−2

V+kwkp−2

Vwith

L0:R→[0,∞). It is also possible to include the case γ= 0 to the assumptions of the

theorem in the sense that we assume u0∈L2(0, T ;H). The proof remains similar. Instead

of applying Lemma 2.2.5 to the terms containing the derivative u0, it then becomes necessary

to apply Lemma 2.2.3.

Proof of Theorem 2.2.6. In the following, let i∈ {1, . . . , N}be fixed. As the exact solution

is an element of Wp(0, T ) its derivative is in Lq(0, T;V∗). Further, u(tθ

i)−Uiis an element

of Lp(Ω; V) and we can write

E(u(tθ

i)−u(tθ

i−1), u(tθ

i)−Ui)H=EhZtθ

tθ

i−1

hu0(t), u(tθ

i)−UiiV∗×Vdti.(2.25)

Here, we technically do not have any point evaluation of uas we consider the expectation

of the equality. The scheme (2.20) tested with u(tθ

i)−Uiyields the equality

E(Ui−Ui−1, u(tθ

i)−Ui)H=Ekihf(ξi)−A(ξi)Ui, u(tθ

i)−UiiV∗×V.(2.26)

Using the identity from Lemma A.1.4, the difference of the left-hand sides of (2.25) and

(2.26) can be written as

E(u(tθ

i)−Ui−u(tθ

i−1) + Ui−1, u(tθ

i)−Ui)H

2Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H+Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H.

The difference of the right-hand sides of (2.25) and (2.26) can be rewritten by adding and

subtracting terms containing A(ξi). This allows us to use the structure of the operator A(ξi)

more efficiently. We then obtain

EhZtθ

tθ

i−1

hu0(t), u(tθ

i)−UiiV∗×Vdti

−Ekihf(ξi)−A(ξi)Ui, u(tθ

i)−UiiV∗×V

=EhZtθ

tθ

i−1

hu0(t), u(tθ

i)−UiiV∗×Vdti

−Ekihf(ξi)−A(ξi)u(ξi), u(tθ

i)−UiiV∗×V

−EkihA(ξi)u(ξi)−A(ξi)u(tθ

i), u(tθ

i)−UiiV∗×V

−EkihA(ξi)u(tθ

i)−A(ξi)Ui, u(tθ

i)−UiiV∗×V

=EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−UiiV∗×Vdti

−EkihA(ξi)u(ξi)−A(ξi)u(tθ

i), u(tθ

i)−UiiV∗×V

−EkihA(ξi)u(tθ

i)−A(ξi)Ui, u(tθ

i)−UiiV∗×V

=: Γ1+ Γ2+ Γ3,

2.2. EXPLICIT ERROR ESTIMATES 37

where

Γ1=EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−UiiV∗×Vdti,

Γ2=−EkihA(ξi)u(ξi)−A(ξi)u(tθ

i), u(tθ

i)−UiiV∗×V,

Γ3=−EkihA(ξi)u(tθ

i)−A(ξi)Ui, u(tθ

i)−UiiV∗×V.

(2.27)

Thus, we obtain that

Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H+Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

= 2Γ1+ 2Γ2+ 2Γ3.(2.28)

Since we added and subtracted the terms containing A(ξi)u(ξi) and A(ξi)u(tθ

i) above, we

can now estimate Γ1, Γ2, and Γ3more easily. For Γ1, we use the regularity of u0. In order

to estimate Γ2, we use the bounded Lipschitz condition and the regularity of u, while for Γ3

we use the monotonicity of A(ξi). Precisely, for Γ1, we obtain that

Γ1=EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−UiiV∗×Vdti

=EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−Ui−u(tθ

i−1) + Ui−1iV∗×Vdti(2.29)

+EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i−1)−Ui−1iV∗×Vdti.(2.30)

Using the Cauchy–Schwarz inequality as well as the weighted Young inequality, we find the

estimate for (2.29)

EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−Ui−u(tθ

i−1) + Ui−1iV∗×Vdti

≤EhZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdt

2kiku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

≤EhkiZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti

2Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

≤1

2EhkiZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti+1

2Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H.

This structure is useful, as we can absorb the second summand in the last row using one of

the terms on the left-hand side of (2.28). Further, we can write for (2.30)

EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i−1)−Ui−1iV∗×Vdti

=EθhEτhZtθ

tθ

i−1

hu0(t), u(tθ

i−1)−Ui−1iV∗×Vdtii

−EθkiEτhu0(ξi), u(tθ

i−1)−Ui−1iV∗×V.

38 CHAPTER 2. RANDOMIZED SCHEMES

In the following, we denote the conditional expectation with respect to Fτ

i−1by Eτ·|Fτ

i−1,

compare Appendix A.3. We notice that Ωτ3ωτ7→ u(tθ

i−1(ωθ))−Ui−1(ωθ, ωτ) is measurable

with respect to the σ-algebra Fτ

i−1for almost every ωθ∈Ωθ, compare Lemma 2.2.2. Thus,

we use the tower property for the conditional expectation to obtain

Eτhu0(ξi), u(tθ

i−1)−Ui−1iV∗×V=EτEτhu0(ξi), u(tθ

i−1)−Ui−1iV∗×V|Fτ

i−1

=EτhEτu0(ξi)|Fτ

i−1, u(tθ

i−1)−Ui−1iV∗×V

almost surely in Ωθ. As the generated σ-algebra σ(τi) is independent of Fτ

i−1, it follows that

Eτu0(ξi)|Fτ

i−1=Eτu0(ξi)almost surely in Ωθ. Then we find that

hEτu0(ξi)|Fτ

i−1, u(tθ

i−1)−Ui−1iV∗×V=hEτu0(ξi), u(tθ

i−1)−Ui−1iV∗×V

kiZtθ

tθ

i−1

hu0(t), u(tθ

i−1)−Ui−1iV∗×Vdt

almost surely in Ω and therefore, in particular,

EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i−1)−Ui−1iV∗×Vdti= 0.

Altogether, this proves a bound for Γ1that is given by

Γ1≤1

2EhkiZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti+1

2Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H.

In order to estimate Γ2, we apply the bounded Lipschitz condition of the operator A(t),

t∈[0, T ], from (2.23). Thus, we see that

Γ2=−EkihA(ξi)u(ξi)−A(ξi)u(tθ

i), u(tθ

i)−UiiV∗×V

≤L(R)Ekiku(ξi)−u(tθ

i)kVku(tθ

i)−UikV

≤c1Ekiku(ξi)−u(tθ

i)kq

V+η

2Ekiku(tθ

i)−Uikp

V,

for c1=(pη)1−qL(R)q

21−qq. Here, we can choose the parameter Rfor (2.23) as R=kukL∞(0,T ;H).

Recall that a weak solution uof (2.1) is an element of Wp(0, T ),→C([0, T]; H). Thus, this

particular Ris finite. If the solution fulfills u∈L∞(0, T ;V), we can choose R=kukL∞(0,T ;V)

and it is sufficient that (2.23) is fulfilled for v, w ∈Vwith kvkV,kwkV≤R. Last, observe

that

Γ3=−EkihA(ξi)u(tθ

i)−A(ξi)Ui, u(tθ

i)−UiiV∗×V≤ −ηEkiku(tθ

i)−Uikp

V

is fulfilled due to the monotonicity condition from (2.24). After an insertion of these bounds

into (2.28), we see that

Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H+Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

= 2Γ1+ 2Γ2+ 2Γ3

≤EhkiZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti+Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

+ 2c1Ekiku(ξi)−u(tθ

i)kq

V−ηEkiku(tθ

i)−Uikp

V.

2.2. EXPLICIT ERROR ESTIMATES 39

This implies, in particular,

Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H+ηEkiku(tθ

i)−Uikp

V

≤EhkiZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti+ 2c1Ekiku(ξi)−u(tθ

i)kq

V.

Summing up the inequality from i= 1 to n∈ {1, . . . , N}, we can make use of the telescopic

sum structure, the fact that u(0) −U0=u(0) −u0= 0, as well as Lemma 2.2.4 and

Lemma 2.2.5 and obtain

Eku(tθ

n)−Unk2

H+η

i=1

Ekiku(tθ

i)−Uikp

V

≤k

i=1

EhZtθ

tθ

i−1

ku0(t)−u0(ξi)k2

Hdti+ 2c1

i=1

Ekiku(ξi)−u(tθ

i)kq

V

≤2k2γ+1|u0|2

Wγ,2(0,T ;H)+ 2c1kqα|u|q

Wα,q(0,T ;V).(2.31)

The next theorem contains a comparable result, where the temporal regularity condition

of u0changes while the spatial regularity condition can be relaxed to V∗.

Theorem 2.2.7. Let Assumptions 2.1.1 and 2.2.1 be fulfilled and let f∈Lq(0, T;V∗),

q=p

p−1, as well as the initial value u0∈Hbe given. Furthermore, let the operator A(t),

t∈[0, T ], fulfill the bounded Lipschitz condition (2.23) and the p-monotonicity condition

(2.24) as in Theorem 2.2.6.

Let the exact solution ube an element of Wα,q(0, T ;V)for α∈(0,1). Further, let the

temporal derivative u0of the exact solution be an element of Wγ,q(0, T ;V∗)for γ∈(0,1).

Then there exists C∈(0,∞)such that for every maximal step size k=T

N,N∈N, the error

estimate

max

n∈{1,...,N}Eku(tθ

n)−Unk2

H+

n=1

Eknku(tθ

n)−Unkp

V≤C(kqγ +kqα)

is fulfilled for the solution (Un)n∈{1,...,N}to (2.20).

Proof. Analogously to the proof of Theorem 2.2.6, we can write

Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H

+Eku(tθ

i)−Ui−u(tθ

i−1) + Ui−1k2

H

= 2Γ1+ 2Γ2+ 2Γ3,

(2.32)

where Γ1, Γ2, and Γ3are given in (2.27). Again, we consider Γ1, Γ2, and Γ3separately,

where we can use analogous bounds as in the proof of Theorem 2.2.6. First, we obtain

Γ2=−EkihA(ξi)u(ξi)−A(ξi)u(tθ

i), u(tθ

i)−UiiV∗×V

≤c1Ekiku(ξi)−u(tθ

i)kq

V+η

4Ekiku(tθ

i)−Uikp

V,

for c1=(pη)1−qL(R)q

41−qq. Again, we choose the parameter Rfor (2.23) as R=kukL∞(0,T ;H).

If the solution fulfills u∈L∞(0, T ;V), we choose R=kukL∞(0,T ;V)and only require that

40 CHAPTER 2. RANDOMIZED SCHEMES

(2.23) is fulfilled for v, w ∈Vwith kvkV,kwkV≤R. For Γ3, we apply the monotonicity

condition (2.24) and find

Γ3≤ −ηEkiku(tθ

i)−Uikp

V.

When estimating Γ1, we now have to use the different regularity assumption on u0. Here,

we apply the weighted Young inequality to see that

Γ1=EhZtθ

tθ

i−1

hu0(t)−u0(ξi), u(tθ

i)−UiiV∗×Vdti

≤EhZtθ

tθ

i−1

ku0(t)−u0(ξi)kV∗ku(tθ

i)−UikVdti

≤c2EhZtθ

tθ

i−1

ku0(t)−u0(ξi)kq

V∗dti+η

4Ekiku(tθ

i)−Uikp

V,

with the constant c2=(pη)1−q

41−qq. Inserting the bounds in (2.32), shows that

Eku(tθ

i)−Uik2

H−Eku(tθ

i−1)−Ui−1k2

H

≤2Γ1+ 2Γ2+ 2Γ3

≤2c2EhZtθ

tθ

i−1

ku0(t)−u0(ξi)kq

V∗dti+ 2c1Ekiku(ξi)−u(tθ

i)kq

V−ηEkiku(tθ

i)−Uikp

V.

The remainder of the proof can be done analogously to the end of the proof of Theorem 2.2.6.

Note that in the proof of Theorem 2.2.7 we do not use the independence of {τn}n∈{1,...,N}.

Thus, here it would also be possible to choose the same random variable τn=τfor every

n∈ {1, . . . , N}, which is uniformly distributed in [0,1].

When we compare the two results from the previous theorems, different error rates

can be seen. If u0is an element of Wγ,2(0, T;H) then the error can be smaller than for

u0∈Wγ,q(0, T ;V∗) if uis smooth enough. Still, the first result is not necessarily stronger.

In [90, Proposition 6.6], it is demonstrated how the temporal regularity decreases when

the spatial regularity becomes higher. Thus, in practice, the two results should lead to

comparable error estimates.

2.3 Example: A Problem of p-Laplacian Type

For a finite end time T∈(0,∞) and a bounded Lipschitz domain D ⊂ Rd,d∈N, we regard











∂tu(t, x)−∇·a(t, x, ∇u(t, x)) = f(t, x),(t, x)∈(0, T )× D,

u(t, x) = 0,(t, x)∈(0, T )×∂D,

u(0, x) = u0(x), x ∈ D.

(2.33)

Here, the mapping a: [0, T ]×D×Rd→Rdfulfills the assumption below and f: [0, T ]×D → R

as well as u0:D → Rwill be specified later.

Assumption 2.3.1. Let p∈[2,∞)be given and q=p

p−1. Let a: [0, T ]×D × Rd→Rd

fulfill the following conditions:

2.3. EXAMPLE: A PROBLEM OF P-LAPLACIAN TYPE 41

(1) The map (t, x)7→ a(t, x, z)is measurable for every z∈Rd, while z7→ a(t, x, z)is

continuous for every t∈[0, T ]and almost every x∈ D.

(2) The map afulfills a monotonicity condition in the sense that the inequality (a(t, x, z)−

a(t, x, ˜z)) ·(z−˜z)≥0is satisfied for every t∈[0, T ], almost every x∈ D, and every

z, ˜z∈Rd.

(3) The map afulfills a growth condition in the sense that there exist d1∈[0,∞)and a

nonnegative function d2∈Lq(D)such that for every t∈[0, T ], almost every x∈ D,

and every z∈Rdthe inequality |a(t, x, z)| ≤ d1|z|p−1+d2(x)is satisfied.

(4) The map afulfills a coercivity condition in the sense that there exist d3∈(0,∞)and

a nonnegative d4∈L1(D)such that for every t∈[0, T ], almost every x∈ D, as well

as every z∈Rdthe condition a(t, x, z)·z≥d3|z|p−d4(x)is satisfied.

Assumption 2.3.2. Let Assumption 2.3.1 be fulfilled. Additionally, there exists d5∈(0,∞)

such that

(a(t, x, z)−a(t, x, ˜z)) ·(z−˜z)≥d5|z−˜z|p

is satisfied for every t∈[0, T ], almost every x∈ D, and every z, ˜z∈Rd.

Assumption 2.3.3. Let Assumption 2.3.1 be fulfilled. Additionally, there exists d6∈[0,∞)

such that

|a(t, x, z)−a(t, x, ˜z)| ≤ d61 + max{|z|p−2,|˜z|p−2}|z−˜z|(2.34)

is satisfied for every t∈[0, T ], almost every x∈ D, and every z, ˜z∈Rd.

A prototype example for the function ais given by a(t, x, z) = a(z) = |z|p−2z. Then

(2.33) is the p-Laplace equation. It is easy to see that afulfills Assumption 2.3.1. In [28,

Chapter I, Lemma 4.4] it is proved that this afulfills Assumption 2.3.2. In order to see that

the function fulfills Assumption 2.3.3, we notice that for z∈Rdwith z6= 0

∂ia(z) = p−2

2d

j=1

jp−4

22ziz+|z|p−2ei= (p−2)|z|p−4ziz+|z|p−2ei

is fulfilled for i∈ {1, . . . , d}and the i-th unit vector eiin Rd. Moreover, we have ∂ia(0) = 0

for every i∈ {1, . . . , d}. Then the Jacobian matrix fulfills

|∇a(z)|=(p−2)|z|p−4zzT+|z|p−2I≤(p−1)|z|p−2

for z∈Rd. An application of the mean value theorem then shows that (2.34) is fulfilled.

In order to formulate the problem (2.33) in a weak formulation, we consider the spaces

V=W1,p

0(D) and H=L2(D), where p∈[2,∞) is chosen as in Assumption 2.3.1. We equip

the spaces with the norms introduced in the notation section in the introduction. Then we

assume that for f: [0, T ]×D → Rthe abstract function [f(t)](x) = f(t, x), (t, x)∈(0, T )×D,

is an element of Lq(0, T ;V∗) and u0∈H. Further, the operator A(t): V→V∗is given by

hA(t)v, wi=ZD

a(t, ·,∇v)· ∇wdx

for t∈[0, T ] and v, w ∈V. Then we consider the variational formulation of (2.33) given by

(u0+Au =fin Lq(0, T ;V∗),

u(0) = u0in H.

42 CHAPTER 2. RANDOMIZED SCHEMES

Theorem 2.3.4. Let Assumption 2.3.1 be fulfilled. Let f∈Lq(0, T ;V∗)and u0∈Hbe

given. Further, let (N`)`∈Nbe a sequence of natural numbers with N`→ ∞ as `→ ∞,

k=T

N`,tn=nk,n∈ {0, . . . , N`}, and consider the corresponding randomly shifted grid

introduced in Assumption 2.1.3. Then the backward Euler scheme

kUn−Un−1+A(tθ

n)Un=f(tθ

n)in Lq(Ωθ;V∗), n ∈ {1, . . . , N`},

U0=u0in H

admits a unique solution (Un)n∈{1,...,N`}in Lp(Ωθ;V). All the convergence results from

Theorem 2.1.11 and Theorem 2.1.12 hold true. In particular, the sequences of piecewise

constant and piecewise linear prolongations of (Un)n∈{1,...,N`}converge to the weak solution

uof (2.33) pointwise strongly in L2(Ωθ;H)as k→0.

If Assumption 2.3.2 is satisfied additionally, then the sequence of piecewise constant

prolongations converges to ustrongly in Lp(0, T ;Lp(Ωθ;V)) as k→0.

Proof. In order to apply Theorem 2.1.11 and Theorem 2.1.12 from Section 2.1, it only

remains to verify that A(t), t∈[0, T], fulfills Assumption 2.1.1. To this end, let v, w ∈Vbe

given. Then we see that

hA(t)v, wiV∗×V=ZD

a(t, ·,∇v)· ∇wdx

≤ZDd1|∇v|p−1+d2|∇w|dx

≤max d1,kd2kLq(D)1 + kvkp−1

VkwkV,(2.35)

which proves both that A(t), t∈[0, T], is well-defined and that the boundedness condition

from Assumption 2.1.1 (4) is fulfilled.

Since t7→ a(t, x, z) is measurable for almost every x∈ D and every z∈Rd, there

exists a sequence (ai)i∈Nof functions ai: [0, T ]×D × Rd→Rd,i∈N, that are simple

with respect to the first argument such that ai(t, x, z)→a(t, x, z) in Rdas i→ ∞ and

|ai(t, x, z)|≤|a(t, x, z)|,i∈N, for almost every (t, x)∈(0, T )× D and every z∈Rd. Then

Ai(t): V→V∗given by

hAi(t)v, wiV∗×V=ZD

ai(t, ·,∇v)· ∇wdx, v, w ∈V

is a simple function with respect to t∈(0, T ). Using a similar bound as in (2.35), it follows

that a(t, ·,∇v)−ai(t, ·,∇v)· ∇wis bounded by a function that is integrable on D. We

can apply Lebesgue’s dominated convergence theorem to obtain that

lim

i→∞hA(t)v−Ai(t)v, wiV∗×V=ZD

lim

i→∞ a(t, ·,∇v)−ai(t, ·,∇v)· ∇wdx= 0

for every v, w ∈Vand almost every t∈(0, T ). This implies that t7→ A(t)vis weakly mea-

surable since V∗is reflexive. As V∗is also separable, the mapping is Bochner measurable.

In order to prove that A(t): V→V∗,t∈[0, T ], is radially continuous, let (si)i∈Nbe a

convergent sequence in [0,1] with the limit s∈[0,1]. Using the fact that (2.35) is finite,

it follows that a(t, ·,∇v+si∇w)· ∇wis bounded by an integrable function on Dfor every

v, w ∈V. Then we can apply Lebesgue’s dominated convergence theorem and it follows

2.3. EXAMPLE: A PROBLEM OF P-LAPLACIAN TYPE 43

that

lim

i→∞hA(t)(v+siw), wiV∗×V= lim

i→∞ ZD

a(t, ·,∇v+si∇w)· ∇wdx

=ZD

lim

i→∞ a(t, ·,∇v+si∇w)· ∇wdx

=ZD

a(t, ·,∇v+s∇w)· ∇wdx

for every v, w ∈Vand t∈[0, T ] due to Assumption 2.3.1 (1).

The monotonicity condition for A(t), t∈[0, T], is a direct consequence of Assump-

tion 2.3.1 (2). This can be seen as

hA(t)v−A(t)w, v −wiV∗×V=ZD

(a(t, ·,∇v)−a(t, ·,∇w)) ·(∇v− ∇w) dx≥0

is fulfilled for every v, w ∈Vand t∈[0, T ]. Analogously, the condition from Assump-

tion 2.3.2 implies that

hA(t)v−A(t)w, v −wiV∗×V≥d5ZD

|∇v− ∇w|pdx=d5kv−wkp

for every v, w ∈Vand t∈[0, T ]. So we see that (2.18) is fulfilled.

It remains to verify the coercivity condition from Assumption 2.1.1 (5). Here, we apply

Assumption 2.3.1 (4) to see that

hA(t)v, viV∗×V≥ZDd3|∇v|p−d4dx=d3kvkp

V− kd4kL1(D)

for every v∈Vand t∈[0, T ]. Therefore, as the operator A(t), t∈[0, T ], fulfills all

the necessary conditions, we can apply Theorem 2.1.11 and Theorem 2.1.12 to finish the

proof.

To prove explicit error bounds, we make an additional regularity assumption on the

solution of (2.33). We do not discuss this condition here. In Section 1.2, more details and

suitable examples can be found.

Theorem 2.3.5. Let Assumptions 2.3.1, 2.3.2, and 2.3.3 be fulfilled and let f∈Lq(0, T ;V∗)

and u0∈Hbe given. For α∈(0,1) and γ∈(0,1), assume that the exact solution uis

an element of Wα,2(0, T ;V)for p= 2 and of L∞(0, T;V)∩Wα,q(0, T ;V)for p∈(2,∞)

while its temporal derivative u0belongs to Wγ,2(0, T ;H). For every N∈N,k=T

N, and the

corresponding random values (ξn)n∈{1,...,N}introduced in Assumption 2.2.1, the scheme

knUn−Un−1+A(ξn)Un=f(ξn)in Lq(Ω; V∗), n ∈ {1, . . . , N},

U0=u0in H

admits a unique solution (Un)n∈{1,...,N}in Lp(Ω; V). Then there exists C∈(0,∞)such

that

max

n∈{1,...,N}Eku(tθ

n)−Unk2

H+

n=1

Eknku(tθ

n)−Unkp

V≤C(kqγ+1 +kqα)

is fulfilled.

44 CHAPTER 2. RANDOMIZED SCHEMES

Proof. The proof can be carried out by applying Theorem 2.2.6. To this end, it only remains

to verify that A(t), t∈[0, T ], fulfills (2.23). The proof of the other conditions for A(t),

t∈[0, T ], has already been done for Theorem 2.3.4. To this end, we use the bounded

Lipschitz condition from Assumption 2.3.3. For p= 2, (2.34) is a global Lipschitz condition.

We obtain that

hA(t)v1−A(t)v2, wiV∗×V=ZDa(t, ·,∇v1)−a(t, ·,∇v2)· ∇wdx

≤2d6ZD

|∇v1− ∇v2||∇w|dx

≤2d6k∇v1− ∇v2kL2(D)dk∇wkL2(D)d= 2d6kv1−v2kVkwkV

for every v1, v2, w ∈Vand t∈[0, T ]. This proves the global Lipschitz condition

kA(t)v1−A(t)v2kV∗≤2d6kv1−v2kV

for every v1, v2∈Vand t∈[0, T ]. In the case p∈(2,∞), we can argue in a similar fashion

but have to handle the Lipschitz constant that depends on the input v1, v2∈V. Here, we

use Lemma A.1.3 to obtain that

hA(t)v1−A(t)v2, wiV∗×V

=ZDa(t, ·,∇v1)−a(t, ·,∇v2)· ∇wdx

≤d6ZD

|∇v1− ∇v2||∇w|dx+d6ZD

max |∇v1|p−2,|∇v2|p−2|∇v1− ∇v2||∇w|dx

≤d6k1kL

p−2(D)k∇v1− ∇v2kLp(D)dk∇wkLp(D)d

+d6ZD

max |∇v1|p,|∇v2|pdxp−2

pk∇v1− ∇v2kLp(D)dk∇wkLp(D)d

for every v1, v2, w ∈Vand t∈[0, T ]. Since p−2

p∈(0,1) for p∈(2,∞) we get that

ZD

max |∇v1|p,|∇v2|pdxp−2

p≤ZD

|∇v1|pdx+ZD

|∇v2|pdxp−2

≤ZD

|∇v1|pdxp−2

p+ZD

|∇v2|pdxp−2

=k∇v1kp−2

Lp(D)d+k∇v2kp−2

Lp(D)d

≤2 max k∇v1kp−2

Lp(D)d,k∇v2kp−2

Lp(D)d.

Thus, for R∈(0,∞) and all v1, v2∈Vwith kv1kV,kv2kV≤R, we obtain the bound

kA(t)v1−A(t)v2kV∗≤d6k1kL

p−2(D)+ 2 max kv1kp−2

V,kv2kp−2

Vkv1−v2kV

=: L(R)kv1−v2kV,

which proves the weaker form of (2.23) that is needed if u∈L∞(0, T ;V).

Theorem 2.3.6. Let Assumptions 2.3.1, 2.3.2, and 2.3.3 be fulfilled and let f∈Lq(0, T ;V∗)

and u0∈Hbe given. For α∈(0,1) and γ∈(0,1), assume that the exact solution uis

an element of Wα,2(0, T ;V)for p= 2 and of L∞(0, T;V)∩Wα,q(0, T ;V)for p∈(2,∞)

2.3. EXAMPLE: A PROBLEM OF P-LAPLACIAN TYPE 45

while its temporal derivative u0belongs to Wγ,q(0, T ;V∗). For every N∈N,k=T

N, and

the corresponding random values (ξn)n∈{1,...,N}introduced in Assumption 2.2.1, the scheme

knUn−Un−1+A(ξn)Un=f(ξn)in Lq(Ω; V∗), n ∈ {1, . . . , N},

U0=u0in H

admits a unique solution (Un)n∈{1,...,N}in Lp(Ω; V). Then there exists C∈(0,∞)such

that

max

n∈{1,...,N}Eku(tθ

n)−Unk2

H+

n=1

Eknku(tθ

n)−Unkp

V≤C(kqγ +kqα)

is fulfilled.

Proof. The proof can be done analogously to the proof of Theorem 2.3.6, where we now use

Theorem 2.2.7.

46 CHAPTER 2. RANDOMIZED SCHEMES

Chapter 3

An Operator Splitting Based Scheme

for Nonlinear, Nonautonomous

Evolution Equations

An operator splitting offers the opportunity to obtain easier solvable subproblems, which in

some settings can even be solved in parallel. Due to modern hardware structures, methods

that are based on parallelization become more and more useful for a faster computation.

To this end, we will present a numerical scheme based on an operator splitting in order to

discretize a nonlinear, nonautonomous evolution equation on a finite time horizon. Precisely,

for T∈(0,∞), we consider

(u0(t) + A(t)u(t) + B(t)u(t) = f(t) in V∗,for almost all t∈(0, T ),

u(0) = u0in H(3.1)

for a Gelfand triple Vd

,→H∼

=H∗d

,→V∗as well as families {A(t)}t∈[0,T ]and {B(t)}t∈[0,T ]

of operators A(t): V→V∗and B(t): H→H. Here, A(t), t∈[0, T ], is an operator

of monotone type and B(t), t∈[0, T ], is Lipschitz continuous. Further, we allow for an

integrable source term f: [0, T ]→V∗and an initial value u0∈H. Standard examples for

our problem class contain p-Laplacian type and porous media type problems with lower order

perturbations. As the solutions of such nonlinear equations usually lack global higher-order

temporal and spatial regularity, we concentrate on a lower-order scheme.

As in the previous chapter, our starting point is to look at the well-known backward

Euler scheme. For N∈N, we consider an equidistant grid 0 = t0< t1<· · · < tN=T

for points tn=nk,n∈ {0, . . . , N}, as well as a step size k=T

N. Then the solution to the

recursion

Un−Un−1

k+AnUn+BnUn=fnin V∗, n ∈ {1, . . . , N},

with U0=uk

0in Hcan be used to obtain an approximation Un≈u(tn), n∈ {1, . . . , N}.

Here, uk

0, (fn)n∈{1,...,N}, (An)n∈{1,...,N}, as well as (Bn)n∈{1,...,N}are approximations of

the data that we assume to be known. This scheme is formally of first order and we want

to present modifications that preserve the order but lead to several subproblems, which

can potentially be solved more efficiently. For the first modification, we notice that from

a numerical point of view, on first sight, it could seem like a good option to exchange

AnUnand BnUnby AnUn−1and BnUn−1, respectively. This leads to a similar scheme

48 CHAPTER 3. OPERATOR SPLITTING

as the forward Euler method. In terms of an implementable full discretization, it has the

advantage that no nonlinear, implicit equation has to be solved. But note that if an operator

An:V→V∗is interpreted as an Hvalued operator in terms of the restriction An

Fgiven

by An

F: dom(An

F)⊂H→H,An

Fv=Anvwith dom(An

F) = {v∈V:An

Fv∈H}, it

can be unbounded. Thus, in a fully discretized scheme, it can become necessary to choose

a temporal discretization parameter that depends on the spatial discretization parameter.

This coupling is highly undesirable in applications. Due to the monotonicity of the operator

A(t), t∈[0, T ], the backward Euler method has better stability properties as the underlying

physical system is dissipative. Here, no coupling of the discretization parameters appears.

Therefore, we do not exchange AnUnby AnUn−1. As the operator Bnis bounded in H,

these problems do not appear. Hence, we use BnUn−1and propose the implicit-explicit

scheme

Un−Un−1

k+AnUn+BnUn−1=fnin V∗, n ∈ {1, . . . , N},

with U0=uk

0in H. Altogether, this yields the equations

(I+kAn)Un=kfn+Un

0in V∗with Un

0= (I−kBn)Un−1in H

for n∈ {1, . . . , N}and U0=uk

0in H. Next, we decompose the operator A(t), t∈[0, T ], and

the source term f. To this end, we assume that there exist M∈Nfamilies {Am(t)}t∈[0,T ]

of operators Am(t): Vm→V∗

mand functions fm: [0, T ]→V∗

mfor m∈ {1, . . . , M}, where

Vm⊂Hand TM

m=1 Vm=V. These operators and functions have to fulfill the sum property

m=1

Am(t)v=A(t)v,

m=1

fm(t) = f(t) in V∗

for every v∈Vand almost every t∈(0, T ). For m∈ {1, . . . , M}and approximations

(fn

m)n∈{1,...,N}, (An

m)n∈{1,...,N}, which also fulfill a corresponding sum property, we use a

product splitting scheme to approximate a backward Euler step containing Anand fn. The

idea of such a scheme is that for real numbers am,m∈ {1, . . . , M}, some basic calculations

show that

1 + k

m=1

am−1−

m=1 1 + kam−1

=k21 + k

m=1

am−1M

j,m=1,

j<m

ajamM

m=1 1 + kam−1

is fulfilled. This suggests that the splitting error, i.e., the difference of one Euler step

I+kAn−1=I+kPM

m=1 An

m−1and the product QM

m=1 I+kAn

m−1, is sufficiently

small. Altogether, this gives rise to consider the following system of equations

0= (I−kBn)Un−1in H

and

(I+kAn

m)Un

m=kfn

m+Un

m−1in V∗

m, m ∈ {1, . . . , M},

for n∈ {1, . . . , N}with

Un=Un

Min H, n ∈ {1, . . . , N},and U0=uk

0in H.

This structure has some similarities to a Runge–Kutta method. In such a scheme, for one

temporal step, the data can be evaluated at different points to receive several explicit or

implicit equations. A linear combination of the solutions of them is used to obtain Un. In

contrast to this, we decompose the data into different parts. The decomposed data is used

to receive several equations that are used to obtain Un.

Under no additional regularity assumptions on the solution, we can prove that se-

quences of piecewise polynomial prolongations of the values (Un)n∈{1,...,N}converge point-

wise strongly to the solution in H. Depending on the monotonicity condition, we can also

show that the sequence of piecewise constant prolongations converges weakly or even strongly

to the solution in Lp(0, T ;VM), where pdepends on the operator A(t), t∈[0, T ].

Under the assumption that the exact solution is more regular and the operators Am(t),

t∈[0, T ] fulfill a bounded Lipschitz condition and a stronger monotonicity condition for

m∈ {1, . . . , M}, we even obtain explicit error estimates. Precisely, there exists C∈(0,∞),

which depends on u, such that

max

n∈{1,...,N}ku(tn)−Unk2

H+k

n=1

ku(tn)−Unkp

VM≤Ck p

p−1α

for all N∈Nand k=T

N, where α∈(0,1] is the exponent of the H¨older space that contains

the solution uand pdepends on A(t), t∈[0, T ]. In particular, we see that the order of

the error bounds can be the same as the convergence rate of the classical backward Euler

scheme for suitable data.

Splitting schemes offer a useful tool in decreasing the computational costs of algorithms.

A general introduction can be found in [70]. A well-known field of applications to operator

splittings is given by evolution equations with different structures. In order to name a few

examples, reaction-diffusion equations have been studied in terms of an operator splitting

in [12, 64, 74], the Riccati differential equation in [65, 108], and the Navier–Stokes problem

in [107]. Another useful way of splitting operators is a dimension splitting, where each

Am(t), t∈[0, T ] and m∈ {1, . . . , M}, contains different partial derivatives, see [62, 106].

A modern alternative to dimension splitting is given by domain decomposition schemes, see

[13, 34, 61, 91, 113]. This approach is even more suitable for a parallel implementation as

the communication between subproblems is smaller. Moreover, in contrast to a dimension

splitting, non-Cartesian spatial domain can be considered in a domain decomposition based

scheme.

The analysis of splitting schemes for evolution equations has mostly been done in a

semigroup framework. General results as presented in [20, 23] can be used to prove the con-

vergence of several schemes. In [34, 61, 64, 66], a convergence analysis with these techniques

for the product splitting, sum splitting, Douglas–Rachford scheme, Peaceman–Rachford

scheme, Crank–Nicolson scheme, and implicit-explicit splitting can be found. Thus, this ap-

proach can be used for many examples. When it comes to a setting, where we want to allow

for a temporal dependence of the operator or a time-dependent source term, the results from

[20, 23] cannot be applied directly anymore. In the spirit of the work of [35, 106], we want

to prove the convergence of a splitting scheme in a variational framework. This enables us

to also look at a nonautonomous problem.

The structure of this work is comparable to [35], where a sum splitting scheme was

analyzed. The product splitting scheme in a variational approach has been considered in

50 CHAPTER 3. OPERATOR SPLITTING

[106]. Both the sum and the product splitting in a mild framework have been analyzed [34].

The main advantage of this work is the additional, possibly non-monotone, operator B(t), t∈

[0, T ], which is handled in terms of a forward Euler step. Similar implicit-explicit splittings

can be found in [2, 5, 17, 24, 64]. In a similar fashion to [35], we prove in Theorem 3.1.18

and Theorem 3.1.19 that sequences of piecewise polynomial prolongations of (Un)n∈{1,...,N}

converge to the solution of the evolution equation (3.1).

In Theorem 3.2.3, we provide error estimates for the method. While there are known

results for this in the semigroup setting, see [61, 64], this is still new within a variational

approach to splitting schemes.

These results are well applicable to dimension splitting and domain decomposition meth-

ods if B≡0. For a nontrivial operator B(t), t∈[0, T ], a compact embedding result remains

to be proved within this context. In the last section of this chapter, we provide an example

where the necessary compact embedding can easily be obtained.

In the first section of this chapter, we begin by introducing the exact assumptions needed

on the data and prove the convergence of the piecewise polynomial prolongations without

any further regularity assumptions made on the solution. In the second section, we show that

under an additional bounded Lipschitz condition and a stronger monotonicity assumption on

the operators Am(t), t∈[0, T ] and m∈ {1, . . . , M}, we can even prove explicit error bounds.

The size of the error depends on an additional regularity assumption on the solution. At

the end of the chapter, we show that the theoretical results from the first two sections can

be applied to a nonlinear parabolic problem.

3.1 Convergence of the Splitting Scheme

In this first section, we focus on proving the convergence of the implicit-explicit product

splitting in a general framework. To this end, we begin to state the exact assumptions that

have to be made on the data. This in mind, we can introduce our scheme and prove that it

is has a unique solution. The solution also fulfills a priori bounds. Using these bounds, we

can argue that the piecewise constant and piecewise linear prolongations of the solution to

the semidiscrete problem are bounded in suitable spaces. Therefore, we can extract weakly

or weakly∗converging subsequences. It remains to identify the limit with the equation,

where among other things the Minty monotonicity trick will be used. The following setting

is similar to both [35] and [106]. Let us begin by introducing the structure of the spaces

which will be used in the following.

Assumption 3.1.1. Let (H, (·,·)H,k·kH)be a real, separable Hilbert space and (V, k·kV)be

a real, separable, reflexive Banach space, which is continuously and densely embedded into H.

Further, there exist a seminorm |·|Von Vand cV∈(0,∞)such that k·kV≤cVk·kH+|·|V

is fulfilled.

For M∈Nand m∈ {1, . . . , M}, let (Vm,k · kVm)be real reflexive Banach spaces, which

are continuously embedded into H, such that TM

m=1 Vm=Vand PM

m=1 k·kVmis equivalent

to k·kV. For every m∈ {1, . . . , M}, there exist a seminorm |·|Vmon Vmand cVm∈(0,∞)

such that k·kVm≤cVmk·kH+|·|Vmand PM

m=1 |·|Vmis equivalent to |·|V.

Note that asking for the existence of the seminorms in the previous assumption is no

additional restriction on the spaces. As Vand Vm,m∈ {1, . . . , M}, are continuously

embedded into Hit is possible to use the full norm as the seminorm. If we consider, for

example, H=L2(D) and V=W1,p(D) for p∈[1,∞) on a bounded domain D ⊂ Rd,d∈N,

it is possible to use the seminorm |v|V=RD|∇v|pdx

p. In this case, the seminorm is

3.1. CONVERGENCE OF THE SPLITTING SCHEME 51

not a full norm. This setting is closely related to [99, Chapter 8] and allows for a different

coercivity condition for the operators defined below.

These spaces in mind, we can identify Hwith its dual space H∗and obtain the Gelfand

triples

,→H∼

=H∗d

,→V∗and Vm

,→H∼

=H∗d

,→V∗

m, m ∈ {1, . . . , M}.

Note that for every m∈ {1, . . . , M},Vmis densely embedded into Hbecause Vm⊇Vand

Vis densely embedded into H. The operator A(t), t∈[0, T ], acts on the spaces defined

above and fulfills the next assumption.

Assumption 3.1.2. Let the spaces Hand Vbe given as stated in Assumption 3.1.1. Fur-

thermore, for T∈(0,∞)as well as p∈[2,∞), let {A(t)}t∈[0,T ]be a family of operators

A(t): V→V∗that satisfy the following conditions:

(1) The mapping Av : [0, T ]→V∗given by t7→ A(t)vis continuous almost everywhere in

(0, T )for all v∈V.

(2) The operator A(t): V→V∗,t∈[0, T ], is radially continuous, i.e., the mapping

s7→ hA(t)(v+sw), wiV∗×Vis continuous on [0,1] for all v, w ∈V.

(3) The operator A(t): V→V∗,t∈[0, T ], is monotone, i.e.,

hA(t)v−A(t)w, v −wiV∗×V≥0

is fulfilled for all v, w ∈V.

(4) The operator A(t): V→V∗,t∈[0, T ], is uniformly bounded such that there exists

β∈[0,∞), which does not depend on t, with

kA(t)vkV∗≤β1 + kvkp−1

V

for all v∈V.

(5) The operator A(t): V→V∗,t∈[0, T ], fulfills a uniform semi-coercivity condition

such that there exist µ∈(0,∞)and λ∈[0,∞), which do not depend on t, with

hA(t)v, viV∗×V+λ≥µ|v|p

for all v∈V.

The assumption p∈[2,∞) can be weakened to p∈(1,∞) for this section, compare

[40, 41, 42] for more details. It will then be necessary to choose some of the appearing

function spaces differently to ensure that the spaces are embedded into each other. In

Section 3.2, we cannot directly consider the case p∈(1,2). Here, we make a stronger

monotonicity condition, compare (3.48) below. There exists no operator that fulfills this

condition for p∈(1,2). Altogether, for simplicity and to keep the assumptions consistent,

we concentrate on the case p∈[2,∞) throughout the entire chapter.

For the existence of a solution, Assumption 3.1.2 (1) can be generalized to assuming that

the mapping Av : [0, T ]→V∗given by t7→ A(t)vis Bochner measurable for every v∈V,

compare [118, Chapter 30]. We use a stronger condition to keep the proof of Lemma 3.1.15

below more simple. Furthermore, we use a semi-coercivity condition instead of a standard

52 CHAPTER 3. OPERATOR SPLITTING

coercivity assumption, which contains the full norm. A similar condition was imposed in

[99, Chapter 8]. As pointed out in [41] such a condition implies, in particular, that

hA(t)v, viV∗×V+ ˜νkvk2

H+˜

λ≥˜µkvk2

is fulfilled. This implies that A(t), t∈[0, T ], fulfills an ordinary coercivity condition but

with different exponents. See (1.1) for the exact definition of the coefficients. We will use

the semi-coercivity condition as this enables us to prove certain bounds in Lp(0, T ;V) while

the coercivity condition above only allows for bounds in L2(0, T ;V).

In the following, we want to decompose the operator A(t), t∈[0, T ], in Moperators

that all have the same structure and act on the spaces Vm,m∈ {1, . . . , M}, from Assump-

tion 3.1.1.

Assumption 3.1.3. For M∈N, let H,Vand Vm,m∈ {1, . . . , M}, be as stated in

Assumption 3.1.1. For T∈(0,∞), let {A(t)}t∈[0,T ]be a family of operators A(t): V→V∗

as given in Assumption 3.1.2. For m∈ {1, . . . , M}, let Am(t): Vm→V∗

m,t∈[0, T ], also

fulfill Assumption 3.1.2, with Vreplaced by Vmsuch that the sum property

m=1

Am(t)v=A(t)vin V∗

is satisfied for all t∈[0, T ]and v∈V.

Remark 3.1.4. Note that the optimal coefficients β, λ, µ for the operators A(t) and Am(t),

t∈[0, T ] and m∈ {1, . . . , M}, do not necessarily have to coincide. For the sake of simplicity,

we assume that the set of coefficients is the same for all appearing operators.

A comparable setting can be found in [83, Chapitre 2, Section 1.7]. Here, the operators

Am(t), t∈[0, T ] and m∈ {1, . . . , M}, fulfill a similar assumption as Assumption 3.1.2. But

pcan be a different value pmfor each operator. This could be an interesting generalization

for our proposed operator splitting.

Additionally, we introduce a Lipschitz continuous operator B(t), t∈[0, T ], stated below.

Assumption 3.1.5. Let Hbe given as stated in Assumption 3.1.1. Furthermore, for T∈

(0,∞), let {B(t)}t∈[0,T ]be a family of operators B(t): H→Hthat satisfy the following

conditions:

(1) The mapping Bv : [0, T ]→Hgiven by t7→ B(t)vis continuous almost everywhere in

(0, T )for every v∈H.

(2) The operator B(t): H→H,t∈[0, T ], fulfills a uniform Lipschitz condition such that

there exists κ∈[0,∞), which does not depend on t, with

kB(t)v−B(t)wkH≤κkv−wkH

for all v, w ∈H.

(3) The operator B(t): H→H,t∈[0, T ], is uniformly bounded in 0∈Hsuch that there

exists ρ∈[0,∞)with kB(t)0kH≤ρfor all t∈[0, T ].

Moreover, if κis strictly larger than zero, then let the space VMfrom Assumption 3.1.1 be

compactly embedded into H.

3.1. CONVERGENCE OF THE SPLITTING SCHEME 53

Remark 3.1.6. Observe that every operator B(t), t∈[0, T], which fulfills Assumption 3.1.5,

also fulfills

kB(t)vkH≤κ(1 + kvkH),(B(t)v, v)H≤κ(1 + kvk2

for all v∈Hand t∈[0, T ] after possibly enlarging κ.

We also want to consider the operators of Assumptions 3.1.2, 3.1.3, and 3.1.5 as operators

acting on Bochner spaces. To this end, we consider their Nemytski˘ı operators and state some

useful properties in the lemma below. A proof can be found in [39, Lemma 8.4.4] or [118,

Section 30].

Lemma 3.1.7. Let the spaces Vand Hbe given as in Assumption 3.1.1. For T∈(0,∞)and

p∈[2,∞), let A(t): V→V∗be an operator as stated in Assumption 3.1.2 and B(t): H→H

as in Assumption 3.1.5 for t∈[0, T ]. Then for q=p

p−1the operators (Av)(t) = A(t)v(t)

and (Bv)(t) = B(t)v(t)map Lp(0, T ;V)into Lq(0, T;V∗)and L2(0, T;H)into L2(0, T ;H),

respectively.

The operator A:Lp(0, T ;V)→Lq(0, T ;V∗)is radially continuous, i.e., the mapping

s7→ hA(v+sw), wiLq(0,T ;V∗)×Lp(0,T ;V)is continuous on [0,1] for all v, w ∈Lp(0, T ;V).

Furthermore, Afulfills a monotonicity, a boundedness, and a coercivity condition such that

it holds true that

hAv −Aw, v −wiLq(0,T ;V∗)×Lp(0,T ;V)≥0,

kAvkLq(0,T ;V∗)≤βT1

q+kvkp−1

Lp(0,T ;V),

hAv, viLq(0,T ;V∗)×Lp(0,T ;V)+µkvkp

Lp(0,T ;H)+λT ≥21−pµc−p

Vkvkp

Lp(0,T ;V)

for all v, w ∈Lp(0, T ;V). The operator B:L2(0, T ;H)→L2(0, T;H)is Lipschitz continu-

ous and bounded at 0∈L2(0, T ;H)such that

kBv −BwkL2(0,T ;H)≤κkv−wkL2(0,T ;H)

kB0kL2(0,T ;H)≤T1

2ρ

is fulfilled for all v, w ∈L2(0, T ;H).

Note that for every m∈ {1, . . . , M}, the Nemytski˘ı operator of Am(t), t∈[0, T ], intro-

duced in Assumption 3.1.3 maps Lp(0, T ;Vm) into Lq(0, T ;V∗

m) and fulfills the same bounds

stated in the previous lemma with Vreplaced by Vm. It remains to state the assumptions

on the source term fand its decomposition.

Assumption 3.1.8. Let Vand Vm,m∈ {1, . . . , M}, be given as in Assumption 3.1.1 and

q=p

p−1, where p∈[2,∞)is the same as in Assumption 3.1.2. Let f∈Lq(0, T ;V∗)be

given and assume that there exist functions fm∈Lq(0, T ;V∗

m),m∈ {1, . . . , M}, such that

m=1

fm(t) = f(t)in V∗,kfm(t)kV∗

m≤ kf(t)kV∗,for almost all t∈(0, T ).

It is also possible to allow for a more general source term f∈Lq(0, T ;V∗)+L1(0, T;H),

compare [106] and [109, Chapter III, Section 1.5]. For simplicity, we only concentrate on

functions from Lq(0, T ;V∗). As discussed in Section 1.1, the evolution equation (3.1) is

uniquely solvable if Assumptions 3.1.2 and 3.1.5 are fulfilled, fis an element of Lq(0, T ;V∗),

and u0∈H.

54 CHAPTER 3. OPERATOR SPLITTING

In order to discretize the equation, we consider an equidistant grid on [0, T ], where

N∈N,k=T

N, and tn=nk,n∈ {0, . . . , N}. For m∈ {1, . . . , M}and n∈ {1, . . . , N}, we

introduce

mv=1

kZtn

tn−1

Am(t)vdtin V∗

m,Bnw=1

kZtn

tn−1

B(t)wdtin H(3.2)

for v∈Vmand w∈Has well as

m=1

kZtn

tn−1

fm(t) dtin V∗

m.(3.3)

We use these values to construct an approximation Un≈u(tn) of the solution uof the

evolution equation (3.1) at the grid points. To this end, we examine the semidiscrete problem

0−Un−1

k+BnUn−1= 0 in H, (3.4)

and

m−Un

m−1

k+An

mUn

m=fn

min V∗

m, m ∈ {1, . . . , M},(3.5)

for n∈ {1, . . . , N}with

Un=Un

Min H, n ∈ {1, . . . , N},and U0=uk

0in H. (3.6)

Depending on the statement, uk

0has to be in VMor H. In order to prove that the approxi-

mation converges to the solution, we require that uk

0→u0in Has k→0. For some results,

we further need that (k1

pkuk

0kVM)k>0is uniformly bounded with respect to k. In order to

see that such a sequence exists, let (ui

0)i∈Nbe a sequence in VMsuch that ui

0→u0in Has

i→ ∞. This sequence exists because VMis densely embedded into H. For the construction

of a sequence that fulfills this boundedness condition, we use a sequence (kj)j∈Nsuch that

kj→0 as j→ ∞. Then we set uk1

0=u1

0in VM. As k−1

j→ ∞ as j→ ∞ there exists j1∈N

such that ku2

0kVM≤k−1

j1. This in mind, we write uk1

0=· · · =ukj1−1

0and ukj1

0=u2

0in VM.

Analogously, there exists j2∈Nsuch that ku3

0kVM≤k−1

j2and we write ukj1

0=· · · =ukj2−1

and ukj2

0=u3

0in VM. Repeating this argument, we obtain an appropriate sequence to

approximate the initial value.

The following two lemmas show that the discrete values (An

m)n∈{1,...,N},m∈ {1, . . . , M},

and (Bn)n∈{1,...,N}fulfill the same properties as their underlying operators Am(t), m∈

{1, . . . , M}, and B(t) do for every t∈[0, T ].

Lemma 3.1.9. Let Assumptions 3.1.1, 3.1.2, and 3.1.3 be fulfilled. For n∈ {1, . . . , N}

and m∈ {1, . . . , M}, the operator An

m:Vm→V∗

mdefined in (3.2) is radially continuous,

i.e., the mapping s7→ hAn

m(v+sw), wiV∗

m×Vmis continuous on [0,1] for all v, w ∈Vm.

Furthermore, it fulfills a monotonicity, a boundedness, and a coercivity condition such that

hAn

mv−An

mw, v −wiV∗

m×Vm≥0,(3.7)

kAn

mvkV∗

m≤β1 + kvkp−1

Vm,(3.8)

hAn

mv, viV∗

m×Vm+λ≥µ|v|p

Vm(3.9)

are fulfilled for all v, w ∈Vm.

3.1. CONVERGENCE OF THE SPLITTING SCHEME 55

Proof. Let (si)i∈Nbe a sequence in [0,1] that converges to s∈[0,1]. Then we see that

hAm(t)(v+siw), wiV∗

m×Vm≤ kAm(t)(v+siw)kV∗

mkwkVm

≤β1 + kv+siwkp−1

VmkwkVm

≤β1+2p−2kvkp−1

Vm+ 2p−2kwkp−1

VmkwkVm.

for v, w ∈Vmdue to the boundedness condition of A(t), t∈[0, T ], from Assumption 3.1.2 (4).

We can then apply the radial continuity of A(t), t∈[0, T], from Assumption 3.1.2 (2) and

Lebesgue’s dominated convergence theorem to obtain

lim

i→∞hAn

m(v+siw), wiV∗

m×Vm= lim

i→∞

kZtn

tn−1

hAm(t)(v+siw), wiV∗

m×Vmdt

kZtn

tn−1

lim

i→∞hAm(t)(v+siw), wiV∗

m×Vmdt

kZtn

tn−1

hAm(t)(v+sw), wiV∗

m×Vmdt

=hAn

m(v+sw), wiV∗

m×Vm

for v, w ∈Vm. Therefore, An

mis radially continuous. In order to prove (3.7)–(3.9), we apply

Assumption 3.1.2 (3)–(5) and obtain the monotonicity condition

hAn

mv−An

mw, v −wiV∗

m×Vm=1

kZtn

tn−1

hAm(t)v−Am(t)w, v −wiV∗

m×Vmdt≥0,

the boundedness condition

kAn

mvkV∗

m≤1

kZtn

tn−1

kAm(t)vkV∗

mdt≤β1 + kvkp−1

Vm,

and the coercivity condition

hAn

mv, viV∗

m×Vm=1

kZtn

tn−1

hAm(t)v, viV∗

m×Vmdt≥1

kZtn

tn−1µ|v|p

Vm−λdt=µ|v|p

Vm−λ

for all v, w ∈Vm.

Lemma 3.1.10. Let Assumptions 3.1.1 and 3.1.5 be fulfilled. The operator Bn:H→H

defined in (3.2) fulfills

kBnv−BnwkH≤κkv−wkH,kBn0kH≤ρ, (3.10)

as well as

kBnvkH≤κ1 + kvkH,(Bnv, v)H≤κ1 + kvk2

H(3.11)

for all v, w ∈Hand n∈ {1, . . . , N}.

We omit the proof of this lemma. It can be done analogously to the proof of Lemma 3.1.9,

where we also use the bounds proposed in Remark 3.1.6. Now, we are well prepared to prove

that the operator splitting scheme (3.4)–(3.6) is uniquely solvable and its solution fulfills a

priori bounds.

56 CHAPTER 3. OPERATOR SPLITTING

Lemma 3.1.11. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled. For a

step size k=T

N,N∈N, and uk

0∈H, the semidiscrete problem (3.4)–(3.6) is uniquely

solvable.

Proof. Let n∈ {1, . . . , N}and m∈ {1, . . . , M}be fixed in the following. Assuming that

Un−1∈Hexists, then Un

0∈His given by the explicit equation

0=I−kBnUn−1in H.

If now Un

m−1∈Hexists, we want to show that there exists a unique element Un

m∈Vmthat

solves

I+kAn

mUn

m=kfn

m+Un

m−1in V∗

m.(3.12)

As proven in Lemma 3.1.9 the operator An

mis radially continuous. This implies that I+kAn

is radially continuous. Applying the monotonicity condition (3.7) from Lemma 3.1.9, it can

be seen that I+kAn

mis strictly monotone. Using the inequality kvkVm≤cVmkvkH+

|v|Vmfor the Vm-norm stated in Assumption 3.1.1 and the coercivity condition (3.9) from

Lemma 3.1.9, we obtain

h(I+kAn

m)v, viV∗

m×Vm

kvkVm

≥kvk2

H+µ|v|p

cVmkvkH+|v|Vm−λ

kvkVm

≥min{1, µ}

cVm

·kvk2

H+|v|p

kvkH+|v|Vm

−λ

kvkVm

→ ∞ as kvkVm→ ∞.

Hence, there exists a unique element Un

m∈Vmthat solves (3.12) due to the Browder–Minty

theorem, see [99, Theorem 2.14].

Lemma 3.1.12. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled and let

0∈Hbe given. Then there exists K∈(0,∞)such that for every step size k=T

N,N∈N,

the unique solution of (3.4)–(3.6) fulfills the a priori estimates

max

n∈{1,...,N}kUnk2

H+

n=1

m=1

kUn

m−Un

m−1k2

H≤K, (3.13)

max

n∈{1,...,N}kUn

0k2

H+

n=1

kUn

0−Un−1k2

H≤K, (3.14)

max

n∈{1,...,N}kUn

mk2

H+k

n=1

kUn

mkp

Vm≤K, m ∈ {1, . . . , M},(3.15)

as well as

k1−q

n=1

kUn−Un−1kq

V∗=k

n=1 



Un−Un−1

k



V∗

≤K. (3.16)

In order to prove the a priori bounds, we follow a similar structure as in [99, 106].

Since we only assumed that Am(t), t∈[0, T], fulfills a semi-coercivity condition for every

m∈ {1, . . . , M}a Gronwall-like-argument becomes necessary. In [99, Lemma 8.6], the

classical Gronwall lemma leads to a step size restriction, which is fdepended. For some

appearing terms, we avoid the classical Gronwall argument and use Lemma A.1.2 instead.

The main advantage of this argumentation is that we do not have a restriction for the step

size k.

3.1. CONVERGENCE OF THE SPLITTING SCHEME 57

Proof of Lemma 3.1.12. Let i∈ {1, . . . , N}and m∈ {1, . . . , M}be fixed in the following.

We test (3.4) with Ui

0∈Hand use the identity from Lemma A.1.4 to obtain that

2kUi

0k2

H− kUi−1k2

H+kUi

0−Ui−1k2

H=−k(BiUi−1,Ui

0)H.(3.17)

An application of the conditions in (3.11) yields

−(BiUi−1,Ui

0)H=−(BiUi−1,Ui−1)H−(BiUi−1,Ui

0−Ui−1)H

≤κ1 + kUi−1k2

H+kkBiUi−1k2

H+1

4kkUi

0−Ui−1k2

≤κ1 + kUi−1k2

H+kκ21 + kUi−1kH2+1

4kkUi

0−Ui−1k2

≤c11 + kUi−1k2

H+1

4kkUi

0−Ui−1k2

for c1=κ+ 2κ2T. We insert this bound into (3.17) to obtain

kUi

0k2

H− kUi−1k2

H+1

2kUi

0−Ui−1k2

H≤2kc11 + kUi−1k2

H.(3.18)

Similarly, we test (3.5) with Ui

m∈Vmand again use the identity from Lemma A.1.4 to find

that

2kUi

mk2

H− kUi

m−1k2

H+kUi

m−Ui

m−1k2

H+khAi

mUi

m,Ui

miV∗

m×Vm

=khfi

m,Ui

miV∗

m×Vm≤kkfi

mkV∗

mkUi

mkVm.

(3.19)

We then multiply the inequality by two, insert the coercivity condition (3.9) stated in

Lemma 3.1.9, and the inequality kvkVm≤cVmkvkH+|v|Vmfor the Vm-norm from As-

sumption 3.1.1 as well as Young’s inequality to obtain that

kUi

mk2

H− kUi

m−1k2

H+kUi

m−Ui

m−1k2

H+ 2kµ|Ui

m|p

≤2kcVmkfi

mkV∗

mkUi

mkH+|Ui

m|Vm+ 2kλ

≤2kcVmkfi

mkV∗

mkUi

mkH+kc2kfi

mkq

V∗

m+kµ|Ui

m|p

Vm+ 2kλ,

with c2=(2cVm)q(pµ)1−q

q. After absorbing the summand containing the Vm-seminorm, it

follows that

kUi

mk2

H− kUi

m−1k2

H+kUi

m−Ui

m−1k2

H+kµ|Ui

m|p

≤2kcVmkfi

mkV∗

mkUi

mkH+kc2kfi

mkq

V∗

m+ 2kλ.

We sum up the inequality from m= 1 to M, add (3.18), and insert Ui

M=Uiin Hto see

that

kUik2

H− kUi−1k2

H+

m=1

kUi

m−Ui

m−1k2

H+1

2kUi

0−Ui−1k2

H+kµ

m=1

|Ui

m|p

≤k

m=1 2cVmkfi

mkV∗

mkUi

mkH+c2kfi

mkq

V∗

m+ 2kλM + 2kc11 + kUi−1k2

H.

58 CHAPTER 3. OPERATOR SPLITTING

Summing up the previous inequality from i= 1 to n∈ {1, . . . , N}, shows that

kUnk2

H− kU0k2

H+

i=1

m=1

kUi

m−Ui

m−1k2

H+1

i=1

kUi

0−Ui−1k2

H+kµ

i=1

m=1

|Ui

m|p

≤k

i=1

m=1 2cVmkfi

mkV∗

mkUi

mkH+c2kfi

mkq

V∗

m+ 2TλM + 2kc1

n−1

i=0 1 + kUik2

H.

(3.20)

The sums containing fi

mcan be bounded using Assumption 3.1.8 as well as H¨older’s inequal-

ity. Then we see that

i=1

m=1

kfi

mkq

V∗

m=k

i=1

m=1 



kZti

ti−1

fm(t) dt



V∗

≤

i=1

m=1 Zti

ti−1

kfm(t)kq

V∗

mdt≤Mkfkq

Lq(0,T ;V∗)

(3.21)

and

kkfi

mkV∗

m≤k



kZti

ti−1

fm(t) dt



V∗

≤Zti

ti−1

kf(t)kV∗dt. (3.22)

Inserting these inequalities and U0=uk

0in Hto (3.20), we obtain that

kUnk2

H+

i=1

m=1

kUi

m−Ui

m−1k2

H+1

i=1

kUi

0−Ui−1k2

H+kµ

i=1

m=1

|Ui

m|p

≤ kuk

0k2

H+k

i=1

m=1 2cVmkfi

mkV∗

mkUi

mkH+c2kfi

mkq

V∗

m+ 2TλM + 2kc1

n−1

i=0 1 + kUik2

H

≤1+2T c1kuk

0k2

H+ 2c3

i=1 Zti

ti−1

kf(t)kV∗dt

m=1

kUi

mkH+c2Mkfkq

Lq(0,T ;V∗)

+ 2T(λM +c1)+2kc1

n−1

i=1

kUik2

≤1+2T c1kuk

0k2

H+ 2c3kfkL1(0,T ;V∗)max

i∈{1,...,N}

m=1

kUi

mkH+c2Mkfkq

Lq(0,T ;V∗)

+ 2T(λM +c1)+2kc1

n−1

i=1

kUik2

where c3= maxm∈{1,...,M}cVm. This can now be estimated using Lemma A.1.1 such that

we arrive at

kUnk2

H+

i=1

m=1

kUi

m−Ui

m−1k2

H+1

i=1

kUi

0−Ui−1k2

H+kµ

i=1

m=1

|Ui

m|p

≤1+2T c1kuk

0k2

H+ 2c3kfkL1(0,T ;V∗)max

i∈{1,...,N}

m=1

kUi

mkH+c2Mkfkq

Lq(0,T ;V∗)

+ 2T(λM +c1)exp(2T c1).

3.1. CONVERGENCE OF THE SPLITTING SCHEME 59

As the right-hand side is independent of n, the inequality is also fulfilled if we take the

maximum over all n∈ {1, . . . , N}on the left-hand side. A telescopic sum argument and

Ui=Ui

Min Himply that

kUi

mk2

H=



Ui−

j=m+1 Ui

j−Ui

j−1



H≤MkUik2

H+

j=m+1

kUi

j−Ui

j−1k2

H

≤MkUik2

H+

j=1

kUi

j−Ui

j−1k2

H

(3.23)

and therefore

m=1

kUi

mkH≤M3

2kUik2

H+

j=1

kUi

j−Ui

j−1k2

H

We then abbreviate the terms

x2= max

n∈{1,...,N}kUnk2

H+

i=1

m=1

kUi

m−Ui

m−1k2

H+1

i=1

kUi

0−Ui−1k2

+kµ

i=1

m=1

|Ui

m|p

Vm,

a=c3M3

2kfkL1(0,T ;V∗)exp(2T c1),

b2=(1 + 2T c1)kuk

0k2

H+c2Mkfkq

Lq(0,T ;V∗)+ 2T(λM +c1)exp(2T c1)

to obtain x2≤2ax +b2. An application of Lemma A.1.2 then yields the bound x≤2a+b.

This means that there exists K1∈(0,∞), which does not depend on the step size, such that

kUnk2

H+

i=1

m=1

kUi

m−Ui

m−1k2

H+1

i=1

kUi

0−Ui−1k2

H+kµ

i=1

m=1

|Ui

m|p

Vm≤K1

(3.24)

for every n∈ {1, . . . , N}. Using (3.18) and (3.23), it follows that

kUn

0k2

H≤2Tc1+1+2Tc1K1,kUn

mk2

H≤MK1(3.25)

for every m∈ {1, . . . , M}.

In order to prove a bound for (Un

m)n∈{1,...,N}in Vmfor every m∈ {1, . . . , M}, we use

the previous estimate as well as the inequality kvkVm≤cVmkvkH+|v|Vmfor the Vm-norm

from Assumption 3.1.1. Then we obtain that

k

i=1

m=1

kUi

mkp

Vm

p≤k

i=1

m=1

VmkUi

mkH+|Ui

m|Vmp

≤c3k

i=1

m=1

kUi

mkp

H

p+c3k

i=1

m=1

|Ui

m|p

Vm

≤c3T1

pM1

2+1

1+c3K1

µ

p=: K2.

(3.26)

60 CHAPTER 3. OPERATOR SPLITTING

In order to prove the a priori bound (3.16), we rewrite the difference Un−Un−1using (3.5)

as well as (3.4) and insert Un

M=Unin Hto obtain

Un−Un−1=

m=1 Un

m−Un

m−1+Un

0−Un−1=k

m=1 fn

m−An

mUn

m−kBnUn−1

in V∗for n∈ {1, . . . , N}. Testing the equation with v∈V, shows that

(Un−Un−1, v)H

m=1

hfn

m−An

mUn

m, viV∗

m×Vm−k(BnUn−1, v)H

≤kc4kvkV

m=1 kfn

mkV∗

m+kAn

mUn

mkV∗

m+kc4kvkVkBnUn−1kH

≤kc4kvkV

m=1 kfn

mkV∗

m+β1 + kUn

mkp−1

Vm+kc4κkvkV1 + kUn−1kH,

where c4∈(0,∞) is the maximal embedding constant from the embeddings of Vinto V∗

m∈ {1, . . . , M}, and Hinto V∗. This implies



Un−Un−1

k



V∗

≤c4

m=1 kfn

mkV∗

m+β1 + kUn

mkp−1

Vm+c4κ1 + kUn−1kH.

Taking the q-power, summing up from n= 1 to N, multiplying by the step size kand again

taking the 1

q-power, it follows that

k

n=1 



Un−Un−1

k



V∗

≤c4k

n=1 M

m=1 kfn

mkV∗

m+β1 + kUn

mkp−1

Vm+c4κ1 + kUn−1kHq

≤c4k

n=1

m=1

kfn

mkq

V∗

m

q+c4k

n=1

m=1

βq

q+c4k

n=1

m=1

βqkUn

mkp

Vm

+c4k

n=1

m=1

κq

q+c4k

n=1

m=1

κqkUn−1kq

H

≤c4M1

qkfkLq(0,T ;V∗)+c4(β+κ)(T M)1

q+c4βK

2+c4κK

1(TM)1

where we used (3.21), (3.24), and (3.26). A combination of (3.24), (3.25), (3.26), and the

previous inequality shows the desired bounds.

For the time discrete solutions (Un)n∈{1,...,N}and (Un

m)n∈{1,...,N},m∈ {1, . . . , M},

to (3.4)–(3.6) corresponding to the grid 0 = t0< t1<· · · < tN=Twith k=T

Nand

tn=nk,n∈ {0, . . . , N}, we construct piecewise polynomial prolongations defined on the

entire interval [0, T]. To this end, we introduce the piecewise constant prolongations for

t∈(tn−1, tn], n∈ {1, . . . , N},

m(t) = An

m, Bk(t) = Bn, fk

m(t) = fn

m(3.27)

3.1. CONVERGENCE OF THE SPLITTING SCHEME 61

with Ak

m(0) = A1

m,Bk(0) = B1, and fk

m(0) = f1

mfor m∈ {1, . . . , M}and

m(t) = Un

m,¯

Uk(t) = Un,Uk(t) = Un−1(3.28)

for m∈ {0, . . . , M}as well as the piecewise affine-linear function

Uk(t) = Un−1+t−tn−1

k(Un−Un−1),(3.29)

with

m(0) = ¯

Uk(0) = Uk(0) = Uk(0) = uk

0in H, m ∈ {0, . . . , M}.(3.30)

In the following, we always consider step sizes k=T

N`, where (N`)`∈Nis a sequence of natural

numbers with N`→ ∞ as `→ ∞. For simplicity, a sequence ( ¯

N`)`∈Nis abbreviated by

(¯

Uk)k>0and analogously for the other functions introduced above.

The function Ukis weakly differentiable. Note that its weak derivative coincides with

the classical derivative at the points where the latter exists. Using (3.4) and (3.5) as well as

Un=Un

Min H, its weak derivative can be rewritten as

(Uk)0(t) = 1

kUn−Un−1=1

m=1 Un

m−Un

m−1+1

kUn

0−Un−1

m=1 fn

m−An

mUn

m−BnUn−1in V∗

for t∈(tn−1, tn), n∈ {1, . . . , N}. Therefore, we see that

((Uk)0+PM

m=1 Ak

m¯

m+BkUk=PM

m=1 fk

min Lq(0, T ;V∗),

Uk(0) = uk

0in H. (3.31)

For every m∈ {1, . . . , M}, the operator Ak

mmaps Lp(0, T ;Vm) into Lq(0, T ;V∗

m), compare

Lemma 3.1.7. Together with the a priori bounds from Lemma 3.1.12 this shows that (3.31)

is well-defined. For the following calculations, it will be helpful to have an integrated version

of (3.31).

Lemma 3.1.13. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled. For

N∈N,k=T

N, and grid points tn=nk,n∈ {0, . . . , N}, as well as uk

0∈H, let the

piecewise constant and piecewise linear prolongations be given as in (3.27)–(3.29). Then

2k¯

Uk(tn)k2

H−1

2k¯

Uk(0)k2

H+

m=1 Ztn

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt

+Ztn

(Bk(t)Uk(t),¯

0(t))Hdt≤

m=1 Ztn

hfk

m(t),¯

m(t)iV∗

m×Vmdt

is fulfilled for every n∈ {1, . . . , N}.

Proof. In order to prove the inequality, we test (3.4) with Ui

0∈Hto get

k(Ui

0−Ui−1,Ui

0)H+ (BiUi−1,Ui

0)H= 0,(3.32)

62 CHAPTER 3. OPERATOR SPLITTING

and (3.5) with Ui

m∈Vmto obtain

k(Ui

m−Ui

m−1,Ui

m)H+hAi

mUi

m,Ui

miV∗

m×Vm=hfi

m,Ui

miV∗

m×Vm(3.33)

for i∈ {1, . . . , N}and m∈ {1, . . . , M}. Summing up (3.33) from m= 1 to Mand adding

(3.32), yields

m=1

(Ui

m−Ui

m−1,Ui

m)H+1

k(Ui

0−Ui−1,Ui

0)H

m=1

hAi

mUi

m,Ui

miV∗

m×Vm+ (BiUi−1,Ui

0)H=

m=1

hfi

m,Ui

miV∗

m×Vm

for i∈ {1, . . . , N}. Another summation of this equality from i= 1 to n∈ {1, . . . , N}and a

multiplication with kshows that

i=1

m=1

(Ui

m−Ui

m−1,Ui

m)H+

i=1

(Ui

0−Ui−1,Ui

0)H(3.34)

i=1

m=1

hAi

mUi

m,Ui

miV∗

m×Vm+k

i=1

(BiUi−1,Ui

0)H(3.35)

i=1

m=1

hfi

m,Ui

miV∗

m×Vm.(3.36)

We can write for (3.34)

i=1

m=1

(Ui

m−Ui

m−1,Ui

m)H+

i=1

(Ui

0−Ui−1,Ui

0)H

≥1

i=1

m=1 kUi

mk2

H− kUi

m−1k2

H+1

i=1 kUi

0k2

H− kUi−1k2

H

i=1 kUi

Mk2

H− kUi

0k2

H+1

i=1 kUi

0k2

H− kUi−1k2

H

2kUnk2

H− kU0k2

H=1

2k¯

Uk(tn)k2

H− k ¯

Uk(0)k2

H,

due to the identity from Lemma A.1.4, the telescopic structure, and Ui

M=Uiin H.

Inserting the definition of the piecewise constant prolongations from (3.27) and (3.28) in

(3.35) and (3.36), yields

2k¯

Uk(tn)k2

H−1

2k¯

Uk(0)k2

H+

m=1 Ztn

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt

+Ztn

(Bk(t)Uk(t),¯

0(t))Hdt≤

m=1 Ztn

hfk

m(t),¯

m(t)iV∗

m×Vmdt.

3.1. CONVERGENCE OF THE SPLITTING SCHEME 63

It remains to look closer at the behavior of the prolongations from (3.27), (3.28), and

(3.29) as the step size ktends to zero. The following lemma shows that the sequences of

such prolongations converge in a suitable sense.

Lemma 3.1.14. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled. Further,

let (N`)`∈Nbe a sequence of natural numbers with N`→ ∞ as `→ ∞, let the step sizes be

given by k=T

N`, and let (uk

0)k>0be a bounded sequence in H. Then for the sequences of

piecewise constant and piecewise linear prolongations as given in (3.28) and (3.29), there

exists a subsequence of step sizes, again denoted by k, such that

m* U in Lp(0, T ;Vm), m ∈ {1, . . . , M},

∗

* U, ¯

Uk∗

* U, Uk∗

* U in L∞(0, T ;H), m ∈ {0, . . . , M},

(Uk)0* U0in Lq(0, T ;V∗)

as k→0. The limit Uis an element of Lp(0, T;V)∩L∞(0, T;H)and its weak derivative

fulfills U0∈Lq(0, T;V∗). This implies, in particular, that Uis an element of Wp(0, T).

Furthermore, let VMbe compactly embedded into H, let uk

0be in VMfor every k > 0,

and let (k1

pkuk

0kVM)k>0be uniformly bounded. Then it additionally follows that

Uk* U in Lp(0, T ;VM),

Uk→U, Uk→U, Uk→Uin L2(0, T ;H)

as k→0.

Note that the compact embedding of VMinto His only necessary to prove that the

sequences of piecewise constant and piecewise linear prolongations converge strongly in

L2(0, T ;H). If Assumption 3.1.2 is generalized to p∈(1,∞), it is still possible to prove

Uk→Uin L2(0, T ;H). We can use the compact embedding argument from the Lions–

Aubin lemma (cf. Lemma A.2.5) to obtain strong convergence in Lp(0, T ;H). Together

with the a priori bound (3.13) from Lemma 3.1.12 and Lemma A.2.3, it follows that the

sequence converges strongly in L2(0, T ;H).

Proof of Lemma 3.1.14. For simplicity, we do not denote the subsequences differently within

this proof and we drop the index `. Using the a priori bound (3.15) from Lemma 3.1.12,

it follows that the sequence ( ¯

m)k>0of piecewise constant prolongations is bounded in

Lp(0, T ;Vm) and L∞(0, T;H) for every m∈ {1, . . . , M}. Since Lp(0, T ;Vm) is a reflexive

Banach space and L∞(0, T ;H) is the dual of the separable Banach space L1(0, T ;H), there

exists an element Um∈Lp(0, T ;Vm)∩L∞(0, T;H) such that

m* Umin Lp(0, T ;Vm),¯

∗

* Umin L∞(0, T ;H)

as k→0 for every m∈ {1, . . . , M}. Analogously, there exists U0∈L∞(0, T ;H) such that

∗

* U0in L∞(0, T ;H) as k→0,

where we use the a priori bound (3.14) from Lemma 3.1.12. In the following, we will prove

that U0=U1=. . . =UM=: Uin Lp(0, T ;V) and L∞(0, T ;H) is fulfilled. To this end,

it is sufficient to show that U0and U1coincide in Lp(0, T;V1) and L∞(0, T;H). The other

equalities follow analogously. For the difference of the two functions, we see that

1(t)−¯

0(t) = Un

1−Un

0=kfn

1−An

1Un

1=Ztn

tn−1f1(s)−Ak

1(s)¯

1(s)ds

64 CHAPTER 3. OPERATOR SPLITTING

in V∗for t∈(tn−1, tn], n∈ {1, . . . , N}, due to the definition of the scheme (3.5). Therefore,

we obtain

k¯

1(t)−¯

0(t)kV∗

1=



Ztn

tn−1f1(s)−Ak

1(s)¯

1(s)ds



V∗

≤Ztn

tn−1

kf1(s)−Ak

1(s)¯

1(s)kV∗

1ds

≤k1

pZtn

tn−1

kf1(s)−Ak

1(s)¯

1(s)kq

V∗

1ds

where we can bound the integral by

Ztn

tn−1

kf1(s)−Ak

1(s)¯

1(s)kq

V∗

1ds

q≤Ztn

tn−1

kf1(s)kq

V∗

1ds

q+k1

qkAn

1Un

1kV∗

≤ kf1kLq(0,T ;V∗

1)+k1

qβ1 + kUn

1kp−1

V1.

This is bounded independently of the step size kand n∈ {1, . . . , N}due the a priori bound

(3.15) from Lemma 3.1.12. Thus, we have proved that k¯

1(t)−¯

0(t)kV∗

1→0 as k→0 for

every t∈[0, T ]. Further, this also shows that k¯

1(t)−¯

0(t)kV∗

1k>0is uniformly bounded

independently of t∈[0, T ]. Thus, we can apply Lebesgue’s dominated convergence theorem

to see

k¯

1−¯

0kLq(0,T ;V∗

1)→0 as k→0.

Hence, U0and U1coincide in Lq(0, T ;V∗

1). Both spaces Lp(0, T ;V1) and L∞(0, T;H) are

embedded into Lq(0, T;V∗

1). This implies that U0=U1is fulfilled in all three spaces as the

embedding is always injective and U0∈L∞(0, T;H) and U1∈Lp(0, T;V1)∩L∞(0, T;H).

The fact that U:= U0=U1=· · · =UMin TM

m=1 Lp(0, T ;Vm) and L∞(0, T;H) can be

proved analogously. Due to Assumption 3.1.1, we know that TM

m=1 Vm=Vand the norm

m=1 k·kVmis equivalent to k·kV. This shows, in particular, that U∈TM

m=1 Lp(0, T ;Vm) =

Lp(0, T ;V). Note that the functions ¯

Mand ¯

Ukcoincide by definition. Therefore, it follows

that ¯

Uk∗

* U in L∞(0, T ;H) as k→0.

Another application of the a priori bound (3.13) from Lemma 3.1.12 shows that (Uk)k>0

is bounded in L∞(0, T ;H). Again, we find a subsequence and ˜

U∈L∞(0, T ;H) such that

Uk∗

*˜

Uin L∞(0, T ;H) as k→0.

Furthermore, the difference of ¯

Ukand Ukconverges to zero in Lq(0, T ;V∗) since

k¯

Uk(t)−Uk(t)kq

V∗dt=

n=1 Ztn

tn−1



Un−Un−1−t−tn−1

k(Un−Un−1)



V∗

n=1

kUn−Un−1kq

V∗Ztn

tn−1

(tn−t)qdt

q+ 1

n=1

kUn−Un−1kq

V∗≤kq

q+ 1K→0 as k→0,

where we used the a priori bound (3.16) from Lemma 3.1.12. Therefore, the limits of

(¯

Uk)k>0and (Uk)k>0coincide in Lq(0, T ;V∗). The limits Uand ˜

Uare elements of the

3.1. CONVERGENCE OF THE SPLITTING SCHEME 65

space L∞(0, T ;H). This space is continuously embedded into Lq(0, T;V∗). It then follows

that ˜

U=Uin L∞(0, T ;H) because the embedding is injective.

The sequence ((Uk)0)k>0is bounded in Lq(0, T ;V∗) due to the a priori bound (3.16)

from Lemma 3.1.12. As this space is a reflexive Banach space, we can extract a subsequence

and find W∈Lq(0, T ;V∗) such that

(Uk)0* W in Lq(0, T ;V∗) as k→0.

In order to prove that the sequence (Uk)0converges to the weak derivative of Uweakly in

Lq(0, T ;V∗), we use that Uk∗

* U in L∞(0, T ;H) as k→0 and see

−ZT

hW(t), viV∗×Vϕ(t) dt=−lim

k→0ZT

h(Uk)0(t), viV∗×Vϕ(t) dt

= lim

k→0ZT

(Uk(t), v)Hϕ0(t) dt=ZT

(U(t), v)Hϕ0(t) dt

for v∈Vand ϕ∈C∞

c(0, T ). Applying [49, Kapitel IV, Lemma 1.7], it follows that W=U0

in Lq(0, T ;V∗) and therefore, in particular, U∈ Wp(0, T).

In the following, we require that VMis compactly embedded into H. We use the a priori

bound (3.15) from Lemma 3.1.12 and Un=Un

Min Hfor every n∈ {1, . . . , N}to see

kUkkLp(0,T ;VM)

=N

n=1 Ztn

tn−1



Un−1+t−tn−1

kUn−Un−1



dt

=N

n=1 Ztn

tn−1



tn−t

kUn−1+t−tn−1

kUn



dt

≤1

n=1

kUn−1kp

VMZtn

tn−1

(tn−t)pdt

p+1

n=1

kUnkp

VMZtn

tn−1

(t−tn−1)pdt

≤k

p+ 1

n=1

kUn−1kp

VM

p+k

p+ 1

n=1

kUnkp

VM

≤k

p+ 1

pkuk

0kVM+ 2K

p+ 1

This is bounded as k1

pkuk

0kVMk>0is uniformly bounded with respect to k. As the weak

limit of a sequence is unique, this implies that Uk* U in Lp(0, T;VM) as k→0, where

we again choose a suitable subsequence if necessary. Since (Uk)0* U0in Lq(0, T ;V∗) as

k→0, it follows that Uk* U in the space

M(0, T ) = {v∈Lp(0, T;VM) : v0exists and v0∈Lq(0, T;V∗)}

as k→0. By assumption, the space VMis separable, reflexive, and compactly embedded

into H. Furthermore, H∗is embedded into the reflexive space V∗and we see that

,→H∼

=H∗d

,→V∗

is fulfilled. Thus, we can apply Lemma A.2.5 and obtain that Wp

M(0, T ) is compactly

embedded into L2(0, T;H). As the embedding is compact, it follows that Uk→Uin

66 CHAPTER 3. OPERATOR SPLITTING

L2(0, T ;H) as k→0. This in mind, we can also prove that ¯

Uk→Uand Uk→Uin

L2(0, T ;H) as k→0. We use the a priori bounds (3.13) and (3.14) from Lemma 3.1.12 and

Un=Un

Min Hto see that

N

n=1

kUn−Un−1k2

H

2=N

n=1 



m=1 Un

m−Un

m−1+Un

0−Un−1



H

≤N

n=1 M

m=1

kUn

m−Un

m−1kH+kUn

0−Un−1kH2

≤N

n=1

m=1

kUn

m−Un

m−1k2

H+kUn

0−Un−1k2

H

≤(2K)1

(3.37)

Considering the difference of ¯

Ukand Ukin the L2(0, T ;H)-norm squared, it follows that

k¯

Uk−Ukk2

L2(0,T ;H)=

n=1 Ztn

tn−1



Un−Un−1−t−tn−1

k(Un−Un−1)



Hdt

n=1

kUn−Un−1k2

HZtn

tn−1

(tn−t)2dt

n=1

kUn−Un−1k2

H≤2k

3K→0 as k→0.

Therefore, we have shown that

k¯

Uk−UkL2(0,T ;H)≤ k ¯

Uk−UkkL2(0,T ;H)+kUk−UkL2(0,T ;H)→0 as k→0.

Similarly, we consider the difference of Ukand ¯

Ukin L2(0, T ;H) to find

k¯

Uk−Ukk2

L2(0,T ;H)=

n=1 Ztn

tn−1

kUn−Un−1k2

Hdt

n=1

kUn−Un−1k2

H≤2kK →0 as k→0,

where we again use the bound from (3.37). The last desired convergence result then is

fulfilled as we have shown

kUk−UkL2(0,T ;H)≤ kUk−¯

UkkL2(0,T ;H)+k¯

Uk−UkL2(0,T ;H)→0 as k→0.

Lemma 3.1.15. Let Assumptions 3.1.1, 3.1.2, and 3.1.3 be fulfilled. Then for every m∈

{1, . . . , M}the operator Ak

mdefined in (3.27) fulfills that

mv→Amvin Lq(0, T ;V∗

as k→0for every v∈Lp(0, T ;Vm).

3.1. CONVERGENCE OF THE SPLITTING SCHEME 67

Proof. Let m∈ {1, . . . , M}be arbitrary but fixed in the following. We want to estimate

mv−Amvwithin the Lq(0, T ;V∗

m)-norm. To this end, we notice that

kAk

mv−Amvkq

Lq(0,T ;V∗

m)=ZT

kAk

m(t)v(t)−Am(t)v(t)kq

V∗

mdt

n=1 Ztn

tn−1



kZtn

tn−1Am(s)v(t)−Am(t)v(t)ds



V∗

dt.

For t∈[0, T ] such that s7→ Am(s)wis continuous for all w∈Vm, we always choose

n∈ {1, . . . , N}such that t∈(tn−1, tn]. Then it follows that



kZtn

tn−1Am(s)v(t)−Am(t)v(t)ds



V∗

≤1

kZtn

tn−1

kAm(s)v(t)−Am(t)v(t)kV∗

mds→0

as k→0. Furthermore, an application of the boundedness condition for Am(t), t∈[0, T ],

from Assumption 3.1.2 (4), shows that



kZtn

tn−1Am(s)v(t)−Am(t)v(t)ds



V∗

≤1

kZtn

tn−1

kAm(s)v(t)−Am(t)v(t)kV∗

mds

≤1

kZtn

tn−1

kAm(s)v(t)kV∗

mds+1

kZtn

tn−1

kAm(t)v(t)kV∗

mds

≤2β1 + kv(t)kp−1

Vm=: g(t)

for almost all t∈(tn−1, tn), n∈ {1, . . . , N}. Since kv(t)k(p−1)q

Vm=kv(t)kp

Vmand v∈

Lp(0, T ;Vm), it follows that g∈Lq(0, T ). Now, we can apply Lebesgue’s dominated conver-

gence theorem and see

lim

k→0kAk

mv−Amvkq

Lq(0,T ;V∗

m)=ZT

lim

k→0kAk

m(t)v(t)−Am(t)v(t)kq

V∗

mdt= 0.

Lemma 3.1.16. Let Assumptions 3.1.1 and 3.1.5 be fulfilled. Then the operator Bkdefined

in (3.27) fulfills that

Bkv→Bv in L2(0, T ;H)

as k→0for every v∈L2(0, T ;H).

We omit the proof, as it can be done analogously to the proof of Lemma 3.1.15.

Lemma 3.1.17. Let Assumptions 3.1.1 and 3.1.8 be fulfilled. For every m∈ {1, . . . , M},

it follows that fk

mdefined in (3.27) fulfills fk

m→fmin Lq(0, T ;V∗

m)as k→0.

Again, we omit the proof as it is essentially the same as the proof of Lemma 2.1.9.

The previous lemmas in mind, we are well prepared to prove that the limit U∈ Wp(0, T )

from Lemma 3.1.14 is the solution to the initial value problem (3.1). At first, we show in

Theorem 3.1.18 that the sequences of piecewise constant and piecewise linear prolongations

68 CHAPTER 3. OPERATOR SPLITTING

of the solution from (3.4)–(3.6) converge to the solution uof the evolution equation (3.1)

in a weak sense. Also, a strong convergence result can be proved. If VMis compactly

embedded into H, we can argue directly that certain strong convergence results are fulfilled.

This compact embedding is only necessary if the Lipschitz continuous, H-valued operator

B(t), t∈[0, T ], is not constantly zero. If this operator is constantly zero and the embedding

from VMinto His not compact, we can still show a pointwise strong convergence result in

Theorem 3.1.19 below.

Theorem 3.1.18. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled and let

u0∈Hbe given. For a sequence (N`)`∈Nof natural numbers with N`→ ∞ as `→ ∞, step

sizes k=T

N`, and a bounded sequence (uk

0)k>0in Hsuch that uk

0→u0in Has k→0,

let the sequences of piecewise constant and piecewise linear prolongations from (3.28) and

(3.29) be given. If κin Assumption 3.1.5 (2) is zero, then

m* u in Lp(0, T ;Vm), m ∈ {1, . . . , M},

∗

* u, ¯

Uk∗

* u, Uk∗

* u in L∞(0, T ;H), m ∈ {0, . . . , M},

(Uk)0* u0in Lq(0, T ;V∗),

m=1

m¯

m* Au in Lq(0, T ;V∗),

BkUk→Bu in L2(0, T ;H)

as k→0. Here, uis the solution of (3.1) and u0its weak derivative. Also, it holds true that

both ¯

Uk(t)* u(t)and Uk(t)* u(t)in Has k→0for every t∈[0, T ].

Furthermore, let (uk

0)k>0be in VM, let k1

pkuk

0kVMk>0be uniformly bounded with re-

spect to k, and let the space VMbe compactly embedded into H. Then it follows for an

arbitrary value κ∈[0,∞)from Assumption 3.1.5 (2) that the results above are fulfilled and

additionally it follows that

Uk* u in Lp(0, T ;VM),

Uk→u, Uk→u, Uk→uin L2(0, T ;H)

as k→0.

It would also be possible to prove Uk(t)* u(t) and ¯

m(t)* u(t), m∈ {0, . . . , M}, in

Has k→0 for every t∈[0, T ]. For simplicity, we concentrate on the sequences ( ¯

Uk(t))k>0

and (Uk(t))k>0.

Proof of Theorem 3.1.18. For simplicity, we do not denote the subsequences differently in

this proof and we drop the index `. In the cases that κfrom Assumption 3.1.5 (2) is zero, it

is easy to see that BkUk=BkU→BU in L2(0, T ;H) as k→0 due to Lemma 3.1.16. If κis

strictly larger than zero and VMis compactly embedded into H, we can apply Lemma 3.1.14

to obtain that Uk→Uin L2(0, T ;H) as k→0. Since the inequality

kBU −BkUkkL2(0,T ;H)≤ kBU −BkUkL2(0,T ;H)+kBkU−BkUkkL2(0,T ;H)(3.38)

is fulfilled for every k > 0, we can consider the two summands separately. For the first

summand, we can apply Lemma 3.1.16 to obtain

kBU −BkUkL2(0,T ;H)→0 as k→0.

3.1. CONVERGENCE OF THE SPLITTING SCHEME 69

For the square of the second summand on the right-hand side of (3.38), we use the Lipschitz

continuity of the operator Bn,n∈ {1, . . . , N}, compare (3.10) of Lemma 3.1.10, and the

fact that Uk→Uin L2(0, T ;H) as k→0. Then we see that

kBkU−BkUkk2

L2(0,T ;H)=

n=1 Ztn

tn−1

kBnU(t)−BnUn−1k2

Hdt

≤κ2

n=1 Ztn

tn−1

kU(t)−Un−1k2

Hdt

=κ2kU−Ukk2

L2(0,T ;H)→0

as k→0. Altogether, this proves that

BkUk→BU in L2(0, T ;H) as k→0.(3.39)

Due to the a priori bound (3.15) from Lemma 3.1.12 and the boundedness condition (3.8)

from Lemma 3.1.9, we find that

kAk

m¯

mkLq(0,T ;V∗

m)=k

n=1

kAn

mUn

mkq

V∗

m

≤k

n=1

βq1 + kUn

mkp−1

Vmq

q≤βT1

q+K1

q.

As Lq(0, T ;V∗

m) is a reflexive Banach space, we can extract a weakly converging subsequence

such that

m¯

m* bmin Lq(0, T ;V∗

m) as k→0

for bm∈Lq(0, T ;V∗

m). Next, we identify the derivative of Uwith the equation. Using

Lemma 3.1.14, Lemma 3.1.17, (3.39), and b:= PM

m=1 bmin Lq(0, T ;V∗), we obtain the

following equality

U0= w-lim

k→0(Uk)0= w-lim

k→0M

m=1 fk

m−Ak

m¯

m−BkUk=f−b−BU

in Lq(0, T ;V∗). By w-lim we denote the limiting process with respect to the weak topology in

Lq(0, T ;V∗). Since U∈ Wp(0, T ) and Wp(0, T ) is continuously embedded into C([0, T ]; H),

we can work with the continuous representative of Uin the following.

Another application of the a priori bound (3.13) from Lemma 3.1.12 shows that the

sequence (Uk(t))k>0,t∈[0, T ], is bounded in H. As His reflexive, for every t∈[0, T] there

exist a subsequence and an element ˜

U(t)∈Hwith

Uk(t)*˜

U(t) in H(3.40)

as k→0. This in mind, we prove U(t) = ˜

U(t) and U(0) = u0for every t∈[0, T ]. First,

we recall that (Uk)0* U0in Lq(0, T ;V∗) and Uk∗

* U in L∞(0, T ;H) as k→0, compare

70 CHAPTER 3. OPERATOR SPLITTING

Lemma 3.1.14. For arbitrary but fixed x∈Vand ϕ∈C1([0, T ]), we then find that

(U(t), x)Hϕ(t)−(U(0), x)Hϕ(0) −Zt

(U(s), x)Hϕ0(s) ds

=Zt

hU0(s), xiV∗×Vϕ(s) ds= lim

k→0Zt

((Uk)0(s), x)Hϕ(s) ds

= lim

k→0(Uk(t), x)Hϕ(t)−(uk

0, x)Hϕ(0) −Zt

(Uk(s), x)Hϕ0(s) ds

= ( ˜

U(t), x)Hϕ(t)−(u0, x)Hϕ(0) −Zt

(U(s), x)Hϕ0(s) ds

for t∈[0, T ]. This implies U(t) = ˜

U(t) and U(0) = u0in H. For the piecewise constant

prolongation ¯

Uk, we see that

k¯

Uk(t)−Uk(t)kq

V∗=



Un−Un−1−t−tn−1

kUn−Un−1



V∗

≤tn−t

kqkUn−Un−1kq

V∗≤

i=1

kUi−Ui−1kq

V∗≤kq−1K

for every t∈(tn−1, tn], n∈ {1, . . . , N}. For the last inequality, we use the a priori bound

(3.16) from Lemma 3.1.12. Thus, it follows that k¯

Uk(t)−Uk(t)kV∗→0 as k→0 for

every t∈[0, T ]. This means that the limits of ( ¯

Uk(t))k>0and (Uk(t))k>0coincide in V∗.

Due to the a priori bound (3.13) from Lemma 3.1.12, ( ¯

Uk(t))k>0is bounded in Hfor every

t∈[0, T ]. Thus, we can extract a subsequence that converges weakly to an element of H.

As His continuously embedded into V∗, the limit has to coincide with U(t) in Has the

embedding is injective. This implies that ¯

Uk(t)* U(t) in Has k→0 for every t∈[0, T].

It remains to prove that b=AU in Lq(0, T ;V∗). Recall that for every m∈ {1, . . . , M},

the sequence ( ¯

m)k>0converges weakly to Uin Lp(0, T ;Vm), ( ¯

0)k>0converges weakly∗to

Uin L∞(0, T ;H), and Uis an element of Wp(0, T ), compare Lemma 3.1.14. Using (3.39),

the statements of Lemma 3.1.13 and Lemma 3.1.17 as well as the lower semi-continuity of

the norm, it follows that

lim sup

k→0M

m=1 ZT

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt

≤lim sup

k→0M

m=1 ZT

hfk

m(t),¯

m(t)iV∗

m×Vmdt−ZT

(Bk(t)Uk(t),¯

0(t))Hdt

2k¯

Uk(0)k2

H−1

2k¯

Uk(T)k2

H

≤

m=1 ZT

hfm(t), U(t)iV∗

m×Vmdt−ZT

(B(t)U(t), U(t))Hdt+1

2kU(0)k2

H−1

2kU(T)k2

=ZT

hf(t)−B(t)U(t), U(t)iV∗×Vdt−ZT

hU0(t), U(t)iV∗×Vdt=ZT

hb(t), U(t)iV∗×Vdt,

which implies

lim sup

k→0

m=1 ZT

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt≤ZT

hb(t), U(t)iV∗×Vdt. (3.41)

3.1. CONVERGENCE OF THE SPLITTING SCHEME 71

Due to the monotonicity condition for Am(t), t∈[0, T ] and m∈ {1, . . . , M}, from Assump-

tion 3.1.2 (3) we can write

m=1 ZT

hAk

m(t)¯

m(t)−Ak

m(t)v(t),¯

m(t)−v(t)iV∗

m×Vmdt≥0

for every v∈Lp(0, T ;V). Therefore, an application of Lemma 3.1.14 and Lemma 3.1.15

shows

m=1 ZT

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt

≥

m=1 ZT

0hAk

m(t)¯

m(t), v(t)iV∗

m×Vm+hAk

m(t)v(t),¯

m(t)−v(t)iV∗

m×Vmdt

k→0

−→

m=1 ZT

0hbm(t), v(t)iV∗

m×Vm+hAm(t)v(t), U(t)−v(t)iV∗

m×Vmdt

=ZT

0hb(t), v(t)iV∗×V+hA(t)v(t), U(t)−v(t)iV∗×Vdt,

which implies

lim inf

k→0

m=1 ZT

hAk

m(t)¯

m(t),¯

m(t)iV∗

m×Vmdt

≥ZT

0hb(t), v(t)iV∗×V+hA(t)v(t), U(t)−v(t)iV∗×Vdt.

Applying (3.41), this yields

hb(t), U(t)−v(t)iV∗×Vdt≥ZT

hA(t)v(t), U(t)−v(t)iV∗×Vdt.

We then choose v=U−sw for s∈(0,1) and w∈Lp(0, T ;V) and apply the Minty

monotonicity trick, see [99, Lemma 2.13], to prove that AU =bin Lq(0, T ;V∗). To this

end, we notice that

hb(t), sw(t)iV∗×Vdt≥ZT

hA(t)U(t)−sw(t), sw(t)iV∗×Vdt,

implies

hb(t), w(t)iV∗×Vdt≥ZT

hA(t)U(t), w(t)iV∗×Vdt.

In the previous step, we divided by s > 0, considered s→0, and used the radial continuity

of A(t), t∈[0, T ], from Assumption 3.1.2 (2). Together with the same argumentation for

s∈(−1,0), this proves that

hb(t), w(t)iV∗×Vdt=ZT

hA(t)U(t), w(t)iV∗×Vdt

72 CHAPTER 3. OPERATOR SPLITTING

for every w∈Lp(0, T ;V). This shows that AU =bin Lq(0, T ;V∗). Therefore, U=uis the

unique solution of the evolution problem (3.1).

Since every converging subsequence of ( ¯

m)k>0converges to the unique solution uof

(3.1), we can apply the subsequence principle, see [116, Proposition 10.13] or [49, Kapi-

tel I, Lemma 5.4], to prove that the original sequence ( ¯

m)k>0converges to the solution u

of (3.1). Analogously, we see that every other convergence result claimed in the theorem is

fulfilled for the original sequence.

Theorem 3.1.19. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled and let

u0∈Hbe given. For a sequence (N`)`∈Nof natural numbers with N`→ ∞ as `→ ∞, step

sizes k=T

N`, and a bounded sequence (uk

0)k>0in Hsuch that uk

0→u0in Has k→0,

let the sequences of piecewise constant and piecewise linear prolongations from (3.28) and

(3.29) be given. If κfrom Assumption 3.1.5 (2) is zero, then it follows that

Uk(t)→u(t), Uk(t)→u(t)in Has k→0,

for every t∈[0, T ], where uis the solution of (3.1). Under the additional assumption that

for m∈ {1, . . . , M}there exists η∈(0,∞), which does not depend on t, with

hAm(t)v−Am(t)w, v −wiV∗

m×Vm≥η|v−w|p

Vm(3.42)

for all v, w ∈Vmthe sequence (¯

m)k>0converges strongly to the solution uof (3.1) in

Lp(0, T ;Vm).

Furthermore, let uk

0be in VMfor every k > 0, let k1

pkuk

0kVMk>0be uniformly bounded

with respect to k, and let the space VMbe compactly embedded into H. Then the statement

above is fulfilled for an arbitrary operator B(t),t∈[0, T ], that fulfills Assumption 3.1.5.

Proof. For simplicity, we drop the index `within this proof. In order to estimate the

error, we split it up in separate parts, which can be handled more easily. We combine

the monotonicity conditions from Assumption 3.1.2 (3) and from (3.42). This can be done

by including η= 0 to (3.42). The case η= 0 is exactly the monotonicity condition from

Assumption 3.1.2 (3). We point out the additional result for η∈(0,∞) at the end of the

proof. Using the condition (3.42), we then obtain that

ku(t)−¯

Uk(t)k2

H+ 2η

m=1 Zt

|u(s)−¯

m(s)|p

Vmds

≤ ku(t)−¯

Uk(t)k2

H+ 2

m=1 Zt

hAk

m(s)u(s)−Ak

m(s)¯

m(s), u(s)−¯

m(s)iV∗

m×Vmds

=ku(t)k2

H+ 2

m=1 Zt

hAk

m(s)u(s), u(s)iV∗

m×Vmds

−2(u(t),¯

Uk(t))H−2

m=1 Zt

hAk

m(s)u(s),¯

m(s)iV∗

m×Vmds

−2

m=1 Zt

hAk

m(s)¯

m(s), u(s)iV∗

m×Vmds

+k¯

Uk(t)k2

H+ 2

m=1 Zt

hAk

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds

=: Γk

1(t)+Γk

2(t)+Γk

3(t)

3.1. CONVERGENCE OF THE SPLITTING SCHEME 73

with

Γk

1(t) = ku(t)k2

H+ 2

m=1 Zt

hAk

m(s)u(s), u(s)iV∗

m×Vmds,

Γk

2(t) = −2(u(t),¯

Uk(t))H−2

m=1 Zt

hAk

m(s)u(s),¯

m(s)iV∗

m×Vmds

−2

m=1 Zt

hAk

m(s)¯

m(s), u(s)iV∗

m×Vmds,

Γk

3(t) = k¯

Uk(t)k2

H+ 2

m=1 Zt

hAk

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds

for every t∈[0, T ]. Recall that in Theorem 3.1.18, we proved that ¯

m* u in Lp(0, T ;Vm)

and PM

m=1 Ak

m¯

m* Au in Lq(0, T ;V∗) as k→0. Applying the result from Lemma 3.1.15,

it follows that

lim

k→0Γk

1(t) = ku(t)k2

H+ 2 Zt

hA(s)u(s), u(s)iV∗×Vds,

lim

k→0Γk

2(t) = −2ku(t)k2

H−4Zt

hA(s)u(s), u(s)iV∗×Vds

for every t∈[0, T ]. In order to estimate Γk

3, we need a few additional arguments. Here, we

assume that t∈(tn−1, tn], n∈ {1, . . . , N}, and obtain

Γk

3(t) = k¯

Uk(t)k2

H+ 2

m=1 Ztn

hAk

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds

−2

m=1 Ztn

hAk

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds.

(3.43)

For the first appearing sum of integrals, we can apply Lemma 3.1.13 to see that

m=1 Ztn

hAk

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds

≤2

m=1 Ztn

hfk

m(s),¯

m(s)iV∗

m×Vmds−2Ztn

(Bk(t)Uk(s),¯

0(s))Hds

−k¯

Uk(tn)k2

H− k ¯

Uk(0)k2

H,

which we can reinsert in (3.43). Together with the fact that ¯

Uk(0) = uk

0and ¯

Uk(tn) = ¯

Uk(t)

in H, we obtain a bound for Γk

3(t) given by

Γk

3(t)≤ kuk

0k2

H+ 2

m=1 Zt

hfk

m(s),¯

m(s)iV∗

m×Vmds

−2Zt

(Bk(s)Uk(s),¯

0(s))Hds−2Ztn

(Bk(s)Uk(s),¯

0(s))Hds

+ 2

m=1 Ztn

hfk

m(s)−Ak

m(s)¯

m(s),¯

m(s)iV∗

m×Vmds

(3.44)

74 CHAPTER 3. OPERATOR SPLITTING

for every t∈(tn−1, tn], n∈ {1, . . . , N}. As proven in Lemma 3.1.17, the sequence (fk

m)k>0

converges strongly to fmin Lq(0, T ;V∗

m) as k→0. Thus, together with Theorem 3.1.18 it

follows that

m=1 Zt

hfk

m(s),¯

m(s)iV∗

m×Vmds→

m=1 Zt

hfm(s), u(s)iV∗

m×Vmds

=Zt

hf(s), u(s)iV∗×Vdsas k→0.

Similarly, we see that

(Bk(s)Uk(s),¯

0(s))Hds→Zt

(B(s)u(s), u(s))Hdsas k→0,

since BkUk→Bu in L2(0, T ;H) and ¯

∗

* u in L∞(0, T;H) as k→ ∞, compare Theo-

rem 3.1.18. For the remaining integrals in (3.44), we notice that the functions gk

m: [0, T ]→

R,m∈ {0, . . . , M}, given by

(Bk(s)Uk(s),¯

0(s))H≤κ1 + kUk(s)kHk¯

0(s)kH=: gk

0(s)

and

hfk

m(s)−Ak

m(s)¯

m(s),¯

m(s)iV∗

m×Vm

≤ kfk

m(s)kV∗

mk¯

m(s)kVm+kAk

m(s)¯

m(s)kV∗

mk¯

m(s)kVm

≤1

qkfk

m(s)kq

V∗

m+1

pk¯

m(s)kp

Vm+βk¯

m(s)kVm+k¯

m(s)kp

Vm=: gk

m(s)

for almost every s∈(0, T ) are bounded by an L1(0, T)-function uniformly in kdue to the a

priori bounds (3.13), (3.14), and (3.15) from Lemma 3.1.12. Thus, the remaining integrals

in (3.44) tend to zero as 0 ≤tn−t≤k→0 and it follows that

lim sup

k→0

Γk

3(t)≤ ku0k2

H+ 2 Zt

hf(s), u(s)iV∗×Vds−2Zt

(B(s)u(s), u(s))Hds.

The previous arguments have shown

lim sup

k→0Γk

1(t)+Γk

2(t)+Γk

3(t)

≤ ku(t)k2

H+ 2 Zt

hA(s)u(s), u(s)iV∗×Vds

−2ku(t)k2

H−4Zt

hA(s)u(s), u(s)iV∗×Vds

+ku0k2

H+ 2 Zt

hf(s)−B(s)u(s), u(s)iV∗×Vds

=−ku(t)k2

H+ku0k2

H+ 2 Zt

hf(s)−A(s)u(s)−B(s)u(s), u(s)iV∗×Vds

=−ku(t)k2

H+ku0k2

H+ 2 Zt

hu0(s), u(s)iV∗×Vds

=−ku(t)k2

H+ku0k2

H+Zt

dtku(s)k2

Hds= 0

3.2. AN EXPLICIT ERROR ESTIMATE 75

for every t∈[0, T ]. We also use the fact that u∈ Wp(0, T ) and therefore a partial integration

rule can be applied. Altogether, this implies the strong convergence of ( ¯

Uk(t))k>0in Hfor

every t∈[0, T ] as we have proved

lim

k→0ku(t)−¯

Uk(t)k2

H+ 2η

m=1 Zt

|u(s)−¯

m(s)|p

Vmds= 0.(3.45)

Recall the definition of ¯

Ukand Ukfrom (3.28) and (3.29), respectively. Then we obtain that

kUk(t)−u(t)kH≤



tn−t

k¯

Uk(t−k)−u(t)



H+



t−tn−1

k¯

Uk(t)−u(t)



H

≤ k ¯

Uk(t−k)−u(t)kH+k¯

Uk(t)−u(t)kH

≤ k ¯

Uk(t−k)−u(t−k)kH+ku(t−k)−u(t)kH+k¯

Uk(t)−u(t)kH

for every t∈[0, T ]. Using that u∈ Wp(0, T ),→C([0, T]; H) and ¯

Uk(t)→u(t) in Has

k→0 for every t∈[0, T ], it also follows that Uk(t)→u(t) in Has k→0.

Now, we consider the case η∈(0,∞) for m∈ {1, . . . , M}and prove ¯

m→uin

Lp(0, T ;Vm) as k→0. The difference of the piecewise constant prolongations ¯

mand

Ukconverges to zero in L2(0, T ;H). This is true as

k¯

Uk−¯

mk2

L2(0,T ;H)=k

n=1

kUn−Un

mk2

H≤kM

n=1

j=1

kUn

j−Un

j−1k2

H→0 as k→0,

where we used the a priori bound (3.13) from Lemma 3.1.12. Since ( ¯

Uk)k>0is bounded

in L2(0, T ;H) and converges to upointwise strongly in H, this shows that ¯

Uk→uin

L2(0, T ;H) as k→0. We see, in particular, that ¯

m→uin L2(0, T ;H) as k→0 due

to the previous estimate. The a priori bound (3.15) from Lemma 3.1.12 even shows that

the sequence ( ¯

m)k>0is bounded in L∞(0, T ;H). Therefore, it converges to ualso in

the space Lp(0, T ;H) as k→0, compare Lemma A.2.3. Using (3.45) and the inequality

kvkVm≤cVmkvkH+|v|Vmfor the Vm-norm stated in Assumption 3.1.1, it follows that

ku−¯

mkLp(0,T ;Vm)=ZT

ku(t)−¯

m(t)kp

Vmdt

≤cVmZT

0ku(t)−¯

m(t)kH+|u(t)−¯

m(t)|Vmpdt

≤cVmku−¯

mkLp(0,T ;H)+cVmZT

|u(t)−¯

m(t)|p

Vmdt

p→0

as k→0. This proves that ¯

m→uin Lp(0, T ;Vm) as k→0 if (3.42) is fulfilled for ηthat

is strictly larger than zero for Am(t), t∈[0, T].

3.2 An Explicit Error Estimate

After a convergence analysis of the implicit-explicit product splitting scheme under no ad-

ditional regularity assumptions on the solution, we now regard the question whether a more

regular solution uof the evolution equation (3.1) will lead to explicit error bounds. In the

following, we assume that for α∈(0,1] the function uis an element of the space of H¨older

76 CHAPTER 3. OPERATOR SPLITTING

continuous functions C0,α([0, T ]; V). Here, we will not go into detail to explain when this

condition is fulfilled. More information about additional regularity of the solution and some

examples that fit this setting can be found in Section 1.2. In the following, we will need a

similar condition for a counterpart to the approximations ¯

m,m∈ {0, . . . , M}. To this end,

we introduce the functions below.

Assumption 3.2.1. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, and 3.1.8 be fulfilled and

let u0∈Vbe given. For α∈(0,1], let the solution uof the evolution equation (3.1) be an

element of C0,α([0, T ]; V). For N∈N, consider tn=kn for n∈ {0, . . . , N}, where k=T

Moreover, for every m∈ {0, . . . , M}, let the function Um: [0, T]→V∗be given by

Um(t) = u(tn−1) +

j=1 Zt

tn−1fj(s)−Aj(s)u(s)ds−Zt

tn−1

B(s)u(s) dsin V∗,

for every t∈(tn−1, tn],n∈ {1, . . . , N}with Um(0) = u0. Suppose that for m∈ {1, . . . , M},

the function fulfills

kUm(t)−u(tn−1)kVm≤Lα,m|t−tn−1|α(3.46)

for every t∈(tn−1, tn],n∈ {1, . . . , N}.

Note that we use the convention P0

j=1 = 0. It does not make sense to assume that

a function Um,m∈ {1, . . . , M}, fulfills a H¨older condition like u. The functions Umdo

not even have to be continuous for m∈ {1...,M −1}as limt&tnUm(t) does not have to

coincide with Um(tn). Thus, we only ask for a corresponding condition on subintervals. This

assumption is not necessary for U0.

When it comes to verifying such a condition in applications, first note that UM=uin

C0,α([0, T]; V). Thus, the condition (3.46) does not impose any additional regularity on the

function UM. For m∈ {1, . . . , M −1}, the condition means that



j=1 Zt

tn−1fj(s)−Aj(s)u(s)ds−Zt

tn−1

B(s)u(s) ds



Vm

≤Lα,m|t−tn−1|α

has to be fulfilled. In our example from Section 3.3, this follows with the help of an em-

bedding argument. Also, a suitable order of appearance of Amand fmcan be helpful when

proving (3.46). These regularity conditions in mind, we can get the following bounds.

Lemma 3.2.2. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, 3.1.8, and 3.2.1 be fulfilled. Then

for every r∈[1,∞)there exists a constant C∈(0,∞)such that

ku(t)−Um(t)kr

Vmdt≤Ckαr

and

n=1 Ztn

tn−1

kUm(t)−Um(tn)kr

Vmdt≤Ckαr

are fulfilled.

3.2. AN EXPLICIT ERROR ESTIMATE 77

Proof. For the first estimate, we use the H¨older continuity of uand the regularity condition

(3.46) from Assumption 3.2.1 to obtain that

ku(t)−Um(t)kVm≤ ku(t)−u(tn−1)kVm+ku(tn−1)−Um(t)kVm

≤c1ku(t)−u(tn−1)kV+ku(tn−1)−Um(t)kVm

≤c1Lα|t−tn−1|α+Lα,m|t−tn−1|α

for every t∈(tn−1, tn]. The constant c1∈(0,∞) is the embedding constant from Vinto

Vmand Lα∈[0,∞) is the H¨older seminorm of u. Thus, it follows

ku(t)−Um(t)kr

Vmdt≤

i=1 Ztn

tn−1

kαrc1Lα+Lα,mrdt=kαrTc1Lα+Lα,mr.

For the second estimate, we again use (3.46) to see that

N

n=1 Ztn

tn−1

kUm(tn)−Um(t)kr

Vmdt

≤N

n=1 Ztn

tn−1

kUm(tn)−u(tn−1)kr

Vmdt

r+N

n=1 Ztn

tn−1

ku(tn−1)−Um(t)kr

Vmdt

≤2N

n=1 Ztn

tn−1

kαrLr

α,m dt

r≤2kαLα,mT1

This auxiliary statement in mind, we can now turn to the main statement of this section.

We prove explicit error bounds for the approximation of a nonlinear evolution equation.

In [86] or [95], a fully nonlinear problem u0(t) = F(t, u(t)) for t∈(0, T ) with an initial

condition is linearized along the exact solution uin order to find an approximation of u.

Then it is possible to use the partial derivative A(t) = ∂uF(t, u) if Fis smooth enough. In

the following, we do not rely on a linear approximation of our nonlinear equation but use

a similar approach as in Section 2.2. Again, we use the analysis from [37] for a globally

Lipschitz continuous operator A(t), t∈[0, T ], as a starting point. We suppose that u

and Um,m∈ {1, . . . , M}, fulfill the additional regularity condition from Assumption 3.2.1.

Further, we assume that for every m∈ {1, . . . , M}the operator Am(t), t∈[0, T ], fulfills a

bounded Lipschitz condition and a p-monotonicity condition. Then we can obtain an explicit

error bound. The size of the error depends both on q=p

p−1and the H¨older exponent αof

the exact solution u. The assumption that an operator A(t), t∈[0, T ], fulfills a bounded

Lipschitz condition is fulfilled in standard examples as the p-Laplacian. Note that in contrast

to Section 2.2, we do not ask for any regularity conditions for u0. This is technically not

necessary since we assume that the integrals from (3.2) and (3.3) are known. To obtain such

values in practice, usually requires a certain temporal regularity condition on the data. We

exchanged this by a regularity condition for u0in Section 2.2.

Note that for p= 2 and α= 1, we have an error bound of order one. This is also the

formal rate of convergence of the standard backward Euler scheme. Thus, the additional

splitting error of our proposed scheme only affects the error in terms of constants, which do

not depend on the step size.

Theorem 3.2.3. Let Assumptions 3.1.1, 3.1.2, 3.1.3, 3.1.5, 3.1.8, and 3.2.1 be fulfilled and

let the initial value u0∈Vbe given.

78 CHAPTER 3. OPERATOR SPLITTING

For every m∈ {1, . . . , M}, let the operator Am(t),t∈[0, T ], fulfill a bounded Lipschitz

condition in the sense that for every R∈(0,∞)there exists L(R)∈[0,∞)such that

kAm(t)v−Am(t)wkV∗

m≤L(R)kv−wkVm(3.47)

is fulfilled for all t∈[0, T ]and v, w ∈Vmwith kvkVm,kwkVm≤R. Furthermore, let every

Am(t),t∈[0, T ], satisfy a p-monotonicity condition such that there exists η∈(0,∞)with

hAm(t)v−Am(t)w, v −wiV∗

m×Vm≥ηkv−wkp

Vm(3.48)

for all v, w ∈Vmand t∈[0, T ]. Then there exists C∈(0,∞)such that for every step size

k=T

N,N∈N, with 2κk ∈[0,1) and uk

0=u0in Vthe solution of (3.4)–(3.6) fulfills that

max

n∈{1,...,N}ku(tn)−Unk2

H+k

n=1

m=1

ku(tn)−Un

mkp

Vm≤Ckαq (3.49)

for q=p

p−1,tn=nk,n∈ {0, . . . , N}.

The bounded Lipschitz condition we require is more general than a Lipschitz condition

on bounded sets in H. Since Vmis continuously embedded into H, there exists a constant

c1∈(0,∞) such that kvkH≤c1kvkVmfor every v∈Vm. Thus, if (3.47) is fulfilled for every

v, w ∈Vmwith kvkH,kwkH≤c1R, it is also fulfilled for v, w ∈Vmwith kvkVm,kwkVm≤R

since kvkH≤c1kvkVm≤c1R.

Proof of Theorem 3.2.3. In the following, let i∈ {1, . . . , N}be fixed. Recalling the defini-

tion of the function Umfrom Assumption 3.2.1, we first notice that Um∈L∞(0, T ;Vm) for

every m∈ {1, . . . , M}since

kUmkL∞(0,T ;Vm)= ess sup

t∈[0,T ]

kUm(t)kVm

≤max

n∈{1,...,N}ess sup

t∈(tn−1,tn]

kUm(t)−u(tn−1)kVm+ku(tn−1)kVm

≤kαLα,m +kukL∞(0,T ;Vm)<∞

due to (3.46) and the fact that uis also bounded in Vm. Further, we can write that

Um(ti)−Um−1(ti) = Zti

ti−1fm(t)−Am(t)u(t)dtin V∗

for m∈ {1, . . . , M}and

U0(ti)−u(ti−1) = −Zti

ti−1

B(t)u(t) dtin H.

These equations can now be tested with the corresponding element Um(ti)−Ui

m∈Vm,

m∈ {1, . . . , M}, and U0(ti)−Ui

0∈H, respectively. Then we obtain

(Um(ti)−Um−1(ti), Um(ti)−Ui

m)H

=Zti

ti−1

hfm(t)−Am(t)u(t), Um(ti)−Ui

miV∗

m×Vmdt(3.50)

3.2. AN EXPLICIT ERROR ESTIMATE 79

for m∈ {1, . . . , M}and

(U0(ti)−u(ti−1), U0(ti)−Ui

0)H=−Zti

ti−1

(B(t)u(t), U0(ti)−Ui

0)Hdt. (3.51)

We now sum up (3.50) from m= 1 to Mand add (3.51). This yields

m=1

(Um(ti)−Um−1(ti), Um(ti)−Ui

m)H+ (U0(ti)−u(ti−1), U0(ti)−Ui

0)H

m=1 Zti

ti−1

hfm(t)−Am(t)u(t), Um(ti)−Ui

miV∗

m×Vmdt

−Zti

ti−1

(B(t)u(t), U0(ti)−Ui

0)Hdt.

(3.52)

We show a corresponding equality for the numerical scheme. Here, we test the equation

(3.5) with Um(ti)−Ui

m∈Vmto obtain

(Ui

m−Ui

m−1, Um(ti)−Ui

m)H=khfi

m−Ai

mUi

m, Um(ti)−Ui

miV∗

m×Vm(3.53)

for m∈ {1, . . . , M}. Further, we test (3.4) with U0(ti)−Ui

0∈Hand get

(Ui

0−Ui−1, U0(ti)−Ui

0)H=−k(BiUi−1, U0(ti)−Ui

0)H.(3.54)

Then we sum up (3.53) from m= 1 to Mand add (3.54) to see

m=1

(Ui

m−Ui

m−1, Um(ti)−Ui

m)H+ (Ui

0−Ui−1, U0(ti)−Ui

0)H

m=1

hfi

m−Ai

mUi

m, Um(ti)−Ui

miV∗

m×Vm−k(BiUi−1, U0(ti)−Ui

0)H

m=1 Zti

ti−1

hfm(t)−Am(t)Ui

m, Um(ti)−Ui

miV∗

m×Vmdt

−Zti

ti−1

(B(t)Ui−1, U0(ti)−Ui

0)Hdt.

(3.55)

In the last step, we inserted the definition of Ai

m,fi

m,m∈ {1, . . . , M}, and Bifrom (3.2).

In order to estimate the error, we consider the difference of (3.52) and (3.55). We can write

for the difference of the left-hand sides

m=1

(Um(ti)−Um−1(ti), Um(ti)−Ui

m)H+ (U0(ti)−u(ti−1), U0(ti)−Ui

0)H

−

m=1

(Ui

m−Ui

m−1, Um(ti)−Ui

m)H+ (Ui

0−Ui−1, U0(ti)−Ui

0)H

m=1 Um(ti)−Ui

m−Um−1(ti)−Ui

m−1, Um(ti)−Ui

mH

+U0(ti)−Ui

0−u(ti−1)−Ui−1, U0(ti)−Ui

0H.

80 CHAPTER 3. OPERATOR SPLITTING

After inserting the identity from Lemma A.1.4, we see that this is equal to

m=1 kUm(ti)−Ui

mk2

H− kUm−1(ti)−Ui

m−1k2

H

m=1

kUm(ti)−Ui

m−Um−1(ti) + Ui

m−1k2

2kU0(ti)−Ui

0k2

H− ku(ti−1)−Ui−1k2

H+kU0(ti)−Ui

0−u(ti−1) + Ui−1k2

H

2ku(ti)−Uik2

H− ku(ti−1)−Ui−1k2

H

m=1

kUm(ti)−Ui

m−Um−1(ti) + Ui

m−1k2

2kU0(ti)−Ui

0−u(ti−1) + Ui−1k2

(3.56)

where we insert UM(ti) = u(ti) and Ui

M=Uiin H. Next, we rewrite the right-hand side

of the difference of (3.52) and (3.55). Then we see that

m=1 Zti

ti−1

hfm(t)−Am(t)u(t), Um(ti)−Ui

miV∗

m×Vmdt

−

m=1 Zti

ti−1

hfm(t)−Am(t)Ui

m, Um(ti)−Ui

miV∗

m×Vmdt

−Zti

ti−1

(B(t)u(t)−B(t)Ui−1, U0(ti)−Ui

0)Hdt

=−

m=1 Zti

ti−1

hAm(t)u(t)−Am(t)Ui

m, Um(ti)−Ui

miV∗

m×Vmdt

−Zti

ti−1

(B(t)u(t)−B(t)Ui−1, U0(ti)−Ui

0)Hdt

=: Γ1+ Γ2+ Γ3+ Γ4,

where

Γ1=−

m=1 Zti

ti−1

hAm(t)u(t)−Am(t)Um(t), Um(ti)−Ui

miV∗

m×Vmdt,

Γ2=−

m=1 Zti

ti−1

hAm(t)Um(t)−Am(t)Um(ti), Um(ti)−Ui

miV∗

m×Vmdt,

Γ3=−

m=1 Zti

ti−1

hAm(t)Um(ti)−Am(t)Ui

m, Um(ti)−Ui

miV∗

m×Vmdt,

Γ4=−Zti

ti−1

(B(t)u(t)−B(t)Ui−1, U0(ti)−Ui

0)Hdt.

We added and subtracted the terms containing Am(t)Um(t) and Am(t)Um(ti) in order to

estimate Γ1, Γ2, Γ3, and Γ4more easily. For Γ1and Γ2, we can use the bounded Lipschitz

3.2. AN EXPLICIT ERROR ESTIMATE 81

condition and the results from Lemma 3.2.2. In order to estimate Γ3, we use the monotonicity

of Am(t), t∈[0, T ], while for Γ4we use the Lipschitz continuity of B(t), t∈[0, T ].

Precisely, for Γ1, we obtain that

Γ1≤

m=1 Zti

ti−1

kAm(t)u(t)−Am(t)Um(t)kV∗

mkUm(ti)−Ui

mkVmdt

≤LA(R)

m=1 Zti

ti−1

ku(t)−Um(t)kVmkUm(ti)−Ui

mkVmdt

≤c1

m=1 Zti

ti−1

ku(t)−Um(t)kq

Vmdt+kη

m=1

kUm(ti)−Ui

mkp

Vm,

where c1=LA(R)q(pη)1−q

41−qqand we choose Rfor the condition (3.47) to be

R= max kukL∞(0,T ;V1),...,kukL∞(0,T ;VM),kU1kL∞(0,T ;V1),...,kUMkL∞(0,T ;VM).(3.57)

For Γ2, we can argue similarly to obtain that

Γ2≤

m=1 Zti

ti−1

kAm(t)Um(t)−Am(t)Um(ti)kV∗

mkUm(ti)−Ui

mkVmdt

≤LA(R)

m=1 Zti

ti−1

kUm(t)−Um(ti)kVmkUm(ti)−Ui

mkVmdt

≤c2

m=1 Zti

ti−1

kUm(t)−Um(ti)kq

Vmdt+kη

m=1

kUm(ti)−Ui

mkp

Vm,

where c2=LA(R)q(pη)1−q

41−qqand Rcan again be chosen as in (3.57). For Γ3, we use the

monotonicity condition from (3.48) to see that

Γ3≤ −kη

m=1

kUm(ti)−Ui

mkp

Vm.

Furthermore, the Lipschitz continuity of B(t), t∈[0, T ], shows that

Γ4≤κZti

ti−1

ku(t)−Ui−1kHkU0(ti)−Ui

0kHdt

≤κ

2Zti

ti−1

ku(t)−Ui−1k2

Hdt+kκ

2kU0(ti)−Ui

0k2

≤κZti

ti−1

ku(t)−u(ti−1)k2

Hdt+kκku(ti−1)−Ui−1k2

+kκkU0(ti)−Ui

0−u(ti−1) + Ui−1k2

H+kκku(ti−1)−Ui−1k2

≤κZti

ti−1

ku(t)−u(ti−1)k2

Hdt+ 2kκku(ti−1)−Ui−1k2

2kU0(ti)−Ui

0−u(ti−1) + Ui−1k2

82 CHAPTER 3. OPERATOR SPLITTING

for 2κk ∈[0,1). We can now combine the calculations for both the left-hand side and the

right-hand side of the difference of (3.52) and (3.55). The difference of left-hand sides can

be found in (3.56), while the difference of the right-hand sides is given by Γ1+ Γ2+ Γ3+ Γ4.

2ku(ti)−Uik2

H− ku(ti−1)−Ui−1k2

H+1

2kU0(ti)−Ui

0−u(ti−1) + Ui−1k2

m=1

kUm(ti)−Ui

m−Um−1(ti) + Ui

m−1k2

= Γ1+ Γ2+ Γ3+ Γ4

≤c1

m=1 Zti

ti−1

ku(t)−Um(t)kq

Vmdt+kη

m=1

kUm(ti)−Ui

mkp

+c2

m=1 Zti

ti−1

kUm(t)−Um(ti)kq

Vmdt+kη

m=1

kUm(ti)−Ui

mkp

−kη

m=1

kUm(ti)−Ui

mkp

+κZti

ti−1

ku(t)−u(ti−1)k2

Hdt+ 2kκku(ti−1)−Ui−1k2

2kU0(ti)−Ui

0−u(ti−1) + Ui−1k2

This inequality can be rearranged to a suitable bound. Further, we multiply by two to

obtain

ku(ti)−Uik2

H− ku(ti−1)−Ui−1k2

H+kη

m=1

kUm(ti)−Ui

mkp

≤2c1

m=1 Zti

ti−1

ku(t)−Um(t)kq

Vmdt+ 2c2

m=1 Zti

ti−1

kUm(t)−Um(ti)kq

Vmdt

+ 2κZti

ti−1

ku(t)−u(ti−1)k2

Hdt+ 4kκku(ti−1)−Ui−1k2

A summation from i= 1 to n∈ {1, . . . , N}and the fat that u(0) = u0=U0in Vthen

implies that

ku(tn)−Unk2

H+kη

i=1

m=1

kUm(ti)−Ui

mkp

≤2c1

m=1 ZT

ku(t)−Um(t)kq

Vmdt+ 2c2

i=1

m=1 Zti

ti−1

kUm(t)−Um(ti)kq

Vmdt

+ 2κ

i=1 Zti

ti−1

ku(t)−u(ti−1)k2

Hdt+ 4kκ

n−1

i=1

ku(ti)−Uik2

As the function uis H¨older continuous with values in V, it is also an element of the space

3.3. EXAMPLE: A NONLINEAR PARABOLIC PROBLEM 83

C0,α([0, T]; H). Together with the results from Lemma 3.2.2, it then follows that

ku(tn)−Unk2

H+kη

i=1

m=1

kUm(ti)−Ui

mkp

Vm≤c3kαq +k2α+ 4kκ

n−1

i=1

ku(ti)−Uik2

for a constant c3∈(0,∞), which does not depend on the step size k. An application of

Lemma A.1.1 shows that

ku(tn)−Unk2

H+kη

i=1

m=1

kUm(ti)−Ui

mkp

Vm≤c3kαq +k2αexp(4κT ).(3.58)

Moreover, applying (3.46) from Assumption 3.2.1 and the H¨older continuity of u, it follows

that

ku(ti)−Ui

mkVm≤ ku(ti)−u(ti−1)kVm+ku(ti−1)−Um(ti)kVm+kUm(ti)−Ui

mkVm

≤cm|u|C0,α([0,T ];V)kα+Lα,mkα+kUm(ti)−Ui

mkVm

≤c5kα+kUm(ti)−Ui

mkVm,

where cm∈(0,∞) is the embedding constant from Vinto Vmand

c5= max

m∈{1,...,M}cm|u|C0,α([0,T ];V)+Lα,m.

Altogether, this implies

i=1

m=1

ku(ti)−Ui

mkp

Vm≤k

i=1

m=1 c5kα+kUm(ti)−Ui

mkVmp

≤2p−1kc5

i=1

m=1

kαp + 2p−1k

i=1

m=1

kUm(ti)−Ui

mkp

≤2p−1c5kαpTM + 2p−1c3kαq +k2αexp(4κT)

η.

Together with (3.58) this finishes the proof as p∈[2,∞).

3.3 Example: A Nonlinear Parabolic Problem

In order to demonstrate that our abstract theory applies to more concrete problems, we

consider a nonlinear parabolic problem and split the equation into the appearing terms.

This enables us to look at different problems that can be solved more efficiently individually.

A similar example was presented in [35]. The abstract theory in the previous two sections

now offers a possibility to allow for a somewhat more general setting. Here, we can also

permit non-monotone lower-order terms due to the additional Lipschitz continuous operator

B(t), t∈[0, T ], from the theory above. Furthermore, we obtain explicit error bounds under

some additional assumptions.

For a finite end time T∈(0,∞) and a bounded Lipschitz domain D ⊂ Rd,d∈N, we

consider the problem











∂tu(t, x) + a1(t, x, u(t, x)) −∇·a2(t, x, ∇u(t, x)) + b(t, x, u(t, x)) = f(t, x),

(t, x)∈(0, T )× D,

u(t, x)=0,(t, x)∈(0, T )×∂D,

u(0, x) = u0(x), x ∈ D.

(3.59)

84 CHAPTER 3. OPERATOR SPLITTING

Further, a1: [0, T ]×D ×R→Rand a2: [0, T ]×D × Rd→Rdfulfill Assumption 3.3.1 below

and b: [0, T ]×D × R→Rfulfills Assumption 3.3.4 below. Moreover, f: [0, T ]× D → R

and u0:D → Rare functions we will be specify later.

Assumption 3.3.1. Let p∈[2,∞)and `∈ {1, d}be given and q=p

p−1. Let a: [0, T ]×

D × R`→R`fulfill the following conditions:

(1) The mapping t7→ a(t, x, z)is continuous almost everywhere in (0, T)for almost every

x∈ D and every z∈R`,x7→ a(t, x, z)is measurable for every t∈[0, T ]and z∈R`,

while z7→ a(t, x, z)is continuous for every t∈[0, T]and almost every x∈ D.

(2) The mapping afulfills a monotonicity condition such that for every t∈[0, T ], almost

every x∈ D, as well as every z, ˜z∈R`the inequality (a(t, x, z)−a(t, x, ˜z))·(z−˜z)≥0

is satisfied.

(3) The mapping afulfills a growth condition in the sense that there exist d1∈[0,∞)and

a nonnegative function d2∈Lq(D)such that for every t∈[0, T ], almost every x∈ D,

as well as every z∈R`the inequality |a(t, x, z)| ≤ d1|z|p−1+d2(x)is satisfied.

(4) The mapping afulfills a coercivity condition in the sense that there exist d3∈(0,∞)

and a nonnegative d4∈L1(D)such that for every t∈[0, T ], almost every x∈ D, as

well as every z∈R`the condition a(t, x, z)·z≥d3|z|p−d4(x)is satisfied.

Assumption 3.3.2. Let Assumption 2.3.1 be fulfilled. Additionally, there exists d5∈(0,∞)

such that

(a(t, x, z)−a(t, x, ˜z)) ·(z−˜z)≥d5|z−˜z|p

is satisfied for every t∈[0, T ], almost every x∈ D, and every z, ˜z∈R`.

Assumption 3.3.3. Let Assumption 2.3.1 be fulfilled. Additionally, there exists d6∈[0,∞)

such that

|a(t, x, z)−a(t, x, ˜z)| ≤ d61 + max{|z|p−2,|˜z|p−2}|z−˜z|

is satisfied for every t∈[0, T ], almost every x∈ D, and every z, ˜z∈R`.

As explained in Section 2.3, a standard example that fulfills all the assumptions above

is a(t, x, z) = a(z) = |z|p−2z.

Assumption 3.3.4. Let b: [0, T ]×D × R→Rfulfill the following conditions:

(1) The mapping t7→ b(t, x, z)is continuous almost everywhere in (0, T )for almost every

x∈ D and every z∈Rand x7→ b(t, x, z)is measurable for every t∈[0, T ]and z∈R.

(2) The mapping bfulfills a Lipschitz condition in the sense that there exists e1∈[0,∞)

such that for every t∈[0, T ], almost every x∈ D, as well as every z, ˜z∈Rthe

inequality |b(t, x, z)−b(t, x, ˜z)| ≤ e1|z−˜z|is satisfied.

(3) The mapping bfulfills a growth condition at the point zero in the sense that there

exists a nonnegative function e2∈L2(D)such that for every t∈[0, T ]and almost

every x∈ D the inequality |b(t, x, 0)| ≤ e2(x)is satisfied.

3.3. EXAMPLE: A NONLINEAR PARABOLIC PROBLEM 85

Next, let H=L2(D) and V=W1,p

0(D) be equipped with the norms introduced in

the notation section in the introduction. The value pis the same as in Assumption 3.3.1.

Further, we consider V1=Lp(D) equipped with the standard norm and V2=Vwith the

same norm as for V. The seminorms are all chosen as the full norm in the corresponding

space.

For t∈[0, T ], the operators Am(t): Vm→V∗

m,m∈ {1,2}, and B(t): H→Hare given

hA1(t)v, wiV∗

1×V1=ZD

a1(t, ·, v)wdx, v, w ∈V1,(3.60)

hA2(t)v, wiV∗

2×V2=ZD

a2(t, ·,∇v)· ∇wdx, w, v ∈V2(3.61)

(B(t)v, w)H=ZD

b(t, ·, v)wdx, v, w ∈H(3.62)

and A(t): V→V∗is given by A(t) = A1(t) + A2(t). We assume that for f: [0, T ]×D → R

the abstract function [f(t)](x) = f(t, x), (t, x)∈(0, T )× D, is an element of Lq(0, T;V∗).

This function is decomposed into f1= 0 and f2=fin Lq(0, T;V∗). For u0∈H, we can

now state (3.59) in a variational formulation given by

(u0+Au +Bu =fin Lq(0, T ;V∗),

u(0) = u0in H. (3.63)

This evolution equation in mind, we obtain convergence results for the product splitting

scheme.

Theorem 3.3.5. Let a1: [0, T ]×D × R→Rand a2: [0, T ]× D × Rd→Rdfulfill Assump-

tion 3.3.1 and let b: [0, T ]×D × R→Rfulfill Assumption 3.3.4. Let f∈Lq(0, T ;V∗)and

u0∈Hbe given.

Furthermore, let (N`)`∈Nbe a sequence of natural numbers with N`→ ∞ as `→ ∞,

k=T

N`,tn=nk,n∈ {0, . . . , N`}, and (uk

0)k>0in Vsuch that uk

0→u0in Has k→0and

(k1

pkuk

0kV)k>0is uniformly bounded with respect to k. Then the scheme,

0−Un−1

k+BnUn−1= 0 in H,

m−Un

m−1

k+An

mUn

m=fn

min V∗

m, m ∈ {1,2},

for n∈ {1, . . . , N`}with Un=Un

2and U0=uk

0admits a unique solution (Un)n∈{1,...,N`}

in H. Here, the discretizations of the data are given by

m=1

kZtn

tn−1

Am(t) dt, m ∈ {1,2},Bn=1

kZtn

tn−1

B(t) dt,

1= 0, and fn

2=1

kRtn

tn−1f(t) dt. Then all the convergence results from Theorem 3.1.18 and

3.1.19 hold true. In particular, the sequences of the piecewise constant and piecewise linear

prolongations of (Un)n∈{1,...,N`}converge to the solution uof (3.63) pointwise strongly in

Has k→0.

If a2also fulfills Assumption 3.3.2, then the sequence of piecewise constant prolongations

of the values (Un)n∈{1,...,N`}converges to ustrongly in Lp(0, T ;V)as k→0.

86 CHAPTER 3. OPERATOR SPLITTING

In our scheme, the order of the appearing operators A1(t) and A2(t), t∈[0, T], is

important. As we need the embedding of V2into Hto be compact, it is not possible to

change their order. Moreover, as the solution (Un)n∈{1,...,N`}is in the space V2, it makes

sense to choose the smallest space as the last. Then we obtain the best regularity result

for our numerical approximation. The choice of (fn

1)n∈{1,...,N`}and (fn

2)n∈{1,...,N`}is not

unique. Choosing one function as zero, seems like a good choice when it comes to computing

a solution. If only a function f∈Lq(0, T ;V∗

1) is given, it is also possible to set fn

2= 0 in

V∗

2and fn

1=1

kRtn

tn−1f(t) dtin V∗

1for n∈ {1, . . . , N`}.

Proof of Theorem 3.3.5. In order to apply Theorem 3.1.18 and Theorem 3.1.19, it only re-

mains to verify that Am(t), t∈[0, T ] and m∈ {1,2}, fulfill Assumption 3.1.2 and 3.1.3 and

B(t), t∈[0, T ], fulfills Assumption 3.1.5. For the decomposition of f, it is easy to see that

Assumption 3.1.8 is fulfilled.

In order to prove that the operator A1(t), t∈[0, T ], is well-defined, we apply Assump-

tion 3.3.1 (3), which yields that

hA1(t)v, wiV∗

1×V1=ZD

a1(t, ·, v)wdx≤ZDd1|v|p−1+d2|w|dx

≤max d1,kd2kLq(D)1 + kvkp−1

V1kwkV1(3.64)

for every v, w ∈V1and t∈[0, T ]. This also proves that A1(t), t∈[0, T ], fulfills the

boundedness condition from Assumption 3.1.2 (4).

Next, we prove the continuity of t7→ A1(t)valmost everywhere in (0, T) for every v∈V1.

To this end, let t∈[0, T] and (ti)i∈Nwith ti→tbe such that a1(ti, x, z)→a1(t, x, z) as

i→ ∞ for almost every x∈ D and every z∈R. An application of H¨older’s inequality yields

hA1(ti)v−A1(t)v, wiV∗

1×V1=ZDa1(ti,·, v)−a1(t, ·, v)wdx

≤ZD

|a1(ti,·, v)−a1(t, ·, v)|qdx

qkwkV1

for every v, w ∈V1and t∈[0, T ]. Similarly to (3.64), we can obtain |a1(ti,·, v)−a1(t, ·, v)|q

is bounded by a function that is integrable on D. Then we can apply Lebesgue’s dominated

convergence theorem and it follows that

lim

i→∞ kA1(ti)v−A1(t)vkV∗

1= lim

i→∞ ZD

|a1(ti,·, v)−a1(t, ·, v)|qdx

q= 0

for every v∈V1and t∈[0, T ].

In order to prove that A1(t): V1→V∗

1,t∈[0, T ], is radially continuous, let (si)i∈Nbe

a convergent sequence in [0,1] with the limit s∈[0,1]. As (3.64) is finite, it follows that

a1(t, ·, v +siw)wis bounded by an integrable function on D. Thus, we can apply Lebesgue’s

dominated convergence theorem and it follows that

lim

i→∞hA1(t)(v+siw), wiV∗

1×V1= lim

i→∞ ZD

a1(t, ·, v +siw)wdx

=ZD

lim

i→∞ a1(t, ·, v +siw)wdx=ZD

a1(t, ·, v +sw)wdx

for every v, w ∈V1and t∈[0, T] due to Assumption 3.3.1 (1). The monotonicity condition

for A1(t), t∈[0, T ], from Assumption 3.1.2 (3) is a consequence of Assumption 3.3.1 (2).

3.3. EXAMPLE: A NONLINEAR PARABOLIC PROBLEM 87

Here, we see that

hA1(t)v−A1(t)w, v −wiV∗

1×V1=ZD

(a1(t, ·, v)−a1(t, ·, w))(v−w) dx≥0

for v, w ∈V1and t∈[0, T]. In order to verify the coercivity condition from Assump-

tion 3.1.2 (5), we apply Assumption 3.3.1 (3) to see that

hA1(t)v, viV∗

1×V1≥ZDd3|v|p−d4dx=d3|v|p

V1− kd4kL1(D).

The proof that A2(t): V2→V∗

2is well-defined and fulfills Assumption 3.1.2 (1)–(5) can be

done analogously to A1(t), t∈[0, T ]. The functions v, w ∈V1just have to be replaced by

∇v, ∇wfor v, w ∈V2.

In order to see that B(t), t∈[0, T ], fulfills Assumption 3.1.5, we notice that the con-

tinuity of t7→ B(t)valmost everywhere in (0, T) for every v∈His a consequence of

Assumption 3.3.4 (1) and can be proved analogously to the corresponding condition for

t7→ A1(t)v,v∈V1. The Lipschitz condition from Assumption 3.1.5 (2) is fulfilled as

(B(t)v1−B(t)v2, w)H=ZDb(t, ·, v1)−b(t, ·, v2)wdx

≤ZD

e1|v1−v2||w|dx≤e1kv1−v2kHkwkH

holds true for all v1, v2, w ∈Hand t∈[0, T ]. Therefore,

kB(t)v1−B(t)v2kH≤e1kv1−v2kH

is fulfilled for all v1, v2∈Hand t∈[0, T ]. In a similar fashion, we can show Assump-

tion 3.1.5 (3) since

(B(t)0, w)H=ZD

b(t, ·,0)wdx≤ZD

|e2||w|dx≤ ke2kL2(D)kwkH

is fulfilled for every w∈Hand t∈[0, T] because of Assumption 3.3.4 (3). This implies

kB(t)0k≤ke2kL2(D)and t∈[0, T ]. A combination of the two inequalities also shows that

the operator is indeed well-defined. Also the space V2=W1,p

0(Ω) is compactly embedded

into H=L2(Ω), compare [1, Theorem 6.3].

Therefore, for t∈[0, T ] the operators A1(t), A2(t), A(t) = A1(t) + A2(t), and B(t) fulfill

all the necessary conditions and we can apply Theorem 3.1.18 and Theorem 3.1.19.

Moreover, if the stronger monotonicity condition from Assumption 3.3.2 is fulfilled, we

see that (3.42) is satisfied for A2(t), t∈[0, T ], since

hA2(t)v−A2(t)w, v −wiV∗

2×V2=ZD

(a2(t, ·,∇v)−a2(t, ·,∇w)) ·(∇v− ∇w) dx

≥d5ZD

|∇v− ∇w|pdx=d5kv−wkp

for v, w ∈V2and t∈[0, T]. Thus, the sequence of piecewise constant prolongations

(¯

Uk)k>0= ( ¯

2)k>0converges strongly to the exact solution uin Lp(0, T ;V2) = Lp(0, T;V)

as k→0 due to Theorem 3.1.19.

88 CHAPTER 3. OPERATOR SPLITTING

It remains to verify that the results from Section 3.2 are also applicable to the nonlinear

parabolic problem. In order to obtain explicit error bounds, we need to make additional

assumptions on the mappings am,m∈ {1,2}, and on the exact solution uof (3.63). At

this point, we do not explain how the additional regularity of ucan be obtained. For more

information about additional regularity and some examples, see Section 1.2.

Theorem 3.3.6. Let all the assumptions from Theorem 3.3.5 be fulfilled and consider the

same scheme. In addition, let pbe either an element of [2,∞)∩[d, ∞)or 2,2d−p

d−p∩[2, d),

where dis the dimension of the underlying space Rd⊃ D. Assume that a1and a2fulfill

Assumptions 3.3.2 and 3.3.3 and let the nonnegative functions d2and e2from the bounded-

ness condition for a1and bin Assumption 3.3.1 (3) and Assumption 3.3.4 (3) be elements

of Lp(D). For α∈(0,1], let the solution uof (3.63) be an element of C0,α([0, T ]; V).

Then the scheme obtains a unique solution (Un)n∈{1,...,N}in Hand there exists a con-

stant C∈(0,∞)such that

max

n∈{1,...,N}ku(tn)−Unk2

H+k

n=1

ku(tn)−Unkp

V≤Ckαq

is fulfilled for every step size k, which is small enough.

Proof. In order to prove this theorem, we will apply Theorem 3.2.3. In the proof of Theo-

rem 3.3.5 we have already seen that A1(t), A2(t), and B(t) fulfill Assumptions 3.1.2, 3.1.3,

and 3.1.5 for t∈[0, T ]. The stronger monotonicity condition from (3.48) is fulfilled as an

application of Assumption 3.3.2 yields

hAm(t)v−Am(t)w, v −wiV∗

m×Vm≥d5kv−wkp

for m∈ {1,2}.

Next, we use Assumption 3.3.3 to prove the bounded Lipschitz condition for A1(t),

t∈[0, T ], from (3.47). Here, we consider two cases, at first we prove the condition for p= 2

and after that for p∈(2,∞). In the case p= 2, we have a global Lipschitz condition.

Inserting the definition of A1(t), t∈[0, T ], we obtain that

hA1(t)v1−A1(t)v2, wiV∗

1×V1=ZDa1(t, ·, v1)−a1(t, ·, v2)wdx

≤2d6ZD

|v1−v2||w|dx≤2d6kv1−v2kV1kwkV1

and therefore

kA1(t)v1−A1(t)v2kV∗

1≤2d6kv1−v2kV1

for every v1, v2, w ∈V1and t∈[0, T ]. For p∈(2,∞), we obtain a Lipschitz constant, which

depends on the inserted functions. Thus, an additional application of Lemma A.1.3 becomes

necessary to obtain

hA1(t)v1−A1(t)v2, wiV∗

1×V1

=ZDa1(t, ·, v1)−a1(t, ·, v2)wdx

≤d6ZD1 + max |v1|p−2,|v2|p−2|v1−v2||w|dx

≤d6k1kL

p−2(D)+ZD

max |v1|p,|v2|pdxp−2

pkv1−v2kLp(D)kwkLp(D)

3.3. EXAMPLE: A NONLINEAR PARABOLIC PROBLEM 89

for every v1, v2, w ∈V1and t∈[0, T ]. Since p−2

p∈(0,1) for p∈(2,∞), it follows that

ZD

max |v1|p,|v2|pdxp−2

p≤ZD

|v1|pdx+ZD

|v2|pdxp−2

≤ZD

|v1|pdxp−2

p+ZD

|v2|pdxp−2

=kv1kp−2

Lp(D)+kv2kp−2

Lp(D)≤2 max kv1kp−2

V1,kv2kp−2

V1.

Thus, for R∈(0,∞) and all v1, v2∈V1with kv1kV1,kv2kV1≤R, we obtain the bound

kA1(t)v1−A1(t)v2kV∗

1≤d6k1kL

p−2(D)+ 2 max kv1kp−2

V1,kv2kp−2

V1kv1−v2kV1

=: L(R)kv1−v2kV1,

which proves (3.47). An analogous bound for A2(t), t∈[0, T], can be proved by replacing

v1,v2,wwith their gradient ∇v1,∇v2,∇w∈Lp(D)d.

In order to prove the required regularity conditions from Theorem 3.2.3, recall that

by assumption uis an element of C0,α([0, T ]; V). The function U2defined in Assump-

tion 3.2.1 coincides with uin C0,α([0, T ]; V). Thus, it follows by the regularity assumption

u∈C0,α([0, T]; V)

kU2(t)−u(tn−1)kV2=ku(t)−u(tn−1)kV≤ |u|C0,α([0,T ];V)|t−tn−1|α

for every t∈(tn−1, tn], n∈ {1, . . . , N}. The function U1defined in Assumption 3.2.1 is

given by

U1(t) = u(tn−1)−Zt

tn−1A1(s)u(s) + B(s)u(s)ds

for every t∈(tn−1, tn], n∈ {1, . . . , N}. We then obtain that

kU1(t)−u(tn−1)kp

V1≤ |t−tn−1|p−1Zt

tn−1

kA1(s)u(s) + B(s)u(s)kp

Lp(D)ds,

where the integrand can be bounded by

kA1(s)u(s) + B(s)u(s)kLp(D)

≤ ka1(s, ·, u(s))kLp(D)+kb(s, ·, u(s))kLp(D)

≤d1k|u(s)|p−1kLp(D)+kd2kLp(D)+e1ku(s)kLp(D)+ke2kLp(D)(3.65)

for every s∈[0, T ]. In order to prove that the last row is finite, we use the fact that d2, e2∈

Lp(D) and the Sobolev embedding theorem, compare [1, Theorem 4.12]. If p∈[d, ∞), then

V=W1,p

0(D) is continuously embedded into Lr(D) for every r∈[p, ∞). Thus, for u(s)∈V

it follows that k|u(s)|p−1kLp(D)and ku(s)kLp(D)are finite. If p∈[2, d), then the space V

is continuously embedded into Lr(D) for r∈p, dp

d−p. As we assume that p∈2,2d−p

d−pin

this case, it follows that

p(p−1) ≤p2d−p

d−p−1=p2d−p

d−p−d−p

d−p=dp

d−p

and therefore the terms containing u(s) in (3.65) are finite. This shows that

kU1(t)−u(tn−1)kV1≤c2|t−tn−1| ≤ c2T1−α|t−tn−1|α

for c2∈(0,∞), which does not depend on the step size k.

90 CHAPTER 3. OPERATOR SPLITTING

Appendix A

Appendix

A.1 Useful Inequalities

In the following, we collect a few inequalities and identities that appear throughout the

analysis in the chapters above. We begin by a discrete Gronwall lemma.

Lemma A.1.1. Let (un)n∈Nand (bn)n∈Nbe two nonnegative sequences that satisfy, for

given a∈[0,∞)and N∈N, that

un≤a+

n−1

i=1

biui, n ∈ {1, . . . , N}.

Then it follows that

un≤aexp n−1

i=1

bi, n ∈ {1, . . . , N}.

In the case that a sum P0

i=1 appears, we use the convention that it is zero. A proof

of this lemma can be found in [22]. In some settings, it is possible to use the following

inequality as an alternative.

Lemma A.1.2. Let a, b, x ∈[0,∞)be given such that x2≤2ax +b2is fulfilled. Then it

also follows that x≤2a+b.

Proof. Since x2≤2ax +b2is fulfilled, it follows that

(x−a)2=x2−2ax +a2≤a2+b2.

Taking the square root on both sides, this yields

|x−a| ≤ pa2+b2≤a+b.

As x−a≤ |x−a|is fulfilled, we obtain the desired bound after adding ato both sides of

the inequality.

The following lemma is a H¨older type inequality. We omit the proof, it can be done by

an inductive application of the ordinary H¨older inequality.

92 APPENDIX A. APPENDIX

Lemma A.1.3. Let D ⊆ Rd,d≥1, and N∈Nbe given. Then for pn∈[1,∞],n∈

{1, . . . , N}, such that PN

n=1 1

pn= 1 it follows that

n=1

|un|dx≤

n=1

kunkLpn(D)

for un∈Lpn(D).

Last, we state an identity, which appears several times throughout the analysis. Again,

we omit the proof. It can be done by rewriting the right-hand side of the equality with the

definition of the norm via the inner product.

Lemma A.1.4. Let (H, (·,·)H,k·kH)be a real Hilbert space. The identity

(v−w, v)H=1

2kvk2

H− kwk2

H+kv−wk2

H

is fulfilled for every v, w ∈H.

A.2 Bochner Integral

We shortly recall the main statements for Bochner integrable functions on a general measure

space. For a complete introduction, we refer the reader to [115, Chapter V, Section 4–5],

[30, Chapter II.2], [96, Section 4.2], and [100, Kapitel 2].

In the following, we assume that (X, k·kX) is a real Banach space and (Ω,F, µ) is a finite

measure space. Then we consider an abstract function v: Ω →Xand call it simple if for a

finite number N∈Nthere exist x1, . . . , xN∈Xand mutually disjoint sets C1, . . . , CN∈ F

such that v=PN

n=1 xnχCn, where χCnis the characteristic function with respect to the

set Cn,n∈ {1,...N}. We call a function vBochner measurable if there exists a sequence

(vn)n∈Nof simple functions such that vn(ω)→v(ω) in Xas n→ ∞ for almost every

ω∈Ω. A function v: Ω →Xis called weakly measurable if ω7→ hf, v(ω)iX∗×Xis Lebesgue

measurable for every f∈X∗. Under suitable assumptions, these two concepts are equivalent

as we see in the next theorem.

Theorem A.2.1. Let (Ω,F, µ)be a finite measure space and let (X, k · kX)be a real,

separable Banach space. For a function v: Ω →X, the following statements are equivalent:

(a) The function vis Bochner measurable.

(b) The function vfulfills that v−1(C)∈ F for all open sets C⊆X.

This theorem is a consequence of Pettis theorem. A proof can be found in [96, The-

orem 4.2.4]. We are foremost interested in functions that are also Bochner integrable.

A Bochner measurable function v: Ω →Xis called Bochner integrable if there exists a

sequence (vn)n∈Nof simple functions such that vn(ω)→v(ω) in Xas n→ ∞ for al-

most every ω∈Ω and for every ε > 0 there exists N∈Nsuch that for all n, m ≥N

the inequality RΩkvn−vmkXdµ < ε is fulfilled. The integral of vis then given by

A.2. BOCHNER INTEGRAL 93

RΩvdµ= limn→∞ RΩvndµ. Moreover, for p∈[1,∞] we introduce the space Lp(Ω; X)

that consists of all Bochner measurable functions v: Ω →Xsuch that

kvkLp(Ω;X)=







ZΩ

kvkp

Xdµ

p, p ∈[1,∞),

ess sup

Ω

kvkX, p =∞(A.1)

is finite. This function space fulfills the following properties.

Lemma A.2.2. Let (Ω,F, µ)be a finite measure space and let (X, k·kX)be a real Banach

space. Then the following properties are fulfilled:

(a) The space Lp(Ω; X)equipped with the norm given in (A.1) is a Banach space for every

p∈[1,∞].

(b) The set of simple X-valued functions is dense in Lp(Ω; X)for p∈[1,∞).

(d) If Xis reflexive, then the space Lp(Ω; X)is reflexive for every p∈(1,∞).

(e) If Xis continuously embedded into (Y, k·kY), then Lq(Ω; X)is continuously embedded

into Lp(Ω; Y)for p, q ∈[1,∞]and p≤q.

These statements can be found in [30] and [96, Proposition 2.3.24, Proposition 4.2.22].

Furthermore, if Xis continuously and densely embedded into a space Y, then the simple X-

valued functions are also dense in Lq(Ω; Y) for every q∈[1,∞). Thus, the space Lp(Ω; X) is

continuously and densely embedded into Lq(Ω; Y) for q∈[1,∞) and p∈[q, ∞]. Moreover, if

Xhas the Radon–Nikodym property, it follows that Lp(Ω; X)∗=Lq(Ω; X∗) for p∈[1,∞)

and q=p

p−1. Note that the Radon–Nikodym property is fulfilled if, for example, Xis

reflexive or separable, compare [96, Theorem 4.2.25 and 4.2.26].

Lemma A.2.3. Let (Ω,F, µ)be a finite measure space, let (X, k · kX)be a real Banach

space, and let p, r ∈[1,∞]be such that p < r. For a sequence (vn)n∈Nthat is bounded in

Lr(Ω; X)and v∈Lp(Ω; X)such that vn→vin Lp(Ω; X)as n→ ∞, it follows that vn→v

in Lq(Ω; X)as n→ ∞ for every q∈[p, r).

Proof. Since 1

q∈(1

r,1

p] there exists θ∈[0,1) such that 1

q=θ

r+1−θ

p. Choosing α=r

θq and

˜α=p

(1−θ)q, it follows that 1

α+1

˜α= 1 and we can apply H¨older’s inequality to

kwkq

Lq(Ω;X)=ZΩ

kwkθq

Xkwk(1−θ)q

Xdµ

≤ZΩ

kwkr

Xdµ

αZΩ

kwkp

Xdµ

˜α=kwkθq

Lr(Ω;X)kwk(1−θ)q

Lp(Ω;X)

for w∈Lr(Ω; X). Then we obtain

kvn−vmkLq(Ω;X)≤ kvn−vmkθ

Lr(Ω;X)kvn−vmk1−θ

Lp(Ω;X)→0 as m, n → ∞,

since the sequence (vn)n∈Nis bounded in Lr(Ω; X) and convergent in Lp(Ω; X). This shows

that (vn)n∈Nis a Cauchy sequence in Lq(Ω; X) and therefore convergent to vdue to the

uniqueness of the limit.

A proof of the next theorem can be found in [96, Proposition 4.2.12].

94 APPENDIX A. APPENDIX

Theorem A.2.4. Let (Ω,F, µ)be a finite measure space and let (X, k·kX)be a real Banach

space. Further, let v: Ω →Xbe a Bochner measurable function. Then vis Bochner

integrable if and only if kvkX: Ω →Ris integrable.

In the special case that Ω = [0, T ] equipped with the Lebesgue sets and the Lebesgue

measure, we can state the lemma of Lions–Aubin. This gives us a compact embedding

argument for Bochner integrable functions. We refer the reader to [99, Lemma 7.7] for a

proof of this statement.

Lemma A.2.5. Let X−1,X0, and X1be real Banach spaces such that X1,→X0,→X−1,

X1is separable, reflexive, and compactly embedded into X0, and X−1is reflexive. For

p∈(1,∞)and q∈[1,∞], the space

W={v∈Lp(0, T ;X1) : v0exists and v0∈Lq(0, T;X−1)}

is compactly embedded into Lp(0, T ;X0).

A.3 Stochastic Background

As we are dealing with a randomized scheme in Chapter 2, we will give a short overview of

the probabilistic results needed. For more details, we refer the reader to [73].

In the following, let (Ω,F,P) be a probability space. For a real, separable Banach space

(X, k·kX), we call a mapping U: Ω →Xa random variable if it is measurable with respect

to the σ-algebra Fand the Borel σ-algebra B(X) in X. Precisely, this means that for every

C∈ B(X) the set

U−1(C) = {ω∈Ω : U(ω)∈C} ⊆ Ω

is an element of F. The expectation, i.e., the integral of a random variable Uwith respect

to the measure Pis denoted by

E[U] = ZΩ

U(ω) dP(ω).

In our theory, we often work with random variables ξ: Ω →[a, b], a, b ∈R,a < b, that are

uniformly distributed. For such a mapping, the density is given by 1

b−aand a substitution

yields that

E[v(ξ)] = ZΩ

v(ξ(ω)) dP(ω) = 1

b−aZb

v(t) dt, (A.2)

where v: [a, b]→Xis a Bochner integrable function. Note that the theory from the previous

section applies as every probability space is, in particular, a measurable space.

Within the theory of Monte Carlo algorithms, it will be important to consider indepen-

dent random variables. To this end, let us recall the concept of independence. We call a

family (Cn)n∈Nof elements in Findependent if for every finite subset I⊂N

P\

n∈I

Cn=Y

n∈I

P(Cn) (A.3)

is fulfilled. Similarly, we call a family (Fn)n∈Nof σ-algebras independent if again for every

finite subset of I⊂Nand (Cn)n∈Isuch that Cn∈ Fn,n∈I, it follows that the family

A.3. STOCHASTIC BACKGROUND 95

(Cn)n∈Iis independent in the sense of (A.3). This at hand, we can now transfer the concept

of independence to a family (Un)n∈I,I⊂Nand finite, of random variables. If the generated

σ-algebras

σ(Un) = {U−1

n(C) : C∈ B(X)}, n ∈I, (A.4)

are independent, we also call (Un)n∈Iindependent.

In some settings, it can be important to consider a certain family (Fn)n∈Nof σ-algebras.

It is called a filtration if Fnis a subset of Fand Fmfor every n, m ∈Nwith n≤m.

A random variable U: Ω →Xcan be measurable with respect to Fmbut not with

respect to Fnfor n < m,n, m ∈N. It can then be helpful to consider the conditional

expectation E[U|Fn]: Ω →Xof Uwith respect to Fn. More precisely, the Fn-measurable

mapping E[U|Fn] is uniquely determined by the relation

E[UχC] = E[E[U|Fn]χC]

for every C∈ Fn. A useful property of the conditional expectation is the tower property.

This states that for two σ-algebras Fnand Fmof the filtration (Fn)n∈Nwith n≤mwe

obtain that

E[E[U|Fn]|Fm] = E[E[U|Fm]|Fn] = E[U|Fn].

If the random variable Uis measurable with respect to Fn, then the conditional expectation

is the function itself, i.e., E[U|Fn] = U. Furthermore, if σ(U) is independent of Fn, we

obtain that E[U|Fn] = E[U].

96 APPENDIX A. APPENDIX

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