# A domain decomposition method for stochastic evolution equations

Evelyn Buckwar  Evelyn Buckwar
Institute of Stochastics
Johannes Kepler University
Altenberger Straße 69, 4040 Linz, Austria<EMAIL_ADDRESS>,  Ana
Djurdjevac  Ana Djurdjevac
Freie Universtät Berlin
Arnimallee 6
14195 Berlin, Germany<EMAIL_ADDRESS>and  Monika Eisenmann
Monika Eisenmann
Centre for Mathematical Sciences
Lund University
P.O. Box 118
221 00 Lund, Sweden<EMAIL_ADDRESS>
###### Abstract.

In recent years, SPDEs have become a well-studied field in mathematics. With
their increase in popularity, it becomes important to efficiently approximate
their solutions. Thus, our goal is a contribution towards the development of
efficient and practical time-stepping methods for SPDEs.

Operator splitting schemes are a powerful tool for deterministic and
stochastic differential equations. An example is given by domain decomposition
schemes, where we split the domain into sub-domains. Instead of solving one
expensive problem on the entire domain, we deal with cheaper problems on the
sub-domains. This is particularly useful in modern computer architectures, as
the sub-problems may often be solved in parallel. While splitting methods have
already been used to study domain decomposition methods for deterministic
PDEs, this is a new approach for SPDEs.

We provide an abstract convergence analysis of a splitting scheme for
stochastic evolution equations and state a domain decomposition scheme as an
application of the setting. The theoretical results are verified through
numerical experiments.

###### Key words and phrases:

Stochastic partial differential equations, operator splitting, domain
decomposition.

###### 2010 Mathematics Subject Classification:

65C30, 90C15, 65M55

Part of this research took place while E. Buckwar was in Lund as Lise-Meitner
Guest Professor. A. Djurdjevac’s research has been partially supported by
Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 Scaling Cascades
in Complex Systems, Project Number 235221301, Project C10. M. Eisenmann has
been partially supported by Vetenskapsrådet (VR) through Grant No. 2023-03930.
The computations were enabled by resources provided by LUNARC

##  1\. Introduction

Modeling with partial differential equations (PDEs) is ubiquitous in many
areas of natural sciences and engineering. Phenomena described by PDEs include
models of vibrations or waves, flows of liquids or the distribution of heat.
In all such models, one may take uncertainty of parameters, natural
variability of evolution or extrinsic noise into account, arriving at random
PDEs (RPDEs) or stochastic PDEs (SPDEs). In the former case, only parameters
or initial or boundary data are described by random variables or processes,
whereas in the latter case the new type of PDEs involves a driving noise,
often an infinite dimensional Wiener process. Applications employing SPDEs
include neuroscience [11, 34], micro-ferro-magnetism [4, 6], reservoir
engineering and geo-statistical climate models [17, 28]. Design and analysis
of numerical methods to approximate solutions of SPDEs or quantities of
interest constitute an active area of research, we refer to early articles
[20, 22, 35], and the book [30]. However, compared to the level of theoretical
and practical development of numerical methods for PDEs, this area is still in
its infancy. In this article, we are in particular interested in the topic of
domain decomposition, which is a standard tool for the parallelisation of
numerical algorithms and thus speed-up, as well as treatment of multi-physics
description systems. To the best of our knowledge, for SPDEs this topic is
only addressed in [8, 23, 39]. While in [39] the problem considered differs
from our work as it is only concerned with stationary models, the papers [8,
23] also consider time-dependent problems. However, their underlying equations
are either less general, in [8] the stochastic heat equation is considered, or
in [23] they consider the nonlinear Schrödinger equation which is of a
different kind. In contrast to the very few works on domain decomposition
approaches for SPDEs, there exist several articles treating RPDEs, e.g., [10,
24].

As stated in [31], domain decomposition methods can be described as a class of
divide-and-conquer methods for the numerical approximation of solutions of
PDEs. A standard approach for a decomposition framework, called hybrid
formulation in [31], consists of the following steps. First the domain of the
PDE is decomposed into (overlapping or non-overlapping) sub-domains, then, on
each sub-domain, a local PDE problem is formulated and neighbouring local
problems are coupled by matching conditions. Each local problem must be well-
defined and uniquely solvable, as well as consistent with the global PDE
problem. Overall, the coupled system of PDEs is required to be equivalent to
the original problem and the global solution can be found by summing up the
local solutions using an appropriate partition of unity. Variations of this
setting include Schwarz iterative methods or Lagrange Multiplier methods in
the context of optimization problems. This approach works directly for time-
independent PDEs, while in the case of parabolic PDEs, usually the first step
is to apply an implicit time discretization method. For each time-step one
obtains a time-independent problem, to which a domain decomposition method can
be applied.

Another divide-and-conquer approach to solving differential equations (usually
time-dependent problems) is a splitting method, we refer to [19, 32] for
overviews in the deterministic case and very exemplary to [1, 3, 7, 36] for
the case of stochastic differential equations.

Domain decomposition operator splittings, which were introduced already in
1989 by Vabishchevich [37] and called regionally additive schemes there, have
been discussed recently in various contexts, see [13, 14, 15, 18, 21]. The
main idea, in contrast to the approach to domain decomposition described
above, is to first decompose or split the differential operators and right-
hand side of the PDE into a sum of component terms, where each one is defined
on a sub-domain. Therefore, each sub-operator represents the vector field of
the global PDE on its corresponding sub-domain. These split operators can then
directly be incorporated in the time-integration through an operator
splitting. Thus, an advantage of this formulation is that the domain
decomposition and its corresponding error is directly combined in the time-
integration strategy. In contrast to this, the previous approach only uses the
domain decomposition as a black box method to solve elliptic equations that
arise in every time step after the application of a time-stepping method. In
this process of solving the arising elliptic equations, a domain decomposition
method is usually used as an iterative method that can require additional
computational costs due to the iterations. These additional iterations do not
appear in our approach.

The purpose of this article is to extend this approach to certain class of
stochastic evolution equations for the first time. In particular, we provide
the theoretical background and analysis for a domain decomposition operator
splitting applied to a stochastic evolution equation

$\displaystyle\begin{cases}\,\mathrm{d}u(t)=\tilde{A}(t,u(t))\,\mathrm{d}t+B(t,u(t))\,\mathrm{d}W(t),\quad
t\in(0,T],\\\ u(0)=u_{0}.\end{cases}$

More precisely, we are considering an abstract Cauchy problem with nonlinear
monotone and coercive drift term, additional Lipschitz nonlinear term and
multiplicative $Q$-Wiener noise. The equation is studied in the variational
context which is well-suited for our numerical analysis, in particular for
deriving a priori bounds. In order to obtain the final error bound for the
method, we require additional regularity of the solution. More precisely, we
need the solution to be Hölder continuous in time. While this cannot be
guaranteed for every admissible solution in our framework, we can state one
slightly more restrictive setting where the needed regularity is available in
the existing literature. Note that these additional assumptions are not
needed, if the Hölder regularity of the solution can be obtained by other
means. The considered semi-discrete operator splitting scheme involves a
decomposition of the space into sub-spaces and splitting all the operators on
these sub-spaces. On every sub-space, the obtained split Cauchy problem is
discretized in time using an implicit-explicit (IMEX) Euler–Maruyama method.

For the method, we prove well-posedness of the stated implicit equations. The
path-wise well-posedness follows directly from standard PDE theory and we also
provide the measurability of the obtained solution. As a particular example of
our scheme, we consider a domain decomposition method. Based on the derived
bound of the semi-discrete solution, we provide mean-square convergence of our
numerical method. In the last part of the paper, we illustrate the obtained
convergence with numerical experiments for a two-dimensional stochastic heat
equation. Here, we consider both a linear equation with a multiplicative noise
term and a semi-linear equation with additive noise.

The paper is structured as follows. In Section 2, we state the stochastic
evolution equation considered in this paper with all the needed assumptions.
Moreover, we specify the solution concept and state a restricted setting where
we can provide addtional regualrity of the solution. With this at hand we can
then state the semi-discretized scheme in Section 3. More precisely, we give a
detailed explanation about the setup of the scheme in Section 3.1, prove the
well-posedness of the implicit equation in Section 3.2 and conclude the
section with a our main application for the splitting which is a domain
decomposition of an SPDE. The main theoretical result of this paper can be
found in Section 4. While we begin with some technical details needed for the
error analysis in the first part of the section, we state our error bound in
Section 4.3. This error bound is exemplified through a numerical experiment in
Section 5. At the end of the paper, in Appendix A, we collect some further
auxiliary results needed in the analysis throughout this manuscript and state
the details of the proof of a stated lemma.

##  2\. Analytical background

In this section, we formally state the equation and gather the main
assumptions concerning the setting we are interested in. In the following, let
$T\in(0,\infty)$ be a given final time point. We then denote the underlying
filtered probability space by
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in[0,T]},\mathcal{P})$, which
satisfies the usual conditions, and we assume that the $Q$-Wiener process $W$
is adapted to the filtration $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$. We then
consider the stochastic abstract Cauchy problem:

(2.1)
$\displaystyle\begin{cases}\,\mathrm{d}u(t)=\tilde{A}(t,u(t))\,\mathrm{d}t+B(t,u(t))\,\mathrm{d}W(t),\quad
t\in(0,T],\\\ u(0)=u_{0},\end{cases}$

where the operator $\tilde{A}(t,\cdot)$ consits of three different parts

$\displaystyle\tilde{A}(t,\cdot)=F(t,\cdot)+G(t)-A(t,\cdot)$

that we will explain in more detail in the following subsection. Decomposing
the operator $\tilde{A}(t,\cdot)$ into these three parts provides a large
degree of freedom for the numerical scheme explained in Section 3.1. We will
use the different parts of the operator to propose an implicit-explicit (IMEX)
time-stepping method in combination with an operator splitting that can be
used to describe a domain decomposition method.

###  2.1. Assumptions

Let $(H,(\cdot,\cdot)_{H},\|\cdot\|_{H})$ be a real, separable Hilbert space
and let $(V,\|\cdot\|_{V})$ be a real, separable, reflexive Banach space such
that $V$ is continuously and densely embedded into $H$. In a standard way, we
obtain the Gelfand triple $V\stackrel{{\scriptstyle
d}}{{\hookrightarrow}}H\cong H^{*}\stackrel{{\scriptstyle
d}}{{\hookrightarrow}}V^{*}$. For $f\in V^{*}$ and $v\in V$ we denote the dual
paring $\langle f,v\rangle_{V^{*}\times V}=f(v)$, which extends the $H$-inner
product. In the following, we state the exact assumptions needed on the
coefficients in (2.1).

######  Assumption 1.

The initial data $u_{0}\colon\Omega\to H$ is $\mathcal{F}_{0}$-measurable and
belongs to $L^{2}(\Omega;H)$.

######  Assumption 2.

Let the nonlinear operator $A\colon[0,T]\times V\times\Omega\to V^{*}$ be
progressively measurable. By writing $A(t,v)$ the map $\omega\mapsto
A(t,v,\omega)$ is meant. Moreover, the following conditions hold:

  * (i)

The operator $A(t,\cdot)$ is monotone and coercive uniformly in $t$, i.e.
there exist constants $\eta,\mu\in(0,\infty)$ such that

$\displaystyle\langle A(t,v)-A(t,w),v-w\rangle_{V^{*}\times
V}\geq\eta\|v-w\|_{V}^{2},\quad\langle A(t,v),v\rangle_{V^{*}\times
V}\geq\mu\|v\|_{V}^{2}$

almost surely for all $t\in[0,T]$, $v,w\in V$.

  * (ii)

The operator $A(t,\cdot)$ is bounded uniformly at $0\in V$, $\nu_{A}$-Hölder
continuous with respect to the first argument and Lipschitz continuous with
respect to the second argument, i.e. there exist constants
$\nu_{A}\in(0,\frac{1}{2}]$ and $K_{A},L_{A}\in(0,\infty)$ such that

$\displaystyle\|A(t,0)\|_{V^{*}}\leq
K_{A},\quad\|A(t,v)-A(\tau,w)\|_{V^{*}}\leq
L_{A}\big{(}|t-\tau|^{\nu_{A}}+\|v-w\|_{V}\big{)}$

almost surely for $t,\tau\in[0,T]$ and $v,w\in V$.

######  Assumption 3.

Let the nonlinear operator $F\colon[0,T]\times H\times\Omega\to H$ and the
function $G\colon[0,T]\times\Omega\to V^{*}$ be progressively measurable.
Again, by writing $F(t,v)$ and $G(t)$ the maps $\omega\mapsto F(t,v,\omega)$
and $\omega\mapsto G(t,\omega)$ are meant. Moreover, the operator $F(t,\cdot)$
is bounded uniformly at $0\in H$, $\nu_{F}$-Hölder continuous with respect to
the first variable and Lipschitz continuous with respect to the second
variable, i.e. there exist constants $\nu_{F}\in(0,\frac{1}{2}]$ and
$K_{F},L_{F}\in[0,\infty)$ such that

$\displaystyle\|F(t,0)\|_{H}\leq K_{F},\quad\|F(t,v)-F(\tau,w)\|_{H}\leq
L_{F}\big{(}|t-\tau|^{\nu_{F}}+\|v-w\|_{H}\big{)}$

almost surely for $t,\tau\in[0,T]$ and $v,w\in H$. Moreover, there exists
$\nu_{G}\in(0,\frac{1}{2}]$ such that the function $G$ is an element of
$C^{\nu_{G}}([0,T];L^{2}(\Omega;V^{*}))$.

In the following, we denote the space of linearly bounded operators from a
Hilbert space $(H_{1},(\cdot,\cdot)_{H_{1}},\|\cdot\|_{H_{1}})$ into a second
Hilbert space $(H_{2},(\cdot,\cdot)_{H_{2}},\|\cdot\|_{H_{2}})$ by
$\mathcal{L}(H_{1},H_{2})$. The space is equipped with the norm
$\|S\|_{\mathcal{L}(H_{1},H_{2})}=\sup_{\|v\|_{H_{1}}=1}\|Sv\|_{H_{2}}$.
Moreover, by $\mathcal{HS}(H_{1},H_{2})$ we denote the Hilbert space of
Hilbert–Schmidt operators equipped with the norm
$\|S\|_{\mathcal{HS}(H_{1},H_{2})}=\sqrt{\text{tr}(S^{*}S)}$, where tr is the
trace operator.

######  Assumption 4.

Let $Q\in\mathcal{L}(H)$ be a self-adjoint, positive operator with finite
trace whose eigenfunctions $\\{\psi_{k}\\}_{k\in\mathbb{N}}$ form an
orthonormal basis of $H$. The corresponding eigenvalues of $Q$ are denoted by
$\\{q_{k}\\}_{k\in\mathbb{N}}$. With a family
$\\{\beta_{k}\\}_{k\in\mathbb{N}}$ of independent scalar Brownian motions on
the probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in[0,T]},\mathcal{P})$, the
$Q$-Wiener noise can be stated through

$\displaystyle
W(t)=\sum_{k=1}^{\infty}q_{k}^{\frac{1}{2}}\psi_{k}\beta_{k}(t),$

almost surely for $t\in[0,T]$.

######  Assumption 5.

For $L_{2}^{0}:=\mathcal{HS}(Q^{\frac{1}{2}}(H),H)$, let the
$B\colon[0,T]\times H\times\Omega\to L_{2}^{0}$ be progressively measurable.
Again, by writing $B(t,v)$ the map $\omega\mapsto B(t,v,\omega)$ is meant.
Moreover, the operator $B(t,\cdot)$ is bounded uniformly at $0\in H$,
$\nu_{B}$-Hölder continuous with respect to the first variable and Lipschitz
continuous with respect to the second variable, i.e. there exist constants
$\nu_{B}\in(0,\frac{1}{2}]$, $K_{B},L_{B}\in[0,\infty)$ such that

$\displaystyle\|B(t,0)\|_{L_{2}^{0}}\leq
K_{B},\quad\|B(t,v)-B(\tau,w)\|_{L_{2}^{0}}\leq
L_{B}\big{(}|t-\tau|^{\nu_{B}}+\|v-w\|_{H}\big{)},$

almost surely for $t,\tau\in[0,T]$ and $v,w\in H$.

###  2.2. Solution concept and additional regularity

With the exact assumptions at hand, we can state that the stochastic evolution
equation (2.1) has a unique variational solution. Moreover, to prove our error
bounds, we also need additional Hölder regularity of the solution. While the
numerical scheme is well-defined in the broader framework explained above,
this setting does not guarantee the needed regularity for the error bounds. We
can provide the additional regularity requirements for a subclass which we
explain later in this subsection.

Let us first recall what is meant by a variational solution, as stated for
example in [29]. An $H$-valued adapted continuous process
$\\{u(t)\\}_{t\in[0,T]}$ is a variational solution to Equation (2.1) on
$[0,T]$ if

(2.2) $\displaystyle\int_{0}^{T}\|u(t)\|^{2}_{V}\,\mathrm{d}t<\infty$

and

$(u(t),v)_{H}=(u_{0},v)_{H}+\int_{0}^{t}\langle\tilde{A}(\tau,u(\tau)),v\rangle_{V^{*}\times
V}\,\mathrm{d}\tau+\int_{0}^{t}(B(\tau,u(\tau))\,\mathrm{d}W(\tau),v)_{H}$

holds almost surely for all $t\in[0,T]$ and $v\in V$.

######  Lemma 2.1.

Let Assumptions 1–5 be fulfilled. Then the stochastic evolution equation (2.1)
has a unique (up to $\mathcal{P}$-indistinguishability) variational solution
that fulfills

(2.3) $\displaystyle\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|u(t)\|_{H}^{2}\Big{]}.$

The proof is a standard result of well-posedness of variational solution of
stochastic evolution equation, that even holds with weaker assumptions on
$\tilde{A}$ and $B$, and can be found for example in [29, Theorem 4.2.4].

Given the next assumption, we can provide additional regularity on the
solution. This regularity is needed in our error bound in Theorem 4.4.

######  Assumption 6.

Let Assumptions 1–5 be fulfilled. Additionally, the following stronger
conditions hold:

  * (i)

Let $A\colon V\to V^{*}$ be a linear operator. Moreover, let $H^{2}:=\\{v\in
V:A(v)\in H\\}$ be a dense subset of $H$ such that it is a complete space with
respect to the norm $\|\cdot\|_{H^{2}}$ and let $A\colon H^{2}\subset H\to H$
be a self-adjoint and positive definite operator. The inverse of $A$ in $H$ is
assumed to be compact.

  * (ii)

The initial value $u_{0}\colon\Omega\to H^{2}$ is $\mathcal{F}_{0}$-measurable
with $\mathbb{E}[\|u_{0}\|_{H^{2}}^{2}]<\infty$.

  * (iii)

There exists $\nu_{r}\in[0,1)$ such that the mapping $B(t,\cdot)$ preserves
additional regularity in the sense that

$\displaystyle\|A^{\frac{\nu_{r}}{2}}B(t,v)\|_{L_{2}^{0}}\leq
C(1+\|A^{\frac{\nu_{r}}{2}}v\|_{H})$

for every $v\in H$ with $\|A^{\frac{\nu_{r}}{2}}v\|_{H}<\infty$.

######  Theorem 2.2.

Let Assumptions 1–6 be fulfilled. Then the solution of Equation (2.1) is
Hölder continuous in time with values in $L^{2}(\Omega;V)$. More precisely,
for $\nu_{r}\in[0,1)$ from Assumption 6, it follows that

$\displaystyle\sup_{t,\tau\in[0,T],t\neq\tau}\frac{(\mathbb{E}[\|u(t)-u(\tau)\|_{V}^{2}])^{\frac{1}{2}}}{|t-\tau|^{\min(\frac{1}{2},\frac{\nu_{r}}{2})}}<\infty.$

A proof for autonomous coefficients can be found in [26, Theorem 4.1] and for
non-autonomous coefficients in [25, Theorem 2.31]. Note that in the cited
papers mild solutions are considered. However, in our setting, results
regarding equivalence of mild and variational solutions hold, see for example
[38, Section 7.5].

##  3\. A semi-discrete operator splitting scheme

In the following subsections, we will explain the scheme we choose for the
approximation of the solution of (2.1). We begin with a derivation and exact
statement of the method in Section 3.1. Here, we will use an IMEX scheme with
respect to the different parts of $\tilde{A}(t,\cdot)$ and decompose both
$\tilde{A}(t,\cdot)$ and $B(t,\cdot)$ further with a domain decomposition in
mind. The derivation of the scheme is followed by a justification of why the
appearing implicit equations are well-posed. In Section 3.3, we exemplify the
splitting with our main application for the method and explain how a domain
decomposition approach fits into the setting.

###  3.1. Derivation of method

In the following, we describe the strategy of the method that we will analyze
in this paper to approximate the solution of (2.1). The main idea behind the
method can be summarized into three steps that we will explain in more detail
throughout this subsection.

  * Step 1:

We split the data. More precisely, we decompose the space int sub-spaces and
all operators into sub-operators on these sub-spaces.

  * Step 2:

We introduce a time discretization for every split abstract Cauchy problem. As
a method, we choose an IMEX-Euler–Maruyama method.

  * Step 3:

We combine the numerical solutions to the sub-problems using the Lie-splitting
scheme and obtain a an approximation of the original problem.

Combining these steps, we can define an operator splitting method for
stochastic evolution equations (2.1). In the following, let us explain Step
1–3 in more detail.

Step 1: We split the equation (2.1) into $s\in\mathbb{N}$ sub-equations. To
this end, we begin by splitting the space $V$ into a sequence of sub-spaces.
More precisely, for $\ell\in\\{1,\dots,s\\}$, let
$(V_{\ell},\|\cdot\|_{V_{\ell}})$ be real, reflexive, seperable Banach spaces,
which are continuously and densely embedded into $H$, such that
$\bigcap_{\ell=1}^{s}V_{\ell}=V$ and $\sum_{\ell=1}^{s}\|\cdot\|_{V_{s}}$ is
equivalent to $\|\cdot\|_{V}$. On these spaces we can now define split
versions of the operators $A$, $F$, $B$ and the function $G$.

######  Assumption 7.

Let the operators $A\colon[0,T]\times V\times\Omega\to V^{*}$,
$F\colon[0,T]\times H\times\Omega\to H$, $G\colon[0,T]\times\Omega\to V^{*}$
and $B\colon[0,T]\times H\times\Omega\to L_{2}^{0}$ be given as in Assumptions
2, 3 and 5, respectively. For $\ell\in\\{1,\dots,s\\}$, let
$A_{\ell}\colon[0,T]\times V_{\ell}\times\Omega\to V_{\ell}^{*}$ also fulfill
Assumption 2, with $V$ replaced by $V_{\ell}$ and $F_{\ell}\colon[0,T]\times
H\times\Omega\to H$, $G_{\ell}\colon[0,T]\times\Omega\to V_{\ell}^{*}$ and
$B_{\ell}\colon[0,T]\times H\times\Omega\to L_{2}^{0}$ fulfill Assumptions 3
and 5, respectively. For simplicity, assume that all the constants from
Assumptions 2, 3 and 5 are the same for both the unsplit and the split
operators.

Moreover, the split operators fulfill the sum property

$\displaystyle\sum_{\ell=1}^{s}A_{\ell}(t,v,\omega)$
$\displaystyle=A(t,v,\omega)\quad\text{in
}V^{*},\quad\sum_{\ell=1}^{s}F_{\ell}(t,w,\omega)=F(t,w,\omega)\quad\text{in
}H,$ $\displaystyle\sum_{\ell=1}^{s}G_{\ell}(t,\omega)$
$\displaystyle=G(t,\omega)\quad\text{in
}V^{*},\quad\sum_{\ell=1}^{s}B_{\ell}(t,w,\omega)=B(t,w,\omega)\quad\text{in
}L_{2}^{0}$

for all $t\in[0,T]$, $v\in V$, $w\in H$ and almost all $\omega\in\Omega$.

Using the split data, we can define $s$ separate stochastic evolution
equations of the form

(3.1)
$\displaystyle\begin{cases}\,\mathrm{d}u_{\ell}(t)=\tilde{A}_{\ell}(t,u_{\ell}(t))\,\mathrm{d}t+B_{\ell}(t,u_{\ell}(t))\,\mathrm{d}W(t),\quad
t\in(0,T],\\\ u_{\ell}(0)=u_{0},\end{cases}$

with $\tilde{A}_{\ell}=F_{\ell}+G_{\ell}-A_{\ell}$.

Note that in the following, we do not split the noise $W$ into $s$ parts as
this is not necessary for the domain decomposition application that we have in
mind. Here, every split operator only acts on a subset of the spatial domain
$\mathcal{D}$ of the given SPDE. Thus, the split diffusion operator $B_{\ell}$
is only non-zero on one part of the spatial domain and the product
$B_{\ell}(t,u(t))\,\mathrm{d}W(t)$ therefore only acts on this part of the
domain even if the noise still acts on the entire domain. We allow that the
noise truncation parameter $K_{\ell}\in\mathbb{N}$ used in Step 2 is different
on every part of the domain. This means that we may approximate the noise more
or less accurately on the different domains.

Step 2: We propose a time-discretization of Equation (2.1) using the IMEX-
Euler–Maruyama method, i.e. the scheme is implicit with respect to $A_{\ell}$
and explicit with respect to $F_{\ell}$ and $B_{\ell}$. We define the time
grid $0=t_{0}<\dots<t_{N}=T$ with $h_{n}=t_{n+1}-t_{n}$ such that $h_{n}\leq
h$ for all $n\in\\{1,\dots,N\\}$, where $h\in(0,T)$ is the maximal step size.
To obtain an approximation of the Wiener process, we consider the truncation
of the series representation of the form

$\displaystyle
W^{K_{\ell}}(t)=\sum_{k=1}^{K_{\ell}}q_{k}^{\frac{1}{2}}\psi_{k}\beta_{k}(t)$

for $t\in[0,T]$ with $K_{\ell}\in\mathbb{N}$. Further, we denote the
increments by $\Delta
W^{n,K_{\ell}}=W^{K_{\ell}}(t_{n})-W^{K_{\ell}}(t_{n-1})$. We apply the IMEX-
Euler–Maruyama scheme for each $\ell$-th sub-problem as

(3.2)
$\displaystyle\begin{cases}U_{\ell}^{n}-U_{\ell}^{n-1}+h_{n}A_{\ell}(t_{n},U_{\ell}^{n})=h_{n}F_{\ell}(t_{n-1},U_{\ell}^{n-1})+h_{n}G_{\ell}(t_{n})\\\
\hskip 142.26378pt+B_{\ell}(t_{n-1},U^{n-1})\Delta W^{n,K_{\ell}}&\text{in
}V_{\ell}^{*},\\\ U_{\ell}^{0}=u_{0}&\text{in }H\end{cases}$

for $n\in\\{1,\dots,N\\}$.

Step 3: The last step is to find an approximation $U^{n}$ of the original
problem (2.1). In (3.2), we so far only approximated the split problems (3.1)
individually but there is no interaction between the sub-problems. This
interaction we will now introduce using the Lie-splitting method. The main
difference compared to (3.2) becomes that we solve the sub-problems after one
another and use the solution of one as the initial value for the next problem.
This can be seen in the second line of the scheme

(3.3)
$\displaystyle\begin{cases}U_{\ell}^{n}-U_{\ell-1}^{n}+h_{n}A_{\ell}(t_{n},U_{\ell}^{n})=h_{n}F_{\ell}(t_{n-1},U_{\ell-1}^{n})+h_{n}G_{\ell}(t_{n})\\\
\hskip 142.26378pt+B_{\ell}(t_{n-1},U^{n-1})\Delta W^{n,K_{\ell}},&\text{in
}V_{\ell}^{*},\\\ U^{n}=U_{s}^{n}=U_{0}^{n+1}&\text{in }H,\\\
U_{0}^{1}=u_{0}&\text{in }H\end{cases}$

for $\ell\in\\{1,\dots,s\\}$ and $n\in\\{1,\dots,N\\}$. In (3.3), we solve $s$
sub-problems after each other within one time step and find an approximation

$\displaystyle U^{n}=U_{s}^{n}\approx u(t_{n})$

for every $n\in\\{1,\dots,N\\}$.

###  3.2. Well-posedness of the implicit equation

Our numerical scheme defined in (3.3) is an implicit method in $U_{\ell}^{n}$.
For such a method, it remains to verify that it admits a unique solution. In
our case, we need to prove that there exists a unique solution
$U_{\ell}^{n}\colon\Omega\to H$ that is $\mathcal{F}_{t_{n}}$-measurable.

The first step is to show that for a fixed $\omega\in\Omega$, there exists a
unique element $U_{\ell}^{n}(\omega)\in V_{\ell}$ that solves the path of
(3.3). For this we can apply a standard tool from deterministic PDE theory.
Since $I+h_{n}A_{\ell}(t_{n},\cdot,\omega)$ is bounded, Lipschitz continuous,
strictly monotone and coercive, we can apply a variant of the Browder–Minty
theorem from [33, Theorem 2.14] and obtain a unique solution
$U_{\ell}^{n}(\omega)\in V_{\ell}$.

However, the Browder-Minty theorem does not provide any measurability results
for the random variable $U_{\ell}^{n}$. Thus, we provide Lemma 3.1 below for
this purpose. Rearranging the first equation in the scheme (3.3), its solution
$U_{\ell}^{n}(\omega)$ is given by the root $U$ of the equation
$\gamma(U,\omega)=0$, where the function $\gamma$ is defined by

(3.4)
$\displaystyle\begin{split}\gamma(U,\omega)&:=U-U_{\ell-1}^{n}(\omega)+h_{n}A_{\ell}(t_{n},U,\omega)-h_{n}F_{\ell}(t_{n-1},U_{\ell-1}^{n}(\omega),\omega)\\\
&\quad-
h_{n}G_{\ell}(t_{n},\omega)-B_{\ell}(t_{n-1},U^{n-1}(\omega),\omega)\Delta
W^{n,K_{\ell}}(\omega).\end{split}$

The result in the following lemma ensures the measurability of the function
that maps $\omega$ onto the root of $\gamma(\cdot,\omega)$.

######  Lemma 3.1.

Given a probability space $(\Omega,\mathcal{F},\mathcal{P})$ and a reflexive,
real, seperable Banach space $\Xi$, let
$\gamma\colon\Omega\times\Xi\to\Xi^{*}$ fulfill the following conditions,
where $\mathcal{N}\subset\Omega$ is a set with $\mathcal{P}(\mathcal{N})=0$:

  * (1)

The mapping $u\mapsto\langle\gamma(u,\omega),v\rangle_{\Xi^{*}\times\Xi}$ from
$\Xi$ into $\mathbb{R}$ is continuous for every $v\in\Xi$ and
$\omega\in\Omega\setminus\mathcal{N}$.

  * (2)

The mapping $\omega\mapsto\gamma(u,\omega)$ from $\Omega$ into $\Xi$ is
$\mathcal{F}$-measurable for every $u\in\Xi$.

  * (3)

For every $\omega\in\Omega\setminus\mathcal{N}$, there exists a unique element
$\mathbf{U}(\omega)\in\Xi$ such that $\gamma(\mathbf{U}(\omega),\omega)=0$.

Consider the mapping $\mathbf{U}\colon\Omega\to\Xi$,
$\omega\mapsto\mathbf{U}(\omega)$, where $\mathbf{U}(\omega)$ is the unique
element in $\Xi$ described in (3) for $\omega\in\Omega\setminus\mathcal{N}$
and $\mathbf{U}(\omega)=0$ for $\omega\in\mathcal{N}$. Then $\mathbf{U}$ is
$\mathcal{F}$-measurable.

A proof of the previous lemma can be found in [12].

######  Lemma 3.2.

Let Assumptions 1–5 and Assumption 7 be fulfilled. Then the scheme (3.3) has a
unique solution $U_{\ell}^{n}$ that is in $V_{\ell}$ almost surely, such that
$\omega\mapsto U_{\ell}^{n}(\omega)$ is $\mathcal{F}_{t_{n}}$-measurable.

###### Proof.

For $n=1$, we begin to verify that for every $\ell\in\\{1,\dots,s\\}$ the
mapping $\omega\mapsto U_{\ell}^{1}(\omega)$ is measurable. This can be done
inductively over $\ell$. First, by Assumption 1, the initial value
$\omega\mapsto u_{0}(\omega)=U_{0}^{1}(\omega)$ is a
$\mathcal{F}_{0}$-measurable function and therefore in particular
$\mathcal{F}_{t_{1}}$-measurable.

Assuming that for a fixed $\ell\in\\{1,\dots,s\\}$, the function
$\omega\mapsto U_{\ell-1}^{1}(\omega)\in H$ exists and is
$\mathcal{F}_{t_{1}}$-measurable, the idea in the following is to apply the
Browder-Minty theorem to prove the existence of $U_{\ell}^{1}(\omega)$ for
fixed $\omega\in\Omega$. With this at hand, the measurablility of
$\omega\mapsto U_{\ell}^{1}(\omega)$ will be shown using Lemma 3.1.

In order to apply the variant of the Browder–Minty theorem, we notice that for
$\omega\in\Omega$, the value $U_{\ell}^{n}(\omega)$ is given implicitly
through the equation

$\displaystyle(I+h_{1}A_{\ell}(t_{1},\cdot,\omega))U_{\ell}^{1}(\omega)=U_{\ell-1}^{1}(\omega)+h_{1}F_{\ell}(t_{0},U_{\ell-1}^{1}(\omega),\omega)+h_{1}G_{\ell}(t_{1},\omega)$
$\displaystyle\hskip 142.26378pt+B_{\ell}(t_{0},u_{0}(\omega),\omega)\Delta
W^{1,K_{\ell}}(\omega)\quad$ $\displaystyle\text{in }V_{1}^{*}.$

Since $I+h_{1}A_{\ell}(t_{1},\cdot,\omega)$ is bounded, Lipschitz continuous,
strictly monotone and coercive, we can apply [33, Theorem 2.14] to ensure the
existence of a unique element $U_{\ell}^{1}(\omega)\in V_{\ell}\subset H$ that
fulfills the equation. We now consider the mapping $\gamma\colon
V_{\ell}\times\Omega\to V_{\ell}^{*}$ as defined in (3.4) for $n=1$. Due to
the Browder–Minty argument above, we have found the unique element
$U_{\ell}^{1}(\omega)\in V_{\ell}$ such that
$\gamma(U_{\ell}^{1}(\omega),\omega)=0$. Since

$\displaystyle\big{|}\langle\gamma(v_{1},\omega),w\rangle_{V_{\ell}^{*}\times
V_{\ell}}-\langle\gamma(v_{2},\omega),w\rangle_{V_{\ell}^{*}\times
V_{\ell}}\big{|}$
$\displaystyle\leq\|\gamma(v_{1},\omega)-\gamma(v_{2},\omega)\|_{V_{\ell}^{*}}\|w\|_{V_{\ell}}\leq\big{(}\|v_{1}-v_{2}\|_{V_{\ell}^{*}}+h_{n}L_{A}\|v_{1}-v_{2}\|_{V_{\ell}}\big{)}\|w\|_{V_{\ell}}$

shows the mapping
$v\mapsto\langle\gamma(v,\omega),w\rangle_{V_{\ell}^{*}\times V_{\ell}}$ is
Lipschitz continuous for every $w\in V_{\ell}$ and $\omega\in\Omega$, it is in
particular continuous. The mapping

$\displaystyle\omega\mapsto\gamma(v,\omega)$
$\displaystyle=v-U_{\ell-1}^{1}(\omega)+h_{1}A(t_{1},v,\omega)-h_{1}F(t_{0},U_{\ell-1}^{1}(\omega),\omega)$
$\displaystyle\quad-h_{1}G(t_{1},\omega)-B(t_{0},u_{0}(\omega),\omega)\Delta
W^{1,K_{\ell}}(\omega)$

is $\mathcal{F}_{t_{1}}$-measurable as a composition of
$\mathcal{F}_{t_{1}}$-measurable functions. Thus, we can now apply Lemma 3.1
and find that the mapping $\omega\mapsto U_{\ell}^{1}(\omega)$ is
$\mathcal{F}_{t_{1}}$-measurable.

Altogether this shows that the function $\omega\mapsto
U_{0}^{2}(\omega)=U_{s}^{1}(\omega)\in H$ exists and is
$\mathcal{F}_{t_{1}}$-measurable. For the following time steps, we can argue
by marching through the $n\in\\{2,\dots,N\\}$ and using the existence and
$\mathcal{F}_{t_{n-1}}$-measurability of $\omega\mapsto
U_{0}^{n}(\omega)=U_{s}^{n-1}(\omega)\in H$ as well as the same argumentation
as above for the $s$ sub-steps. ∎

###  3.3. Application: Domain decomposition

The abstract framework from the previous section can be applied to allow for a
domain decomposition setting similar to [5, 13, 15, 21] for deterministic
nonlinear evolution equations. In the following, we will summarize this
setting for the convenience of the reader.

To provide an example for our abstract equation (2.1), we consider the setting
where the operator $A$ is of Laplace type. Then, let
$\mathcal{D}\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, be a bounded domain with
a Lipschitz boundary $\partial\mathcal{D}$. We consider a stochastic semi-
linear heat equation with homogeneous Dirichlet boundary conditions

(3.5) $\displaystyle\begin{cases}\,\mathrm{d}u-\alpha(t)\Delta
u\,\mathrm{d}t=(\tilde{F}(t,x,u)+\tilde{G}(t,x))\,\mathrm{d}t+\tilde{B}(t,x,u)\,\mathrm{d}W(t,x),\\\
\hskip 227.62204ptt\in[0,T],x\in\mathcal{D},\\\ u(t,x)=0,\hskip
177.82971ptt\in[0,T],x\in\partial\mathcal{D},\\\ u(0,x)=u_{0}(x),\hskip
157.91287ptx\in\mathcal{D},\end{cases}$

where $\alpha\colon[0,T]\times\Omega\to\mathbb{R}$ and
$u_{0}\colon\mathcal{D}\times\Omega\to\mathbb{R}$. Moreover, the notation
$\tilde{F}$, $\tilde{G}$ and $\tilde{B}$ are used to differentiate between the
functions
$\tilde{F}\colon[0,T]\times\mathcal{D}\times\mathbb{R}\times\Omega\to\mathbb{R}$
and the abstract function $F$ on $[0,T]\times L^{2}(\mathcal{D})\times\Omega$
that it gives rise to through $[F(t,u,\omega)](x)=\tilde{F}(t,x,u,\omega)$ and
analogously $\tilde{G}$ and $\tilde{B}$.

In the following, we will use a variational formulation of (3.5) in order to
introduce the domain decomposition. For this formulation, we use the concept
of Sobolev spaces. The Sobolev space of all functions $v\in
L^{p}(\mathcal{D})$ such that every weak partial derivative $\partial_{i}v$,
$i\in\\{1,\dots,d\\}$, exists and is an element of $L^{p}(\mathcal{D})$ is
denoted by $W^{1,p}(\mathcal{D})$. Additionally, for $p\in[1,\infty]$ we
consider the space $W_{0}^{1,p}(\mathcal{D})$, which is the closure of the
infinitely many times differentiable functions with a compact support
$C_{c}^{\infty}(\mathcal{D})$ with respect to the norm of
$W^{1,p}(\mathcal{D})$. For an introduction and more details, we refer the
reader to [2].

In order to split the domain, we decompose $\mathcal{D}$ into $s\in\mathbb{N}$
overlapping sub-domains $\\{\mathcal{D}_{\ell}\\}_{\ell=1}^{s}$. We assume
that each of these subsets $\mathcal{D}_{\ell}$ has a Lipschitz boundary and
that their union is $\mathcal{D}$, i.e.
$\bigcup_{\ell=1}^{s}\mathcal{D}_{\ell}=\mathcal{D}$. Further, let the
partition of unity $\\{\chi_{\ell}\\}_{\ell=1}^{s}\subset
W^{1,\infty}(\mathcal{D})$ be given such that for $\ell\in\\{1,\dots,s\\}$

$\displaystyle\chi_{\ell}(x)>0\text{ for all
}x\in\mathcal{D}_{\ell},\quad\chi_{\ell}(x)=0\text{ for all
}x\in\mathcal{D}\setminus\mathcal{D}_{\ell}\quad\text{and}\quad\sum_{\ell=1}^{s}\chi_{\ell}=1.$

Together with the weight functions
$\\{\chi_{\ell}\\}_{\ell\in\\{1,\dots,s\\}}$, we can now define suitable
function spaces and operators. First, let the pivot space
$\left(H,(\cdot,\cdot)_{H},\|\cdot\|_{H}\right)$ be the space
$L^{2}(\mathcal{D})$ of square integrable functions on $\mathcal{D}$ with the
usual norm and inner product. The space $V$ is given by

$\displaystyle
V=\text{clos}_{\|\cdot\|_{V}}\big{(}C_{c}^{\infty}(\mathcal{D})\big{)}=W_{0}^{1,2}(\mathcal{D}),\quad\text{with}\quad\|\cdot\|_{V}^{2}=\|\cdot\|_{H}^{2}+\|\nabla\cdot\|_{L^{2}(\mathcal{D})^{d}}^{2},$

where $L^{2}(\mathcal{D})^{d}$ is the space of all measurable functions
$v\colon\mathcal{D}\to\mathbb{R}^{d}$ such that the components are in
$L^{2}(\mathcal{D})$.

In order to state the spaces $V_{\ell}$, $\ell\in\\{1,\dots,s\\}$, we use the
weighted Lebesgue space $L^{2}(\mathcal{D}_{\ell},\chi_{\ell})^{d}$ that
consists of all measurable functions
$v=(v_{1},\dots,v_{d})\colon\mathcal{D}\to\mathbb{R}^{d}$ such that

$\displaystyle\|(v_{1},\ldots,v_{d})\|_{L^{2}(\mathcal{D}_{\ell},\chi_{\ell})^{d}}=\Big{(}\int_{\mathcal{D}_{\ell}}\chi_{\ell}|(v_{1},\ldots,v_{d})|^{2}\,\mathrm{d}x\Big{)}^{\frac{1}{2}}$

is finite. Then $V_{\ell}$, $\ell\in\\{1,\dots,s\\}$, is given by

$\displaystyle
V_{\ell}=\text{clos}_{\|\cdot\|_{V_{\ell}}}\big{(}C_{c}^{\infty}(\mathcal{D})\big{)},\quad\text{with}\quad\|\cdot\|_{V_{\ell}}^{2}=\|\cdot\|_{H}^{2}+\|\nabla\cdot\|_{L^{2}(\mathcal{D}_{\ell},\chi_{\ell})^{d}}^{2}.$

The spaces $V_{\ell}$, $\ell\in\\{1,\dots,s\\}$, are densely embedded into $H$
and their intersection $\bigcap_{\ell=1}^{s}V_{\ell}$ is the original space
$V$. The operators $A(t,\cdot,\omega)\colon V\to V^{*}$,
$F(t,\cdot,\omega)\colon H\to H$, $G(t,\omega)\in V^{*}$,
$B(t,\cdot,\omega)\colon H\to H$, and the split operators
$A_{\ell}(t,\cdot,\omega)\colon V_{\ell}\to V^{*}_{\ell}$,
$F_{\ell}(t,\cdot,\omega)\colon H\to H$, $G_{\ell}(t,\omega)\in V_{\ell}^{*}$,
$B_{\ell}(t,\cdot,\omega)\colon H\to H$, $\ell\in\\{1,\dots,s\\}$,
$t\in[0,T]$, are given by

(3.6) $\displaystyle\begin{split}\langle A(t,v,\omega),w\rangle_{V^{*}\times
V}&=\int_{\mathcal{D}}\alpha(t,\omega)\nabla v\cdot\nabla w\,\mathrm{d}x,\\\
\langle A_{\ell}(t,v_{\ell},\omega),w_{\ell}\rangle_{V_{\ell}^{*}\times
V_{\ell}}&=\int_{\mathcal{D}_{\ell}}\chi_{\ell}\alpha(t,\omega)\nabla
v_{\ell}\cdot\nabla w_{\ell}\,\mathrm{d}x,\end{split}$ (3.7)
$\displaystyle\langle G(t,\omega),v\rangle_{V^{*}\times
V}=\int_{\mathcal{D}}\tilde{G}(t,\cdot,\omega)$ $\displaystyle
v\,\mathrm{d}x,\quad\langle
G_{\ell}(t,\omega),v_{\ell}\rangle_{V_{\ell}^{*}\times
V_{\ell}}=\int_{\mathcal{D}_{\ell}}\chi_{\ell}\tilde{G}(t,\cdot,\omega)v_{\ell}\,\mathrm{d}x$

for $v,w\in V$, $v_{\ell},w_{\ell}\in V_{\ell}$ and $\omega\in\Omega$ as well
as

(3.8) $\displaystyle(F(t,v,\omega),w)_{H}$
$\displaystyle=\int_{\mathcal{D}}\tilde{F}(t,\cdot,v,\omega)w\,\mathrm{d}x,\
(F_{\ell}(t,v,\omega),w)_{H}=\int_{\mathcal{D}_{\ell}}\chi_{\ell}\tilde{F}(t,\cdot,v,\omega)w\,\mathrm{d}x$
(3.9) $\displaystyle(B(t,v,\omega),w)_{H}$
$\displaystyle=\int_{\mathcal{D}}\tilde{B}(t,\cdot,v,\omega)v\,\mathrm{d}x,\
(B_{\ell}(t,v,\omega),w)_{H}=\int_{\mathcal{D}_{\ell}}\chi_{\ell}\tilde{B}(t,\cdot,v,\omega)w\,\mathrm{d}x$

for $v,w\in H$ and $\omega\in\Omega$. In order to fulfill the rest of the
assumptions of the previous section, we further assume that $\alpha\in
C^{\nu_{A}}([0,T];L^{\infty}(\Omega;\mathbb{R}))$, $u_{0}\in
L^{2}(\mathcal{D};L^{2}(\Omega))$, $\tilde{F}(\cdot,\cdot,v,\cdot)\in
C^{\nu_{F}}([0,T];L^{\infty}(\mathcal{D}\times\Omega;\mathbb{R}))$,
$\tilde{G}\in C^{\nu_{G}}([0,T];L^{2}(\Omega;(W_{0}^{1,2}(\mathcal{D}))^{*}))$
as well as $\tilde{B}(\cdot,\cdot,v,\cdot)\in
C^{\nu_{B}}([0,T];L^{\infty}(\mathcal{D}\times\Omega;\mathbb{R}))$ for
$v\in\mathbb{R}$ such that $\tilde{F}$ and $\tilde{B}$ are Lipschitz
continuous in the third argument. A proof that the setting stated in this
section fulfills the Assumptions 1–5 and 7 can be carried through in an
analogous manner to [15, Lemma 3.1] but it remains to show that the operator
$B(t,v)$ is an element of $L_{2}^{0}$ almost surely. To this extend, we choose
the orthonormal basis $\\{\psi_{k}\\}_{k\in\mathbb{N}}$ of $H$ from Assumption
4. We can then write

$\displaystyle\|B(t,v)\|_{L_{2}^{0}}^{2}=\sum_{k=1}^{\infty}\|B(t,v)(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}=\sum_{k=1}^{\infty}q_{k}\int_{\mathcal{D}}|\tilde{B}(t,\cdot,v)\psi_{k}|^{2}\,\mathrm{d}x,$

which is bounded if $\tilde{B}(t,\cdot,v)\in L^{\infty}(\mathcal{D})$ for
every $v\in H$. Alternatively, if the orthonormal basis
$\\{\psi_{k}\\}_{k\in\mathbb{N}}$ is even bounded in
$L^{\infty}(\mathcal{D})$, it is instead sufficient if
$\tilde{B}(t,\cdot,v)\in H$ for $v\in H$ since

$\displaystyle\|B(t,v)\|_{L_{2}^{0}}^{2}\leq\sum_{k=1}^{\infty}q_{k}\|\psi_{k}\|_{\infty}^{2}\int_{\mathcal{D}}|\tilde{B}(t,\cdot,v)|^{2}\,\mathrm{d}x=\|\tilde{B}(t,\cdot,v)\|_{H}^{2}\sum_{k=1}^{\infty}q_{k}\|\psi_{k}\|_{\infty}^{2}$

is then bounded.

In order to obtain the additional regularity from Theorem 2.2, we need the
setting to fit into Assumption 6. To this end, we assume additionally that
$\alpha\equiv 1$, $H^{2}=H^{2}(\mathcal{D})\cap H_{0}^{1}(\mathcal{D})$,
$u_{0}\in H^{2}(\mathcal{D};L^{2}(\Omega))\cap
H_{0}^{1}(\mathcal{D};L^{2}(\Omega))$ and the operator $B(t,\cdot)$ preserves
the regularity of an element $v$ as stated in Assumption 6 (iii). This last
condition is fulfilled for additive noise $B(t,v)=I_{H}$ (where $I_{H}$ is the
identity operator in $H$) if the basis $\\{\psi_{k}\\}_{k\in\mathbb{N}}$
fulfills that they are eigenfunctions of $A$ to the corresponding eigenvalues
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ and
$\\{q_{k}\lambda_{k}^{\nu}\\}_{k\in\mathbb{N}}$ is summable. Then it follows
that

$\displaystyle\|A^{\frac{\nu}{2}}B(t,v)\|_{L_{2}^{0}}^{2}=\sum_{k=1}^{\infty}\|A^{\frac{\nu}{2}}I(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}=\sum_{k=1}^{\infty}q_{k}\int_{\mathcal{D}}|A^{\frac{\nu}{2}}\psi_{k}|^{2}\,\mathrm{d}x=\sum_{k=1}^{\infty}q_{k}\lambda_{k}^{\nu}$

is finite. This condition is more difficult to prove for multiplicative noise.

##  4\. Error analysis

In this section, we state a formal analysis for our method (3.3). This
analysis contains several steps. We begin to show that the solution to our
time-discrtization has bounded moments. This we will state in Lemma 4.1.
Moreover, we will present a result for the regularity of an auxiliary function
that we will need for the following analysis. We conclude this section with
our main result Theorem 4.4, where we provide an explicit error bound of the
scheme.

###  4.1. Boundedness of the numerical solution

As stated in (2.2) and (2.3) the variational solution to the problem (2.1)
fulfills certain moment bounds. More precisely, there exists
$C_{\textup{bound, cont.}}\in(0,\infty)$ such that the exact solution $u$
fulfills

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\big{[}\|u(t)\|_{H}^{2}\big{]}+\int_{0}^{T}\mathbb{E}\big{[}\|u(t)\|_{V}^{2}\big{]}\,\mathrm{d}t\leq
C_{\textup{bound, cont.}}.$

A first step is to replicate this bound for the numerical solution. In the
following lemma, we provide an a priori bound for our numerical approximation.
In particular, this lemma ensures that the numerical solution fulfills the
analogous moment bounds as the exact solution.

######  Lemma 4.1.

Let Assumptions 1–5 and Assumption 7 be fulfilled and let $U_{\ell}^{n}$,
$\ell\in\\{1,\dots,s\\}$, $n\in\\{1,\dots,N\\}$, be the unique solution of the
scheme (3.3). Then there exists a constant $C_{\textup{bound}}\in(0,\infty)$
such that for every maximal step size $h\in(0,T)$ the following a priori bound
holds

$\displaystyle\max_{n\in\\{1,\dots,N\\}}\mathbb{E}\big{[}\|U^{n}\|_{H}^{2}\big{]}+\sum_{i=1}^{N}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{]}+\sum_{i=1}^{N}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}\leq
C_{\textup{bound}}.$

The proof follows with standard arguments used for a priori bounds for
evolution equation as for example in [33, Lemma 8.6]. Moreover, the structure
is similar to the proof of our main result in Theorem 4.4. For the sake
completeness, the proof of the Lemma 4.1 can be found in Appendix B.

###  4.2. Regularity of the domain decomposition

From our numerical scheme, we obtain an approximation $U^{n}$ of $u(t_{n})$
after every time step. Every time step consists of $s$ sub-steps that take the
different operators $\tilde{A}_{\ell}$ and $B_{\ell}$,
$\ell\in\\{1,\dots,s\\}$, into account. This means that for a sub-solution
$U_{\ell}^{n}$ we consider the operators $\tilde{A}_{j}$ and $B_{j}$ for $j$
between $1$ and $\ell$. In the following we introduce functions $u_{\ell}$ to
compare these sub-solutions to. These functions are defined as

(4.1)
$\displaystyle\begin{split}u_{\ell}(t)&=u(t_{n-1})+\sum_{j=1}^{\ell}\Big{(}\int_{t_{n-1}}^{t}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}\\\
&=u_{\ell-1}(t_{n})+\int_{t_{n-1}}^{t}\tilde{A}_{\ell}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t}B_{\ell}(\tau,u(\tau))\,\mathrm{d}W(\tau)\end{split}$

for $t\in(t_{n-1},t_{n}]$, $n\in\\{1,\dots,N\\}$ and with
$\tilde{A}_{\ell}=F_{\ell}+G_{\ell}-A_{\ell}$ for $\ell\in\\{1,\dots,s\\}$. In
order to prove that these auxiliary functions do not differ too much from the
exact solution $u$, we additionally need the following assumption.

######  Assumption 8.

Let Assumptions 2–5 and 7 be fulfilled. There exists
$C_{\textup{split}}\in(0,\infty)$ such that for all $t,\tau\in[0,T]$ it holds
that

$\displaystyle\begin{split}&\Big{\|}\sum_{j=1}^{s}\Big{(}\int_{\tau}^{t}\tilde{A}_{j}(\sigma,v(\sigma))\,\mathrm{d}\sigma+\int_{\tau}^{t}B_{j}(\sigma,v(\sigma))\,\mathrm{d}W(\sigma)\Big{)}\Big{\|}_{V}\\\
&\geq
C_{\textup{split}}\Big{\|}\sum_{j=1}^{\ell}\Big{(}\int_{\tau}^{t}\tilde{A}_{j}(\sigma,v(\sigma))\,\mathrm{d}\sigma+\int_{\tau}^{t}B_{j}(\sigma,v(\sigma))\,\mathrm{d}W(\sigma)\Big{)}\Big{\|}_{V_{\ell}}\\\
&\quad+C_{\textup{split}}\Big{\|}\sum_{j=\ell+1}^{s}\Big{(}\int_{\tau}^{t}\tilde{A}_{j}(\sigma,v(\sigma))\,\mathrm{d}\sigma+\int_{\tau}^{t}B_{j}(\sigma,v(\sigma))\,\mathrm{d}W(\sigma)\Big{)}\Big{\|}_{V_{\ell}}\end{split}$

almost surely for every $v\in L^{2}(0,T;V)$ and $\ell\in\\{1,\dots,s\\}$.

Note that in the case of the orthogonality of the integral operators of Lie
splitting operators $\tilde{A}_{\ell}$ and $B_{\ell}$, the previous inequality
becomes equality and $C_{\textup{split}}=1$. Similar types of orthogonality
conditions are common in multi-grid methods.

######  Lemma 4.2.

Let Assumptions 2–5, 7 and 8 be fulfilled. Then it follows that

$\displaystyle
C_{\textup{split}}\|u(t_{n})-u_{\ell}(t_{n})\|_{V_{\ell}}+C_{\textup{split}}\|u_{\ell}(t_{n})-u(t_{n-1})\|_{V_{\ell}}\leq\|u(t_{i})-u(t_{i-1})\|_{V}.$

###### Proof.

The difference of the function $u$ at two grid points $t_{n}$ and $t_{n-1}$
can be split up into two parts

$\displaystyle u(t_{n})-u(t_{n-1})$
$\displaystyle=\Big{(}\sum_{j=1}^{\ell}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\sum_{j=1}^{s}\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}$
$\displaystyle\quad+\Big{(}\sum_{j=\ell+1}^{s}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\sum_{j=1}^{s}\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}.$

Using this decomposition and the definition of the function $u_{\ell}(t_{n})$,
we find that

$\displaystyle u_{\ell}(t_{n})$
$\displaystyle=u(t_{n})-\sum_{j=\ell+1}^{s}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}$
$\displaystyle=u(t_{n-1})+\sum_{j=1}^{\ell}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}.$

This then enables us to find a representation of $u(t_{n})-u_{\ell}(t_{n})$
and $u_{\ell}(t_{n})-u(t_{n-1})$ given by

$\displaystyle u(t_{n})-u_{\ell}(t_{n})$
$\displaystyle=\sum_{j=\ell+1}^{s}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}$

and

$\displaystyle u_{\ell}(t_{n})-u(t_{n-1})$
$\displaystyle=\sum_{j=1}^{\ell}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}.$

Applying Assumption 8, it then follows that

$\displaystyle
C_{\textup{split}}\|u(t_{n})-u_{\ell}(t_{n})\|_{V_{\ell}}+C_{\textup{split}}\|u_{\ell}(t_{n})-u(t_{n-1})\|_{V_{\ell}}$
$\displaystyle=C_{\textup{split}}\Big{\|}\sum_{j=1}^{\ell}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}\Big{\|}_{V_{\ell}}$
$\displaystyle\quad+C_{\textup{split}}\Big{\|}\sum_{j=\ell+1}^{s}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}\Big{\|}_{V_{\ell}}$
$\displaystyle\leq\Big{\|}\sum_{j=1}^{s}\Big{(}\int_{t_{n-1}}^{t_{n}}\tilde{A}_{j}(\tau,u(\tau))\,\mathrm{d}\tau+\int_{t_{n-1}}^{t_{n}}B_{j}(\tau,u(\tau))\,\mathrm{d}W(\tau)\Big{)}\Big{\|}_{V}$
$\displaystyle=\|u(t_{n})-u(t_{n-1})\|_{V}.$

∎

###  4.3. Error bound

In this section, we provide a result on the mean-square convergence of the
scheme (3.3) which is a combination of an IMEX-Euler–Maruyama method and a Lie
splitting scheme for the stochastic evolution equation (2.1).

In the following section, we define the full error $e^{n}=u(t_{n})-U^{n}$ as
well as as the sub-step error $e_{\ell}^{n}=u_{\ell}(t_{n})-U_{\ell}^{n}$ for
$n\in\\{1,\dots,N\\}$ and $\ell\in\\{1,\dots,s\\}$. We then prove that
$\mathbb{E}[\|e^{n}\|_{H}^{2}]$ tends to zero as the maximal time step size
$h\to 0$. In order to prove this error bound, we summarize the necessary
assumption on the setting:

  * (H1)

Let Assumptions 1–5 be fulfilled which ensures that the stochastic evolution
equation (2.1) has a unique variational solution $u$.

  * (H2)

For $\nu_{u}\in(0,\frac{1}{2})$, let the exact solution $u$ of the stochastic
evolution equation (2.1) be an element of
$C^{\nu_{u}}([0,T];L^{2}(\Omega;V))$. The Hölder exponent $\nu_{u}$ determines
the order of the error.

  * (H3)

Let Assumption 7 be fulfilled to state the decomposed problem. Further, let
Assumption 8 be fulfilled to ensure that the auxiliary function $u_{\ell}$
defined in (4.1) stays sufficiently close to the solution $u$ of (2.1).

  * (H4)

Assume that for every $\ell\in\\{1,\dots,s\\}$ there exists
$\varepsilon_{\ell}\in(0,\infty)$ such that
$\sum_{k>K_{\ell}}\mathbb{E}\big{[}\|B_{\ell}(t,v)(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}\big{]}\leq\varepsilon_{\ell}\big{(}1+\mathbb{E}\big{[}\|v\|_{H}^{2}\big{]}\big{)}$
for every $v\in L^{2}(\Omega;H)$. This ensures that the truncation error
$\varepsilon=\sum_{\ell=1}^{s}\varepsilon_{\ell}$ is sufficiently small.

######  Remark 4.3.

The assumption
$\sum_{k>K_{\ell}}\mathbb{E}\big{[}\|B_{\ell}(t,v)(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}\big{]}\leq\varepsilon_{\ell}\big{(}1+\mathbb{E}\big{[}\|v\|_{H}^{2}\big{]}\big{)}$
on the truncation is fulfilled, if the eigenvalues
$\\{q_{k}\\}_{k\in\mathbb{N}}$ of $Q$ are summable and

$\displaystyle\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},v)\big{(}\psi_{k}\big{)}\big{\|}_{H}^{2}\big{]}\leq
C_{\textup{truncation}}\big{(}1+\mathbb{E}\big{[}\|v\|_{H}^{2}\big{]}\big{)}$

is fulfilled. The value $\varepsilon_{\ell}$ is then given by
$\varepsilon_{\ell}=C_{\textup{truncation}}\sum_{k>K_{\ell}}q_{k}$. An example
for this setting is if $B_{\ell}(t,v)\in\mathcal{L}(H)$, $t\in[0,T]$. But also
weaker assumptions can be made if the eigenfunctions
$\\{\psi_{k}\\}_{k\in\mathbb{N}}$ are more regular.

######  Theorem 4.4 (Bound for the error).

Let $U^{n}$ and $U_{\ell}^{n}$ for $n\in\\{1,\dots,N\\}$ and
$\ell\in\\{1,\dots,s\\}$ be given by the split IMEX-Euler–Maruyama method
(3.3). Under the assumptions stated in (H1)–(H4), there exists a constant
$C_{\textup{bound}}\in(0,\infty)$ such that the errors $e^{n}=u(t_{n})-U^{n}$
and $e_{\ell}^{n}=u_{\ell}(t_{n})-U_{\ell}^{n}$ of the scheme (3.3) can be
bounded by

$\displaystyle\max_{n\in\\{1,\dots,N\\}}\mathbb{E}\big{[}\|e^{n}\|_{H}^{2}\big{]}+\sum_{i=1}^{N}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}+\sum_{i=1}^{N}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq C_{\textup{bound}}\big{(}h^{2\nu}+\varepsilon\big{)},$

where $\nu=\min\\{\nu_{A},\nu_{F},\nu_{B},\nu_{G},\nu_{u}\\}$.

###### Proof.

The proof is based on techniques from PDE analysis in a variational framework.
In the following, we will state an equation for the error, which we will test
with the error itself. The remaining terms can either be compensated by, for
example, the monotonicty of $A_{\ell}(t_{i},\cdot)$ or are small due to the
regularity assumptions.

We begin to subtract the numerical approximation $U_{\ell}^{i}$ from
$u_{\ell}(t_{i})$. Applying the abbreviation
$e_{\ell}^{i}=u_{\ell}(t_{i})-U_{\ell}^{i}$, we then obtain

$\displaystyle e_{\ell}^{i}$
$\displaystyle=e_{\ell-1}^{i}+\int_{t_{i-1}}^{t_{i}}\big{(}F_{\ell}(t,u(t))-F_{\ell}(t_{i-1},U_{\ell-1}^{i})\big{)}\,\mathrm{d}t+\int_{t_{i-1}}^{t_{i}}\big{(}G_{\ell}(t)-G_{\ell}(t_{i-1})\big{)}\,\mathrm{d}t$
$\displaystyle\quad-\int_{t_{i-1}}^{t_{i}}\big{(}A_{\ell}(t,u(t))-A_{\ell}(t_{i},U_{\ell}^{i})\big{)}\,\mathrm{d}t$
$\displaystyle\quad+\int_{t_{i-1}}^{t_{i}}\big{(}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\big{)}\,\mathrm{d}W(t)$
$\displaystyle\quad+B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta W^{i}-\Delta
W^{i,K_{\ell}}\big{)}.$

Next, we test the equation with $e_{\ell}^{i}$ and use the identity
$(a-b,a)_{H}=\frac{1}{2}\big{(}\|a\|_{H}^{2}-\|b\|_{H}^{2}+\|a-b\|_{H}^{2}\big{)}$
for $a,b\in H$. After summing up the equation from $\ell=1$ to $s$ and using
the telescopic sum argument, we obtain

$\displaystyle\frac{1}{2}\big{(}\|e^{i}\|_{H}^{2}-\|e^{i-1}\|_{H}^{2}+\sum_{\ell=1}^{s}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{)}$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t,u(t))-F_{\ell}(t_{i-1},U_{\ell-1}^{i}),e_{\ell}^{i}\big{)}_{H}\,\mathrm{d}t$
$\displaystyle\quad+\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\langle
G_{\ell}(t)-G_{\ell}(t_{i-1}),e_{\ell}^{i}\rangle_{V_{\ell}^{*}\times
V_{\ell}}\,\mathrm{d}t$
$\displaystyle\quad-\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\langle
A_{\ell}(t,u(t))-A_{\ell}(t_{i},U_{\ell}^{i}),e_{\ell}^{i}\rangle_{V_{\ell}^{*}\times
V_{\ell}}\,\mathrm{d}t$
$\displaystyle\quad+\sum_{\ell=1}^{s}\Big{(}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\,\mathrm{d}W(t),e_{\ell}^{i}\Big{)}_{H}$
$\displaystyle\quad+\sum_{\ell=1}^{s}\big{(}B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta
W^{i}-\Delta
W^{i,K_{\ell}}\big{)},e_{\ell}^{i}\big{)}_{H}=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\Gamma_{4}+\Gamma_{5}.$

The main idea will be to split the terms such that they can be compensated by
$\sum_{\ell=1}^{s}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}$, the monotoicity
condition on $A_{\ell}(t_{i},\cdot)$ or Grönwall’s inequality. The remaining
terms will be bounded using the regularity assumptions stated on the solution
and Assumption 8. First note that by assumption the exact solution $u$ is in
$C^{\nu}([0,T];L^{2}(\Omega;V))$. This implies, in particular, that it lies in
$C^{\nu}([0,T];L^{2}(\Omega;H))$ and $C^{\nu}([0,T];L^{2}(\Omega;V_{\ell}))$
for every $\ell\in\\{1,\dots,s\\}$. Moreover, by Lemma 4.2 and the fact that
$V_{\ell}$ is continuously embedded into $H$, the distance of $u(t)$ and
$u_{\ell}(t)$ is also bounded in the $V_{\ell}$ and $H$-norm. For simplicity
of notation, we will use in the following that there exits a constant
$C_{\textup{reg}}\in(0,\infty)$ such that

(4.2)
$\displaystyle\begin{split}\mathbb{E}\big{[}\|u(t)-u(\tau)\|_{H}^{2}\big{]}\leq
C_{\textup{reg}}|t-\tau|^{2\nu},\quad\mathbb{E}\big{[}\|u(t)-u(\tau)\|_{V_{\ell}}^{2}\big{]}\leq
C_{\textup{reg}}|t-\tau|^{2\nu},\\\
\mathbb{E}\big{[}\|u(t)-u_{\ell}(\tau)\|_{H}^{2}\big{]}\leq
C_{\textup{reg}}|t-\tau|^{2\nu},\quad\mathbb{E}\big{[}\|u(t)-u_{\ell}(\tau)\|_{V_{\ell}}^{2}\big{]}\leq
C_{\textup{reg}}|t-\tau|^{2\nu}\end{split}$

for all $t,\tau\in[0,T]$. We now bound each term $\Gamma_{j}$,
$j\in\\{1,\dots,5\\}$, separately. For $\Gamma_{1}$, we begin by adding and
subtracting $e_{\ell-1}^{i}$ to the right side of the inner product. We the
find that

$\displaystyle\Gamma_{1}$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t,u(t))-F_{\ell}(t_{i-1},U_{\ell-1}^{i}),e_{\ell}^{i}-e_{\ell-1}^{i}\big{)}_{H}\,\mathrm{d}t$
$\displaystyle\quad+\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t,u(t))-F_{\ell}(t_{i-1},U_{\ell-1}^{i}),e_{\ell-1}^{i}\big{)}_{H}\,\mathrm{d}t=\Gamma_{1,1}+\Gamma_{1,2}.$

We will now bound $\Gamma_{1,1}$ by applying Cauchy–Schwarz inequality and
Lemma A.1 with $\xi=h_{i}9$ on the following three summands and inserting
Assumption 3. We then obtain

$\displaystyle\Gamma_{1,1}$
$\displaystyle=\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t,u(t))-F_{\ell}(t_{i-1},u(t_{i-1})),e_{\ell}^{i}-e_{\ell-1}^{i}\big{)}_{H}\,\mathrm{d}t$
$\displaystyle\quad+\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t_{i-1},u(t_{i-1}))-F_{\ell}(t_{i-1},u_{\ell-1}(t_{i})),e_{\ell}^{i}-e_{\ell-1}^{i}\big{)}_{H}\,\mathrm{d}t$
$\displaystyle\quad+\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}F_{\ell}(t,u_{\ell-1}(t_{i}))-F_{\ell}(t_{i-1},U_{\ell-1}^{i}),e_{\ell}^{i}-e_{\ell-1}^{i}\big{)}_{H}\,\mathrm{d}t$
$\displaystyle\leq
h_{i}9L_{F}^{2}\Big{(}s\int_{t_{i-1}}^{t_{i}}\big{(}|t-t_{i-1}|^{2\nu}+\|u(t)-u(t_{i-1})\|_{H}^{2}\big{)}\,\mathrm{d}t$
$\displaystyle\quad+h_{i}\sum_{\ell=1}^{s}\|u(t_{i-1})-u_{\ell-1}(t_{i})\|_{H}^{2}+h_{i}\sum_{\ell=1}^{s}\|e_{\ell-1}^{i}\|_{H}^{2}\Big{)}+\frac{1}{12}\sum_{\ell=1}^{s}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}.$

Similarly for the next term $\Gamma_{1,2}$, we apply Lemma A.1 with
$\xi=\frac{3}{4}$ and obtain

$\displaystyle\Gamma_{1,2}$
$\displaystyle\leq\frac{3}{4}L_{F}^{2}\Big{(}s\int_{t_{i-1}}^{t_{i}}\big{(}|t-t_{i-1}|^{2\nu}+\|u(t)-u(t_{i-1})\|_{H}^{2}\big{)}\,\mathrm{d}t$
$\displaystyle\quad+h_{i}\sum_{\ell=1}^{s}\|u(t_{i-1})-u_{\ell-1}(t_{i})\|_{H}^{2}\Big{)}+h_{i}\Big{(}\frac{3}{4}L_{F}^{2}+1\Big{)}\sum_{\ell=1}^{s}\|e_{\ell-1}^{i}\|_{H}^{2}.$

Due to the regularity condition from (4.2), it then follows that

$\displaystyle\mathbb{E}\big{[}\Gamma_{1}\big{]}$ $\displaystyle\leq
h_{i}^{1+2\nu}\Big{(}h\frac{9(1+C_{\textup{reg}})L_{F}^{2}s}{1+2\nu}+h9C_{\textup{reg}}L_{F}^{2}s+\frac{3(1+C_{\textup{reg}})L_{F}^{2}s}{4+8\nu}+\frac{3}{4}C_{\textup{reg}}L_{F}^{2}s\Big{)}$
$\displaystyle\quad+\frac{1}{12}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}+h_{i}\Big{(}h_{i}9L_{F}^{2}+\frac{3}{4}L_{F}^{2}+1\Big{)}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell-1}^{i}\|_{H}^{2}\big{]}$
$\displaystyle=:h_{i}^{1+2\nu}C_{\Gamma_{1},1}+\frac{1}{12}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}+h_{i}C_{\Gamma_{1},2}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell-1}^{i}\|_{H}^{2}\big{]}.$

Using the Cauchy–Schwarz inequality for the dual pairing and Lemma A.1 with
$\xi=\frac{3}{2\eta}$, where $\eta$ comes from the monotonicity condition on
$A_{\ell}(t_{i},\cdot)$, we find for $\mathbb{E}\big{[}\Gamma_{2}\big{]}$ that

$\displaystyle\mathbb{E}\big{[}\Gamma_{2}\big{]}$
$\displaystyle\leq\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|G_{\ell}(t)-G_{\ell}(t_{i})\|_{V_{\ell}^{*}}\|e_{\ell}^{i}\|_{V_{\ell}}\big{]}\,\mathrm{d}t$
$\displaystyle\leq\frac{3}{2\eta}\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|G_{\ell}(t)-G_{\ell}(t_{i})\|_{V_{\ell}^{*}}^{2}\big{]}\,\mathrm{d}t+h_{i}\frac{\eta}{6}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq
h_{i}^{1+2\nu}C_{\Gamma_{2}}+h_{i}\frac{\eta}{6}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]},$

since $G_{\ell}\in C^{\nu}([0,T];L^{2}(\Omega;V_{\ell}^{*}))$. For the third
term $\Gamma_{3}$, we extend it by adding and subtracting both
$A_{\ell}(t_{i},u(t_{i}))$ and $A_{\ell}(t_{i},u_{\ell}(t_{i}))$ to the left
side of the dual pairing. The resulting terms can be bounded using
Cauchy–Schwarz inequality, the monotonicity condition and the Lipschitz
condition from Assumption 2, as well as Lemma A.1 with $\xi=\frac{3}{2\eta}$.
It then follows that

$\displaystyle\mathbb{E}\big{[}\Gamma_{3}\big{]}$
$\displaystyle\leq\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|A_{\ell}(t,u(t))-A_{\ell}(t_{i},u(t_{i}))\|_{V_{\ell}^{*}}\|e_{\ell}^{i}\|_{V_{\ell}}\big{]}\,\mathrm{d}t$
$\displaystyle\quad+h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|A_{\ell}(t_{i},u(t_{i}))-A_{\ell}(t_{i},u_{\ell}(t_{i}))\|_{V_{\ell}^{*}}\|e_{\ell}^{i}\|_{V_{\ell}}\big{]}-h_{i}\eta\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq\frac{3L_{A}^{2}}{2\eta}\Big{(}\int_{t_{i-1}}^{t_{i}}\sum_{\ell=1}^{s}\big{(}|t-t_{i}|^{2\nu}+\mathbb{E}\big{[}\|u(t)-u(t_{i})\|_{V_{\ell}}^{2}\big{]}\big{)}\,\mathrm{d}t$
$\displaystyle\qquad+h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|u(t_{i})-u_{\ell}(t_{i})\|_{V_{\ell}}^{2}\big{]}\Big{)}-h_{i}\frac{2\eta}{3}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq
h_{i}^{1+2\nu}\frac{3L_{A}^{2}s}{2\eta}\Big{(}\frac{1+C_{\textup{reg}}}{1+2\nu}+C_{\textup{reg}}\Big{)}-h_{i}\frac{2\eta}{3}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle=:h_{i}^{1+2\nu}C_{\Gamma_{3}}-h_{i}\frac{2\eta}{3}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]},$

where we again used the regularity condition from (4.2). For the term
$\Gamma_{4}$, we add and subtract $e^{i-1}$ to the right side of the inner
product. Using Lemma A.1 with $\xi=3s^{2}$, we find that

$\displaystyle\Gamma_{4}$
$\displaystyle=\sum_{\ell=1}^{s}\Big{(}\int_{t_{i-1}}^{t_{i}}(B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1}))\,\mathrm{d}W(t),e_{\ell}^{i}-e^{i-1}\Big{)}_{H}$
$\displaystyle\quad+\sum_{\ell=1}^{s}\Big{(}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\,\mathrm{d}W(t),e^{i-1}\Big{)}_{H}$
$\displaystyle\leq
3s^{2}\sum_{\ell=1}^{s}\Big{\|}\int_{t_{i-1}}^{t_{i}}\big{(}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\big{)}\,\mathrm{d}W(t)\Big{\|}_{H}^{2}$
$\displaystyle\quad+\frac{1}{12s^{2}}\sum_{\ell=1}^{s}\|e_{\ell}^{i}-e^{i-1}\|_{H}^{2}$
$\displaystyle\quad+\sum_{\ell=1}^{s}\Big{(}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\,\mathrm{d}W(t),e^{i-1}\Big{)}_{H}$
$\displaystyle=\Gamma_{4,1}+\Gamma_{4,2}+\Gamma_{4,3}.$

In expectation, the term $\Gamma_{4,1}$ can be bounded by first using the Itô-
isometry and then Assumption 5 to obtain

$\displaystyle\mathbb{E}\big{[}\Gamma_{4,1}\big{]}$
$\displaystyle=3s^{2}\sum_{\ell=1}^{s}\int_{t_{i-1}}^{t_{i}}\mathbb{E}\big{[}\big{\|}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\big{\|}_{L_{2}^{0}}^{2}\big{]}\,\mathrm{d}t$
$\displaystyle\leq
6s^{2}\Big{(}\sum_{\ell=1}^{s}\int_{t_{i-1}}^{t_{i}}\mathbb{E}\big{[}\big{\|}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},u(t_{i-1}))\big{\|}_{L_{2}^{0}}^{2}\big{]}\,\mathrm{d}t$
$\displaystyle\qquad+h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},u(t_{i-1})-B_{\ell}(t_{i-1},U^{i-1})\big{\|}_{L_{2}^{0}}^{2}\big{]}\Big{)}$
$\displaystyle\leq
6L_{B}^{2}s^{2}\Big{(}\int_{t_{i-1}}^{t_{i}}\big{(}|t-t_{i-1}|^{2\nu}+\mathbb{E}\big{[}\|u(t)-u(t_{i-1})\|_{H}^{2}\big{]}\big{)}\,\mathrm{d}t+h_{i}\mathbb{E}\big{[}\big{\|}e^{i-1}\big{\|}_{H}^{2}\big{]}\Big{)}$
$\displaystyle\leq
6L_{B}^{2}s^{2}\Big{(}h_{i}^{1+2\nu}\frac{1+C_{\textup{reg}}}{1+2\nu}+h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\big{\|}e_{\ell-1}^{i}\big{\|}_{H}^{2}\big{]}\Big{)}$
$\displaystyle=:h_{i}^{1+2\nu}C_{\Gamma_{4},1}+h_{i}C_{\Gamma_{4},2}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\big{\|}e_{\ell-1}^{i}\big{\|}_{H}^{2}\big{]},$

where we used the regularity condition from (4.2) and the fact that
$\|e^{i-1}\|_{H}^{2}=\|e_{0}^{i}\|_{H}^{2}\leq\sum_{\ell=1}^{s}\|e_{\ell-1}^{i}\|_{H}^{2}$.
For the second term $\Gamma_{4,2}$, we apply a telescopic sum argument and
Hölder’s inequality for sums. Then we find that

$\displaystyle\Gamma_{4,2}$
$\displaystyle=\frac{1}{12s^{2}}\sum_{\ell=1}^{s}\Big{\|}\sum_{j=1}^{\ell}\big{(}e_{j}^{i}-e_{j-1}^{i}\big{)}\Big{\|}_{H}^{2}\leq\frac{1}{12s^{2}}\sum_{\ell=1}^{s}\ell\sum_{j=1}^{\ell}\big{\|}e_{j}^{i}-e_{j-1}^{i}\big{\|}_{H}^{2}$
$\displaystyle\leq\frac{1}{12s^{2}}\sum_{\ell=1}^{s}s\sum_{j=1}^{s}\big{\|}e_{j}^{i}-e_{j-1}^{i}\big{\|}_{H}^{2}=\frac{1}{12}\sum_{j=1}^{s}\big{\|}e_{j}^{i}-e_{j-1}^{i}\big{\|}_{H}^{2}.$

The last term $\Gamma_{4,3}$ disappears in expectation. This can be seen by
the stochastic independence and properties of the Itô integral,

$\displaystyle\mathbb{E}\big{[}\Gamma_{4,3}\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\sum_{\ell=1}^{s}\Big{(}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\,\mathrm{d}W(t),e^{i-1}\Big{)}_{H}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\sum_{\ell=1}^{s}\Big{(}\mathbb{E}\Big{[}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t,u(t))-B_{\ell}(t_{i-1},U^{i-1})\,\mathrm{d}W(t)\Big{|}\mathcal{F}_{t_{i}}\Big{]},e^{i-1}\Big{)}_{H}\Big{|}\mathcal{F}_{t_{i-1}}\Big{]}\Big{]}=0.$

It remains to bound $\Gamma_{5}$. We begin to add and subtract $e^{i-1}$ in
the right side of the inner product, insert the expansion for the $Q$-Wiener
process and apply Lemma A.1 for $\xi=3s^{2}$. It then follows that

$\displaystyle\Gamma_{5}$
$\displaystyle=\sum_{\ell=1}^{s}\big{(}B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta
W^{i}-\Delta W^{i,K_{\ell}}\big{)},e_{\ell}^{i}-e^{i-1}\big{)}_{H}$
$\displaystyle\quad+\sum_{\ell=1}^{s}\big{(}B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta
W^{i}-\Delta W^{i,K_{\ell}}\big{)},e^{i-1}\big{)}_{H}$ $\displaystyle\leq
3s^{2}\sum_{\ell=1}^{s}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta
W^{i}-\Delta
W^{i,K_{\ell}}\big{)}\big{\|}_{H}^{2}+\frac{1}{12s^{2}}\sum_{\ell=1}^{s}\|e_{\ell}^{i}-e^{i-1}\|_{H}^{2}$
$\displaystyle\quad+\sum_{\ell=1}^{s}\big{(}B_{\ell}(t_{i-1},U^{i-1})\big{(}\Delta
W^{i}-\Delta
W^{i,K_{\ell}}\big{)},e^{i-1}\big{)}_{H}=\Gamma_{5,1}+\Gamma_{5,2}+\Gamma_{5,3}.$

For the first term $\Gamma_{5,1}$, we note that we can exchange $\Delta
W^{i}-\Delta W^{i,K_{\ell}}$ by the sum
$\sum_{k>K_{\ell}}\int_{t_{i-1}}^{t_{i}}\,\mathrm{d}\beta_{k}(t)$ of scalar
Wiener processes and recal that
$\\{q_{k}^{\frac{1}{2}}\psi_{k}\\}_{k\in\mathbb{N}}$ is an orthonormal basis
of $Q^{\frac{1}{2}}(H)$. Further, we recall that all these summands are
stochastically independent, apply Itô’s isometry, the bound on the truncation
error and the a prioiri from Lemma 4.1 to find that

$\displaystyle\mathbb{E}\big{[}\Gamma_{5,1}\big{]}$
$\displaystyle=3s^{2}\sum_{\ell=1}^{s}\mathbb{E}\Big{[}\Big{\|}\sum_{k>K_{\ell}}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\,\mathrm{d}\beta_{k}(t)\Big{\|}_{H}^{2}\Big{]}$
$\displaystyle=3s^{2}\sum_{\ell=1}^{s}\sum_{k>K_{\ell}}\mathbb{E}\Big{[}\Big{\|}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\,\mathrm{d}\beta_{k}(t)\Big{\|}_{H}^{2}\Big{]}$
$\displaystyle=h_{i}3s^{2}\sum_{\ell=1}^{s}\sum_{k>K_{\ell}}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\big{\|}_{H}^{2}\big{]}$
$\displaystyle\leq
h_{i}3s^{2}\Big{(}\sum_{\ell=1}^{s}\varepsilon_{\ell}\Big{)}\big{(}1+\mathbb{E}\big{[}\|U^{i-1}\|_{H}^{2}\big{]}\big{)}\leq
h_{i}3s^{2}(1+C_{\textup{bound}})\sum_{\ell=1}^{s}\varepsilon_{\ell}=:h_{i}\varepsilon
C_{\Gamma_{5}}.$

Analogously to $\Gamma_{4,2}$ and $\Gamma_{4,3}$, it follows that
$\Gamma_{5,2}\leq\frac{1}{12}\sum_{\ell=1}^{s}\big{\|}e_{\ell}^{i}-e_{\ell-1}^{i}\big{\|}_{H}^{2}$
and $\mathbb{E}\big{[}\Gamma_{5,3}\big{]}=0$. We can now combine the previous
bounds to the following,

$\displaystyle\frac{1}{2}\Big{(}\mathbb{E}\big{[}\|e^{i}\|_{H}^{2}\big{]}-\mathbb{E}\big{[}\|e^{i-1}\|_{H}^{2}\big{]}+\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}\Big{)}$
$\displaystyle\leq
h_{i}^{1+2\nu}\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{2}}+C_{\Gamma_{3}}+C_{\Gamma_{4},1}\big{)}+h_{i}\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{4},2}\big{)}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell-1}^{i}\|_{H}^{2}\big{]}$
$\displaystyle\quad-
h_{i}\frac{\eta}{2}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}+h_{i}\varepsilon
C_{\Gamma_{5}}+\frac{1}{4}\sum_{j=1}^{s}\mathbb{E}\big{[}\big{\|}e_{j}^{i}-e_{j-1}^{i}\big{\|}_{H}^{2}\big{]}.$

After multiplying by $2$, rearranging some terms and summing up from $i=1$ to
$n$, we find

$\displaystyle\mathbb{E}\big{[}\|e^{n}\|_{H}^{2}\big{]}-\mathbb{E}\big{[}\|e^{0}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{i=1}^{n}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}+\eta\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq
h^{2\nu}2T\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{2}}+C_{\Gamma_{3}}+C_{\Gamma_{4},1}\big{)}+2\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{4},2}\big{)}\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell-1}^{i}\|_{H}^{2}\big{]}$
$\displaystyle+\varepsilon 2TC_{\Gamma_{5}}.$

In the last step, we use that the initial error
$\mathbb{E}\big{[}\|e^{0}\|_{H}^{2}\big{]}$ is zero and it then remains to
apply Grönwall’s inequality for sums. We abbreviate the terms as follows

$\displaystyle u_{n,\ell}$
$\displaystyle=\mathbb{E}\big{[}\|e_{\ell}^{n}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{\ell}\mathbb{E}\big{[}\|e_{j}^{i}-e_{j-1}^{i}\|_{H}^{2}\big{]}+\eta\sum_{i=1}^{n}h_{i}\sum_{j=1}^{\ell}\mathbb{E}\big{[}\|e_{j}^{i}\|_{V_{j}}^{2}\big{]},$
$\displaystyle a$
$\displaystyle=h^{2\nu}2T\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{2}}+C_{\Gamma_{3}}+C_{\Gamma_{4},1}\big{)}+\varepsilon
2TC_{\Gamma_{5}},\quad
b_{i,\ell}=h_{i}2\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{4},2}\big{)}.$

Recall that
$\mathbb{E}\big{[}\|e_{s}^{n}\|_{H}^{2}\big{]}=\mathbb{E}\big{[}\|e^{n}\|_{H}^{2}\big{]}$.
Applying Lemma A.2 then shows that $u_{n,s}\leq
a\exp\big{(}\sum_{i=1}^{n-1}\sum_{\ell=1}^{s}b_{i,\ell}\big{)}$ or
equivalently

$\displaystyle\mathbb{E}\big{[}\|e^{n}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{i=1}^{n}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}-e_{\ell-1}^{i}\|_{H}^{2}\big{]}+\eta\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|e_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq\exp\big{(}2sT\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{4},2}\big{)}\big{)}\big{(}h^{2\nu}2T\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{2}}+C_{\Gamma_{3}}+C_{\Gamma_{4},1}\big{)}+\varepsilon
2TC_{\Gamma_{5}}\big{)},$

which is our desired bound. ∎

##  5\. Numerical experiment

In our numerical experiment, we consider a stochastic heat equation on a
domain $\mathcal{D}=(0,1)^{2}$ in the two dimensional space $\mathbb{R}^{2}$.
We allow for a nonlinear perturbation and possibly multiplicative noise. More
exactly, we look at an SPDE of the form

(5.1) $\displaystyle\begin{cases}\,\mathrm{d}u(t,x)-\alpha(t)\Delta
u(t,x)\,\mathrm{d}t=\big{(}\tilde{F}(t,x,u)+\tilde{G}(t,x)\big{)}\,\mathrm{d}t\\\
\hskip 128.0374pt+\tilde{B}(t,x,u)\,\mathrm{d}W(t,x),\quad
t\in(0,1],x\in\mathcal{D},\\\ u(t,x)=0,\hskip
180.67517ptt\in[0,1],x\in\partial\mathcal{D},\\\ u(0,x)=u_{0}(x),\hskip
159.3356ptx\in\mathcal{D}.\end{cases}$

In the following, we denote the Hilbert spaces $V=W_{0}^{1,2}(\mathcal{D})$,
$H=L^{2}(\mathcal{D})$ and $V^{*}=\big{(}W_{0}^{1,2}(\mathcal{D})\big{)}^{*}$.
The coefficients $\alpha$, $\tilde{F}$, $\tilde{G}$ and $\tilde{B}$, will be
explained in more detail in Sections 5.1 and 5.2. We then consider the
abstract functions and operators as defined in (3.6)–(3.9). These operators
fulfill the Assumptions 1, 2, 3 and 5 as explained in Sections 3.3 and [15,
Lemma 3.1].

The noise $W$ is a $Q$-Wiener process where the eigenvalues
$\\{q_{k}\\}_{k\in\mathbb{N}^{2}}$ and eigenfunctions
$\\{\psi_{k}\\}_{k\in\mathbb{N}^{2}}$ of $Q$ are given by

$\displaystyle
q_{k}=(k_{1}^{2}+k_{2}^{2})^{-\frac{1}{2}-\delta}\quad\text{and}\quad\psi_{k}(x)=2\sin(k_{1}\pi
x_{1})\sin(k_{2}\pi x_{2})$

for $\delta\in(0,\infty)$. Then we can represent $W$ and the truncated noise
$W^{K}$ through the Karhunen-Loève expansion (cf. [30, Section 10.2])

$\displaystyle
W(t,x)=\sum_{k\in\mathbb{N}^{2}}q_{k}^{\frac{1}{2}}\beta_{k}(t)\psi_{k}(x)\quad\text{and}\quad
W^{K}(t,x)=\sum_{|k|\leq K}q_{k}^{\frac{1}{2}}\beta_{k}(t)\psi_{k}(x),$

where $\\{\beta_{k}(t)\\}_{k\in\mathbb{N}^{2}}$ are independent and
identically distributed $\mathcal{F}_{t}$-Wiener processes,
$|k|=|(k_{1},k_{2})|=\max\\{|k_{1}|,|k_{2}|\\}$ and $K\in\mathbb{N}$. The
functions $\\{\psi_{k}\\}_{k\in\mathbb{N}^{2}}$ form an orthonormal basis of
$H=L^{2}(\mathcal{D})$ and they are eigenfunctions of the Laplace operator
with homogeneous Dirichlet boundary conditions. The noise therefore fits into
the setting explained in Assumption 4.

For $\alpha\equiv 1$ and sufficiently smooth $\tilde{F}$, $\tilde{B}$ and
$u_{0}$, we can verify that the exact solution lies in
$C^{\nu}([0,T];L^{2}(\Omega;V))$ for $\nu\in(0,\delta)$ using Theorem 2.2.
When choosing a very small value $\delta$ close to zeros, we should therefore
only observe a convergence order close to zero. This is a certain gap between
our error bound and the numerical results in the coming subsections. While
similar regularity assumptions have been made in [27], they seem to observe
the same gap as we do. A possible explanation could be that the example still
provides a more regular solution or that the same bound could also be observed
without the Hölder condition with values in $V$ for the exact solution. This
is a question left for future research.

For the domain decomposition, we divide the domain $(0,1)^{2}$ into four
strips in $x_{1}$-direction. We choose the weight functions $\chi_{\ell}$,
$\ell\in\\{1,\dots,s\\}$, as piecewise linear functions with an overlap that
is of the size $1/10$, compare Figure 1 for a visualisation.

Figure 1. Visuablization of the weight functions $\chi_{1},\chi_{2},\chi_{3}$
and $\chi_{4}$

Using these weight functions, we can introduce the split data as previously
defined in (3.6)–(3.9).

In our numerical experiment, we discretize (5.1) using the time discrtization
scheme (3.3). Moreover, to discretize the equation in space, we use a standard
finite difference approximation, where we subdivide the domain in both $x_{1}$
and $x_{2}$ direction in $128$ points. The noise is truncated with $K=128$ and
we choose regularisation parameter $\delta=0.001$. In this setting, the
spatial error and truncation error are small enough such that the temporal
error will be dominating the error. In order to model the expectation involved
in the mean-square error bound, we use $50$ Monte Carlo iterations.

###  5.1. Experiment 1

In the first example, we choose the functions

$\displaystyle u_{0}(x)$
$\displaystyle=5x_{1}^{2}(x_{1}-1)^{3}x_{2}^{2}(x_{2}-1)^{3},\quad\alpha(t)=0.1(1+\exp(-t))$
$\displaystyle\tilde{G}(t,x)$
$\displaystyle=0,\quad\tilde{F}(t,x,u)=\sin(u),\quad\tilde{B}(t,x,u)=1.$

The function $u_{0}$ is a smooth and bounded function on $\mathcal{D}$ and
therefore in particular in $H$. As it does not depend on $\omega$, it follows
directly that $u_{0}\in L^{2}(\Omega;H)$. Moreover, the function $\alpha$ is a
smooth function and the functions $\tilde{F}$, $\tilde{G}$ and $\tilde{B}$ do
not depend on $t$ in the following.

Thus, all functions satisfy the given assumptions on the time-regularity
stated in Theorem 4.4. The abstract function $G$ is zero as can be seen as
$\tilde{G}$ is zero which is inserted into (3.7). Moreover, since $\sin$ is a
Lipschitz continuous function, it follows that the abstract function $F$ given
through $\tilde{F}$ in (3.8) is Lipschitz continuous in the second argument.
Similarly, the abstract function $B$ given through $\tilde{B}$ in (3.9) is
Lipschitz continuous in the second argument as it is constant. Moreover, since
the sequence
$\\{(k_{1}^{2}+k_{2}^{2})^{-\frac{1}{2}-\delta}\\}_{k\in\mathbb{N}^{2}}$ is
summable, it follows that for every $\varepsilon_{\ell}>0$, we can find a
$K_{\ell}\in\mathbb{N}$ such that

$\displaystyle\sum_{|k|>K_{\ell}}\mathbb{E}\big{[}\|B_{\ell}(t,v)(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}\big{]}\leq\sum_{|k|>K_{\ell}}q_{k}=\sum_{|k|>K_{\ell}}(k_{1}^{2}+k_{2}^{2})^{-\frac{1}{2}-\delta}\leq\varepsilon_{\ell}.$

for every $v\in L^{2}(\Omega;H)$.

Altogether, we have a semi-linear equation with additive noise that fits into
the setting of Theorem 4.4 and observe a convergence rate of $\frac{1}{2}$ in
the numerical test presented on the left side of Figure 2.

Figure 2. Comparison of an IMEX-Euler–Maruyama scheme with and without a
domain decomposition. The left-hand side shows the result of Experiment 1 and
right-hand side the results of Experiment 2.

###  5.2. Experiment 2

In the second example, we consider a linear equation with multiplicative noise
with the coefficients

$\displaystyle u_{0}(x)$
$\displaystyle=5x_{1}^{2}(x_{1}-1)^{3}x_{2}^{2}(x_{2}-1)^{3},\quad\alpha(t)=0.1,\quad\tilde{G}(t,x)=0,$
$\displaystyle\tilde{F}(t,x,u)$
$\displaystyle=(3\sin(5x_{1})+2\cos(7x_{2}))(\cos(t)+\sin(4t)),\quad\tilde{B}(t,x,u)=u.$

We can argue in a similar fashion to that in Section 5.2 that the coefficient
fit into the setting required in Theorem 4.4. The coefficients are all smooth.
In contrast to Section 5.2 the corresponding diffusion operator $B$ is not
constant in the second argument but still linear and bounded. In order to
fulfill the truncation assumption (H4), we first note that
$\|\psi_{k}\|_{\infty}\leq 2$ for every $k\in\mathbb{N}^{2}$. Since
$\\{(k_{1}+k_{2})^{-\frac{1}{2}-\delta}\\}_{k\in\mathbb{N}^{2}}$ is summable,
we can then again argue that for every $\varepsilon_{\ell}>0$, there exists
$K_{\ell}\in\mathbb{N}$ such that

$\displaystyle\sum_{|k|>K_{\ell}}\mathbb{E}\big{[}\|B_{\ell}(t,v)(q_{k}^{\frac{1}{2}}\psi_{k})\|_{H}^{2}\big{]}\leq
4\sum_{|k|>K_{\ell}}(k_{1}^{2}+k_{2}^{2})^{-\frac{1}{2}-\delta}\mathbb{E}\big{[}\|v\|_{H}^{2}\big{]}\leq\varepsilon_{\ell}\mathbb{E}\big{[}\|v\|_{H}^{2}\big{]}.$

for every $v\in L^{2}(\Omega;H)$. Thus, the truncation assumption from (H4) is
fulfilled. Again, we observe a convergence order of $\frac{1}{2}$ in the right
error plot from Figure 2.

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##  Appendix A Basic inequalities

######  Lemma A.1.

For all $x,y\in\mathbb{R}$, $\xi>0$ it follows that
$|xy|\leq\xi|x|^{2}+\frac{1}{4\xi}|y|^{2}$.

A proof can be found [16, B.2. Elementary inequalities]. Further, we need a
form of Grönwall’s inequality in this paper. A proof for the standard
statement can be found in [9]. We use a variant that is based on a
rearrangement of the terms, where the summands are subdivided into two sums.

######  Lemma A.2.

Let $(u_{n,\ell})_{n\in\mathbb{N},\ell\in\\{1,\dots,s\\}}$ and
$(b_{n,\ell})_{n\in\mathbb{N},\ell\in\\{1,\dots,s\\}}$ be two nonnegative
sequences that satisfy, for given $a\in[0,\infty)$ and $N\in\mathbb{N}$, that

$\displaystyle u_{n,s}\leq
a+\sum_{i=1}^{n-1}\sum_{\ell=1}^{s}b_{i,\ell}u_{i,\ell},\quad
n\in\\{1,\dots,N\\}.$

Then it follows that

$\displaystyle u_{n,s}\leq
a\exp\Big{(}\sum_{i=1}^{n-1}\sum_{\ell=1}^{s}b_{i,\ell}\Big{)},\quad
n\in\\{1,\dots,N\\}.$

##  Appendix B Proof of moment bound

For the sake of completeness, we state the proof of Lemma 4.1 in the
following.

###### Proof of Lemma 4.1.

The idea of this proof follows the idea of an a priori bound that is commonly
used for the Rothe methode for deterministic (nonlinear) evolution equations.
A common step for variational solutions, we start by testing the equation
given by the scheme with the solution and use the coercivity and Lipschitz
continuity of the operators. A similar proof structure can be found in [33,
Lemma 8.6]. First, we consider a sub-step of the numerical scheme

$\displaystyle
U_{\ell}^{i}-U_{\ell-1}^{i}=h_{i}G_{\ell}(t_{i})+h_{i}F_{\ell}(t_{i-1},U_{\ell-1}^{i})-h_{i}A_{\ell}(t_{i},U_{\ell}^{i})+B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}},$

which we test with the solution $U_{\ell}^{i}$. Then we use the identity
$(a-b,a)_{H}=\frac{1}{2}\big{(}\|a\|_{H}^{2}-\|b\|_{H}^{2}+\|a-b\|_{H}^{2}\big{)}$
for $a,b\in H$ and sum up the equation from $\ell=1$ to $s$ to obtain

$\displaystyle\frac{1}{2}\sum_{\ell=1}^{s}\big{(}\|U_{\ell}^{i}\|_{H}^{2}-\|U_{\ell-1}^{i}\|_{H}^{2}+\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{)}=\sum_{\ell=1}^{s}(U_{\ell}^{i}-U_{\ell-1}^{i},U_{\ell}^{i})_{H}$
$\displaystyle=h_{i}\sum_{\ell=1}^{s}(F_{\ell}(t_{i-1},U_{\ell-1}^{i}),U_{\ell}^{i})_{H}+h_{i}\sum_{\ell=1}^{s}\langle
G_{\ell}(t_{i}),U_{\ell}^{i}\rangle_{V_{\ell}^{*}\times V_{\ell}}$
$\displaystyle\quad-h_{i}\sum_{\ell=1}^{s}\langle
A_{\ell}(t_{i},U_{\ell}^{i}),U_{\ell}^{i}\rangle_{V_{\ell}^{*}\times
V_{\ell}}+\sum_{\ell=1}^{s}(B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}},U_{\ell}^{i})_{H}$
$\displaystyle=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\Gamma_{4}.$

In the following, we bound the single terms separately. By adding and
subtracting the term $U_{\ell-1}^{i}$ in the right side of the inner product
and using the Cauchy-Schwarz inequality, the first term $\Gamma_{1}$ can be
bounded by

$\displaystyle\Gamma_{1}$ $\displaystyle\leq
h_{i}\sum_{\ell=1}^{s}\|F_{\ell}(t_{i-1},U_{\ell-1}^{i})\|_{H}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}+h_{i}\sum_{\ell=1}^{s}\|F_{\ell}(t_{i-1},U_{\ell-1}^{i})\|_{H}\|U_{\ell-1}^{i}\|_{H}.$

Further applying Lemma A.1 with $\xi=2h$ and $\xi=\frac{1}{4}$ respectively,
as well as the Lipschitz continuity of $F_{\ell}$ and boundedness of
$F_{\ell}(t_{i-1},0)$ in the $H$-norm, we can bound $\Gamma_{1}$ by

$\displaystyle\Gamma_{1}$ $\displaystyle\leq
h_{i}\sum_{\ell=1}^{s}\Big{(}2h\|F_{\ell}(t_{i-1},U_{\ell-1}^{i})\|_{H}^{2}+\frac{1}{8h}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\Big{)}$
$\displaystyle\quad+h_{i}\sum_{\ell=1}^{s}\Big{(}\frac{1}{4}\|F_{\ell}(t_{i-1},U_{\ell-1}^{i})\|_{H}^{2}+\|U_{\ell-1}^{i}\|_{H}^{2}\Big{)}$
$\displaystyle\leq
h_{i}\Big{(}4h+\frac{1}{2}\Big{)}\sum_{\ell=1}^{s}\|F_{\ell}(t_{i-1},U_{\ell-1}^{i})-F_{\ell}(t_{i-1},0)\|_{H}^{2}+h_{i}\Big{(}4h+\frac{1}{2}\Big{)}sK_{F}^{2}$
$\displaystyle\quad+\frac{1}{8}\sum_{\ell=1}^{s}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}+h_{i}\sum_{\ell=1}^{s}\|U_{\ell-1}^{i}\|_{H}^{2}$
$\displaystyle\leq
h_{i}\Big{(}4hL_{F}^{2}+\frac{L_{F}^{2}}{2}+1\Big{)}\sum_{\ell=1}^{s}\|U_{\ell-1}^{i}\|_{H}^{2}+h_{i}\Big{(}4h+\frac{1}{2}\Big{)}sK_{F}^{2}+\frac{1}{8}\sum_{\ell=1}^{s}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2},$

where we abbreviate $C_{\Gamma_{1},1}:=4hL_{F}^{2}+\frac{L_{F}^{2}}{2}+1$ and
$C_{\Gamma_{1},2}:=(4h+\frac{1}{2})sK_{F}^{2}$. Furthermore, for the second
summand $\Gamma_{2}$, we start by applying Cauchy–Schwarz inequality for the
duality pairing. We can then apply Lemma A.1 to decompose the product into two
summands, one containing the term $\|G_{\ell}(t_{i})\|_{V_{\ell}^{*}}$ and one
that we will compensate through the coercivity of $A_{\ell}(t_{i},\cdot)$.
More precesely, we apply Lemma A.1 with the constant $\xi=\frac{1}{2\mu}$
where $\mu$ is connected to the coercivity of $A_{\ell}(t_{i},\cdot)$ and then
obtain

$\displaystyle\Gamma_{2}\leq
h_{i}\sum_{\ell=1}^{s}\|G_{\ell}(t_{i})\|_{V_{\ell}^{*}}\|U_{\ell}^{i}\|_{V_{\ell}}\leq
h_{i}\frac{1}{2\mu}\sum_{\ell=1}^{s}\|G_{\ell}(t_{i})\|_{V_{\ell}^{*}}^{2}+h_{i}\frac{\mu}{2}\sum_{\ell=1}^{s}\|U_{\ell}^{i}\|_{V_{\ell}}^{2},$

where we abbreviate
$C_{\Gamma_{2}}:=\frac{1}{2\mu}\sum_{\ell=1}^{s}\|G_{\ell}(t_{i})\|_{V_{\ell}^{*}}^{2}$
in the following. To bound the summand $\Gamma_{3}$, we now apply the
coercivity of $A_{\ell}(t_{i},\cdot)$ and find

$\displaystyle\Gamma_{3}=-h_{i}\sum_{\ell=1}^{s}\langle
A_{\ell}(t_{i},U_{\ell}^{i}),U_{\ell}^{i}\rangle_{V_{\ell}^{*}\times
V_{\ell}}\leq-h_{i}\mu\sum_{\ell=1}^{s}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}.$

It remains to bound the term $\Gamma_{4}$, which includes the Wiener process.
We then add and subtract the term
$\sum_{\ell=1}^{s}(B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}},U^{i-1})_{H}$ and again apply Lemma A.1 with $\xi=2s^{2}$ to
obtain

$\displaystyle\Gamma_{4}$ $\displaystyle\leq
2s^{2}\sum_{\ell=1}^{s}\|B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}}\|_{H}^{2}+\frac{1}{8s^{2}}\sum_{\ell=1}^{s}\|U_{\ell}^{i}-U^{i-1}\|_{H}^{2}$
$\displaystyle\quad+\sum_{\ell=1}^{s}(B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}},U^{i-1})_{H}=\Gamma_{4,1}+\Gamma_{4,2}+\Gamma_{4,3}.$

Next, we begin to consider the expectation of $\Gamma_{4,1}$ more closely. To
this extend, we first start to bound the expectation of the summands
$\|B_{\ell}(t_{i-1},U^{i-1})\Delta W^{i,K_{\ell}}\|_{H}^{2}$ separately.
Inserting the definition of the $Q$-Wiener process, applying the independence
of the Brownian motions $\beta_{k}$ and an application of the Itô isometry,
yields that

$\displaystyle\mathbb{E}\big{[}\|B_{\ell}(t_{i-1},U^{i-1})\Delta
W^{i,K_{\ell}}\|_{H}^{2}\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\Big{\|}\sum_{k\leq
K_{\ell}}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\,\mathrm{d}\beta_{k}(t)\Big{\|}_{H}^{2}\Big{]}$
$\displaystyle=\sum_{k\leq
K_{\ell}}\mathbb{E}\Big{[}\Big{\|}\int_{t_{i-1}}^{t_{i}}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\,\mathrm{d}\beta_{k}(t)\Big{\|}_{H}^{2}\Big{]}$
$\displaystyle=\sum_{k\leq
K_{\ell}}\int_{t_{i-1}}^{t_{i}}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\big{\|}_{H}^{2}\big{]}\,\mathrm{d}t$
$\displaystyle=h_{i}\sum_{k\leq
K_{\ell}}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{(}q_{k}^{\frac{1}{2}}\psi_{k}\big{)}\big{\|}_{H}^{2}\big{]}\leq
h_{i}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{\|}_{L_{2}^{0}}^{2}\big{]},$

where in the last step we used the fact that
$\\{q_{k}^{\frac{1}{2}}\psi_{k}\\}_{k\in\mathbb{N}}$ is an orthonormal basis
of $Q^{\frac{1}{2}}(H)$. With this in mind, we can find a suitable bound for
the expectation of $\Gamma_{4,1}$. We apply the boundedness of
$B_{\ell}(t_{i-1},0)$ and the Lipschitz continuity of
$B_{\ell}(t_{i-1},\cdot)$ in $L_{2}^{0}$ from Assumption 5 as well as the fact
that

$\displaystyle\|U^{i-1}\|_{H}^{2}=\|U_{0}^{i}\|_{H}^{2}\leq\|U_{0}^{i}\|_{H}^{2}+\|U_{1}^{i}\|_{H}^{2}+\dots+\|U_{s}^{i}\|_{H}^{2}=\sum_{\ell=1}^{s}\|U_{\ell-1}^{i}\|_{H}^{2},$

to find

$\displaystyle\mathbb{E}\big{[}\Gamma_{4,1}\big{]}$ $\displaystyle\leq
h_{i}2s^{2}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\big{\|}B_{\ell}(t_{i-1},U^{i-1})\big{\|}_{L_{2}^{0}}^{2}\big{]}\leq
h_{i}4L_{B}^{2}s^{3}\mathbb{E}\big{[}\|U^{i-1}\|_{H}^{2}\big{]}+h_{i}4s^{3}K_{B}^{2}$
$\displaystyle\leq
h_{i}4L_{B}^{2}s^{3}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell-1}^{i}\|_{H}^{2}\big{]}+h_{i}4K_{B}^{2}s^{3},$

where we abbreviate $C_{\Gamma_{4},1}:=4L_{B}^{2}s^{3}$ and
$C_{\Gamma_{4},2}:=4K_{B}^{2}s^{3}$. Next we bound the expectation of
$\Gamma_{4,2}$. This can be done by inserting a telescopic sum, which we then
bound by an application of Hölder’s inequality for sums. Then it follows that

$\displaystyle\mathbb{E}\big{[}\Gamma_{4,2}\big{]}$
$\displaystyle=\frac{1}{8s^{2}}\sum_{\ell=1}^{s}\mathbb{E}\Big{[}\Big{\|}\sum_{j=1}^{\ell}\big{(}U_{j}^{i}-U_{j-1}^{i}\big{)}\Big{\|}_{H}^{2}\Big{]}\leq\frac{1}{8s^{2}}\sum_{\ell=1}^{s}\mathbb{E}\Big{[}\ell\sum_{j=1}^{\ell}\big{\|}U_{j}^{i}-U_{j-1}^{i}\big{\|}_{H}^{2}\Big{]}$
$\displaystyle\leq\frac{1}{8s^{2}}\sum_{\ell=1}^{s}\mathbb{E}\Big{[}s\sum_{j=1}^{s}\big{\|}U_{j}^{i}-U_{j-1}^{i}\big{\|}_{H}^{2}\Big{]}=\frac{1}{8}\sum_{j=1}^{s}\mathbb{E}\big{[}\big{\|}U_{j}^{i}-U_{j-1}^{i}\big{\|}_{H}^{2}\big{]}.$

Due to the stochastic independence of $B(t_{i-1},U^{i-1})$ and $\Delta
W^{i,K_{\ell}}$, we observe that the last term $\Gamma_{4,3}$ disappears in
expectation. Thus, combining all the previous bounds, it follows that

$\displaystyle\frac{1}{2}\Big{(}\mathbb{E}\big{[}\|U^{i}\|_{H}^{2}\big{]}-\mathbb{E}\big{[}\|U^{i-1}\|_{H}^{2}\big{]}+\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{]}\Big{)}$
$\displaystyle\leq
h_{i}\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{4},1}\big{)}\sum_{\ell=1}^{s}\|U_{\ell-1}^{i}\|_{H}^{2}+h_{i}\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{2}}+C_{\Gamma_{4},2}\big{)}$
$\displaystyle-
h_{i}\frac{\mu}{2}\sum_{\ell=1}^{s}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}+\frac{1}{4}\sum_{j=1}^{s}\mathbb{E}\big{[}\big{\|}U_{j}^{i}-U_{j-1}^{i}\big{\|}_{H}^{2}\big{]}.$

Rearranging the terms and multiplying the result by $2$, shows that

$\displaystyle\mathbb{E}\big{[}\|U^{i}\|_{H}^{2}\big{]}-\mathbb{E}\big{[}\|U^{i-1}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{]}+h_{i}\mu\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq
h_{i}2\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{4},1}\big{)}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell-1}^{i}\|_{H}^{2}\big{]}+h_{i}2\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{2}}+C_{\Gamma_{4},2}\big{)}.$

We sum up the inequality from $i=1$ to $n$ and use a telescoping sum argument
to obtain

$\displaystyle\mathbb{E}\big{[}\|U^{n}\|_{H}^{2}\big{]}-\mathbb{E}\big{[}\|U^{0}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{i=1}^{n}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{]}+\mu\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq
2\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{4},1}\big{)}\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell-1}^{i}\|_{H}^{2}\big{]}+2T\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{2}}+C_{\Gamma_{4},2}\big{)}.$

For the next step, we will apply Grönwall’s inequality analogously to the
proof of Theorem 4.4. Then we find

$\displaystyle\mathbb{E}\big{[}\|U^{n}\|_{H}^{2}\big{]}+\frac{1}{2}\sum_{i=1}^{n}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}-U_{\ell-1}^{i}\|_{H}^{2}\big{]}+\mu\sum_{i=1}^{n}h_{i}\sum_{\ell=1}^{s}\mathbb{E}\big{[}\|U_{\ell}^{i}\|_{V_{\ell}}^{2}\big{]}$
$\displaystyle\leq\exp\big{(}2sT\big{(}C_{\Gamma_{1},1}+C_{\Gamma_{4},1}\big{)}\big{)}\big{(}\mathbb{E}\big{[}\|U^{0}\|_{H}^{2}\big{]}+2T\big{(}C_{\Gamma_{1},2}+C_{\Gamma_{2}}+C_{\Gamma_{4},2}\big{)}\big{)},$

which is finite. ∎