labelkeyrgb0.6,0,1 violetrgb0.580,0.,0.827

# Generic Numerical Analysis of Stochastic Reaction Diffusion Model with
applications in excitable media

Yahya Alnashri  Department of Mathematics, Al-Qunfudah University College, Umm
Al-Qura University, Saudi Arabia<EMAIL_ADDRESS>and  Hasan Alzubaidi
Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura
University, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

The stochastic reaction-diffusion model driven by a multiplicative noise is
examined. We construct the gradient discretisation method (GDM), an abstract
framework combining several numerical method families. The paper provides the
discretisation and proves the convergence of the approximate schemes using a
compactness argument that works under natural assumptions on data. We also
investigate, using a finite volume method, known as the hybrid mixed mimetic
(HMM) approach, the effects of multiplicative noise on the dynamics of the
travelling waves in the excitable media displayed by the model. Particularly,
we consider how sufficiently high noise can cause waves to backfire or fail to
propagate.

###### Key words and phrases:

Stochastic reaction-diffusion model, Multiplicative noise, A gradient
discretisation method (GDM), Gradient schemes, Convergence analysis, Non
conforming numerical methods, Finite volume schemes, Hybrid mixed mimetic
(HMM) method, Wave propagation failure, Backfiring of waves.

##  1\. Introduction

A very active research area, stochastic reaction-diffusion models are now
considered essential modelling instruments for describing and analysing wave
phenomena that occur in excitable media [24, 35, 13]. A case in point is the
electrical activity waves that travel through cardiac muscle to produce a
heartbeat. These technologies are based on the premise that spiral waves and
the onset of arrhythmias are caused by noise and irregularities [30]. Noise-
induced patterns in excitable media also include the generation and
propagation of travelling pulses along nerve cells [4, 27], and the waves of
excitation in chemical reaction systems [31].

####

In the current work, we look on the following stochastic reaction diffusion
model driven by a multiplicative noise:

$\displaystyle\,\mathrm{d}\bar{v}-\mathop{\rm
div}(\nabla\bar{v})\,\mathrm{d}t=S(\bar{v})\,\mathrm{d}t+g(\bar{v})\,\mathrm{d}B(t)\quad\mbox{
in }\quad\Omega_{T}:={\mathbb{D}}\times(0,T),$ (1.1a)
$\displaystyle\bar{v}=0,\quad\mbox{ on }\quad\partial{\mathbb{D}}\times(0,T),$
(1.1b) $\displaystyle\bar{v}(\cdot,0)=v_{0}\quad\mbox{ in }\quad{\mathbb{D}},$
(1.1c)

where $T>0$, ${\mathbb{D}}\subset{\mathbb{R}}^{d}$, $d=\\{1,2,3\\}$, with an
initial condition $v_{0}\in L^{2}({\mathbb{D}})$. Moreover,
$B:=\\{B(t):t\in[0,T]\\}$ is a Brownian motion on a Hilbert space
${\mathcal{H}}$ with a covariance operator ${\mathcal{Q}}$. Precise
assumptions on data will be detailed in the subsequent section.

A wealthy body of theoretical framework has been built for stochastic reaction
diffusion equations; see for example [25, 26]. However, the irregularity of
the noise term that perturbs the model makes developing an analytical solution
for such a model difficult. Consequently, in this case, accurate numerical
solutions are essential. Models of this nature have been studied and analysed
using a variety of numerical approaches, including finite difference methods
[29, 28, 23], spectral methods [30, 19, 34] and, finite element methods [5,
33, 20]. Most earlier research, it appears, concentrated only on conforming
numerical schemes, which are unable to maintain the model’s physical
properties. Therefore, the main objective of our work is to develop a generic
numerical analysis of the equation (1.1) by means of the gradient
discretisation method (GDM) [7]. The GDM approach involves a broad class of
both conforming and non-conforming schemes. For instance, hybrid high-order
and virtual elements methods, hybrid mimetic mixed methods, mixed finite
element methods, and conforming and non-conforming finite element methods. Our
findings are applicable to all these families of methods. As this approach
removes the need to prove convergence for each framework, it greatly
simplifies convergence analysis research.

A convergence analysis based on the GDM framework has been described in the
literature for the deterministic version of the stochastic reaction diffusion
model ($g(\bar{v})=0$). Some numerical examples based upon the finite volume
approach that fit in the GDM approach were also presented to verify the
theoretical outcomes [1, 2]. To the best of our knowledge, the extant
literature has not yet presented a generic numerical analysis of the
stochastic reaction diffusion model (1.1). The GDM analysis was presented for
a general stochastic Stefan problem with multiplicative noise [9] and a
general stochastic evolution problem based on a Leray-Lions type operator [8].
In this work, we provide a comprehensive and unified convergence analysis of
the stochastic reaction diffusion equation (1.1) using the GDM method. To the
best of our knowledge, this analysis for such a model yields the first results
applicable to various conforming and non-conforming approaches.

By using a finite volume method, known as the hybrid mimetic mixed (HMM) [11],
we also include some numerical tests to investigate and evaluate the behaviour
of the travelling wave solutions of the investigated model under the influence
of the multiplicative noise. Noise generally has several effects on excitable
media; see [17] and cited works within for a thorough analysis. In this work,
we investigate two noise-induced travelling wave dynamics phenomena that the
investigated model displays using the HMM approach. The first is referred to
as the phenomenon of wave propagation failure because of high noise, where no
waves can be observed in this case [18]. Thus, when the noise rises, a
travelling waveform at first starts to lose its shape. Thereafter, the wave is
destroyed when the noise becomes strong enough [16]. The second wave dynamics
phenomenon that the model under study displays is wave backfiring. In this
regime, noise-related increases in travelling wave widths. The wave can be
split into two halves that travel in opposite directions at the same speed if
the transition from the excited to the quiescent states is induced in the
middle of the wave for sufficiently large noise [14, 22].

The remainder of this paper is presented as follows. We present the weak
martingale solutions and mathematical formulation of the model in Section 2.
The approximate scheme is covered in Section 3. Section 4 establishes abound
on numerical solutions and the noise term, which are essential to obtain the
convergence results. In Section 5, we study and prove the main results,
Theorem 5.2. Section 6 gives some numerical tests that investigate the
behaviour of the stochastic Allen-Cahn equation solutions as an example of the
stochastic reaction-diffusion model (1.1). Moreover, we investigate the
influences of the multiplicative noise on the dynamics of such a model’s
travelling wave solutions. The last section contains our final thoughts and
observations.

##  2\. Preliminaries

### Notations and assumptions:

Let the spatial domain ${\mathbb{D}}\subset{\mathbb{R}}^{d}\;(d\geq 1)$ be an
open bounded connected set and the triple
$(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a probability space, where
$({\mathcal{F}}_{t})_{t\in[0,T]}$ is a filtration satisfying the classical
conditions. Given $T\in(0,\infty)$, let
$(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\in[0,T]},{\mathbb{P}})$ be a
stochastic basis. Respect to this stochastic basis, let $B=\\{B(t):0\leq t\leq
T\\}$ denotes an ${\mathcal{H}}$-valued ${\mathcal{Q}}$-Brownian motion with
covariance operator ${\mathcal{Q}}$ such that
$\mbox{Tr}({\mathcal{Q}})<\infty$, and which is of the form:

$B(t)=\displaystyle\sum_{k=0}^{\infty}q_{K}B_{K}(t)e_{K},$ (2.1)

where $\\{e_{K}\\}_{K\in{\mathbb{N}}}$ is an orthonormal basis of
${\mathcal{H}}$ consisting of eigenvectors of ${\mathcal{Q}}$ with finite sum
of non negative eigenvalues $q_{K}$. Here $\\{B_{k}\\}_{k\in{\mathbb{N}}}$ is
a family of independent ${\mathcal{F}}_{t}$–adapted scalar Brownian motion.
The Banach space
${\mathcal{L}}({\mathcal{H}},L^{2}({\mathbb{D}})):=\\{{\mathcal{L}}:{\mathcal{H}}\to
L^{2}({\mathbb{D}})\;:\;{\mathcal{L}}\mbox{ is a bounded linear operator
}\\}$, and it is endowd with the operator norm denoted by
$\|\cdot\|_{{\mathcal{L}}({\mathcal{H}},L^{2}({\mathbb{D}}))}$. Throughout
this paper, $\langle\cdot,\cdot\rangle_{L^{2}}$ denotes the inner product in
$L^{2}({\mathbb{D}})$, and ${\mathbb{E}}[\cdot]$ denotes the expectation
operator. Moreover, we assume the following standard conditions:

######  Hypothesis 2.1.

The model’s data is assumed to satisfy the following:

  * •

Let ${\mathbb{D}}\subset{\mathbb{R}}^{d}\;(d\geq 1)$ be an open bounded
connected set, and $T>0$.

  * •

Let $g:L^{2}({\mathbb{D}})\to{\mathcal{L}}({\mathcal{H}},L^{2}({\mathbb{D}}))$
be a continuous functional satisfying the following condition:

$\displaystyle\exists g_{1},g_{2}>0\mbox{ such that, for all }\varphi\in
L^{2}({\mathbb{D}}),$ (2.2)
$\displaystyle\|g(\varphi)\|_{{\mathcal{L}}({\mathcal{H}},L^{2}({\mathbb{D}}))}^{2}\leq
g_{1}\|\varphi\|_{L^{2}({\mathbb{D}})}^{2}+g_{2}.$

  * •

Let the reaction function $S$ satisfy the Lipschitz continuity condition with
a positive Lipschitz constant $L$.

  * •

$v_{0}\in L^{2}({\mathbb{D}})$.

### Weak formulation:

Under the above assumptions on model data, a solution to the stochastic
equation (1.1) can be understood in a variational form. Let the solution be a
stochastic process taking values in $H_{0}^{1}({\mathbb{D}})$. This means

$\bar{v}:[0,T]\times{\mathbb{D}}\to
H_{0}^{1}({\mathbb{D}}),\quad(t,\omega)\mapsto\bar{v}(t,\cdot,s).$

Multiplying the equation (1.1) with a test function $\varphi\in
H_{0}^{1}({\mathbb{D}})$, integrating over the time-space domain and
performing integration by parts over the spatial variable generates the weak
solution of our model, as in the following definition.

######  Definition 2.2.

Assume that $v_{0}\in
L^{2}(\Omega,{\mathcal{F}}_{0},{\mathbb{P}},L^{2}({\mathbb{D}}))$. A
continuous $\bar{v}:[0,T]\times{\mathbb{D}}\to H_{0}^{1}({\mathbb{D}})$ is a
weak solution of (1.1) if

  * •

the process $\bar{v}$ is a progressively measurable,

  * •

$\bar{v}(\cdot,\theta)\in C([0,T];L_{\rm w}^{2}({\mathbb{D}}))$ for
$\mathbb{P}$-a.s. $\theta\in\Omega$, where $L_{\rm w}^{2}({\mathbb{D}})$ is
equipped with a weak topology,

  * •

${\mathbb{E}}\Big{[}\displaystyle\sup_{t\in[0,T]}\|\bar{v}(t)\|_{L^{2}({\mathbb{D}})}^{2}\Big{]}<\infty$
and
${\mathbb{E}}\Big{[}\|\bar{v}\|_{L^{2}(0,T;H_{0}^{1}({\mathbb{D}}))}^{2}\Big{]}<\infty$,

  * •

the following inequality holds $\mathbb{P}$-almost surely for all $t\in[0,T]$
and for all $\varphi\in H_{0}^{1}({\mathbb{D}})$,

$\displaystyle\langle\bar{v}(t),\varphi\rangle_{L^{2}}$ $\displaystyle-\langle
v_{0},\varphi\rangle_{L^{2}}+\int_{0}^{t}\langle\nabla\bar{v}(s),\nabla\varphi\rangle_{L^{2}}\,\mathrm{d}s$
(2.3) $\displaystyle=\int_{0}^{t}\langle
g(\bar{v})(s,\cdot)\,\mathrm{d}B(s),\varphi\rangle_{L^{2}}+\int_{0}^{t}\langle
S(\bar{v}(s)),\varphi\rangle_{L^{2}}\,\mathrm{d}s,$

in which the first integral in the RHS is the Itô integral in
$L^{2}({\mathbb{D}})$.

##  3\. Approximate Scheme

Here, we use the gradient discretisation method to build our generic analysis.
To construct this framework, we start by defining some discrete space and
operators as follows:

######  Definition 3.1.

Let ${\mathbb{D}}$ be an open subset of ${\mathbb{R}}^{d}$ (with $d\geq 1$)
and $T>0$. A gradient discretisations considered here is
${\mathcal{D}}=(X_{{\mathcal{D}},0},\Pi_{\mathcal{D}},\nabla_{\mathcal{D}},J_{\mathcal{D}},(t^{(n)})_{n=0,...,N}))$,
where

  1. (1)

the set of discrete unknowns $X_{{\mathcal{D}},0}$ is a finite dimensional
vector space on ${\mathbb{R}}$,

  2. (2)

the linear mapping $\Pi_{\mathcal{D}}:X_{{\mathcal{D}},0}\to
L^{2}({\mathbb{D}})$ is the reconstructed function,

  3. (3)

the linear mapping $\nabla_{\mathcal{D}}:X_{{\mathcal{D}},0}\to
L^{2}({\mathbb{D}})^{d}$ is a reconstructed gradient, which must be chosen
such that

$\|\varphi\|_{\mathcal{D}}=\|\nabla_{\mathcal{D}}\varphi\|_{L^{2}({\mathbb{D}})^{d}}$
(3.1)

is a norm on $X_{{\mathcal{D}},0}$,

  4. (4)

the linear continuous mapping $J_{\mathcal{D}}:L^{2}({\mathbb{D}})\to
X_{{\mathcal{D}},0}$ is an interpolation operator for the initial condition,

  5. (5)

$t^{(0)}=0<t^{(1)}<....<t^{(N)}=T$ are time steps.

For $(u^{(n)})_{n\in{\mathbb{N}}}\subset X_{{\mathcal{D}},0}$, the functions
$\Pi_{\mathcal{D}}u:(0,T]\to L^{2}({\mathbb{D}})$,
$\nabla_{\mathcal{D}}u:(0,T]\to L^{2}({\mathbb{D}})^{d}$, and
$\delta_{\mathcal{D}}u:(0,T]\to L^{2}({\mathbb{D}})$ are piecewise constant in
time and given by: For a.e $x\in{\mathbb{D}}$, for all $n\in\\{0,...,N-1\\}$
and for all $t\in(t^{(n)},t^{(n+1)}]$,

$\displaystyle\Pi_{\mathcal{D}}u(x,0):=\Pi_{\mathcal{D}}u^{(0)}(x),\quad\Pi_{\mathcal{D}}u(x,t):=\Pi_{\mathcal{D}}u^{(n+1)}(x),$
$\displaystyle\nabla_{\mathcal{D}}u(x,t):=\nabla_{\mathcal{D}}u^{(n+1)}(x),\quad\delta_{\mathcal{D}}u(t)=\delta_{\mathcal{D}}^{(n+\frac{1}{2})}u:=\frac{\Pi_{\mathcal{D}}(u^{(n+1)}-u^{(n)})}{\delta
t^{(n+\frac{1}{2})}},$

with takink $\delta t^{(n+\frac{1}{2})}=t^{(n+1)}-t^{(n)}$ and $\delta
t_{\mathcal{D}}=\max_{n=0,...,N-1}\delta t^{(n+\frac{1}{2})}$.

Using the elements of ${\mathcal{D}}$ in the problem (2.3) yields an
approximate method called a gradient scheme.

######  Definition 3.2.

The related gradient scheme for the continuous problem (2.3) is to seek $v\in
X_{{\mathcal{D}},0}$ satisfying the following inequality:

$\displaystyle\langle
d_{\mathcal{D}}^{(n+\frac{1}{2})}v,\Pi_{\mathcal{D}}\varphi\rangle_{L^{2}}+\delta
t_{\mathcal{D}}\langle\nabla_{\mathcal{D}}v^{(n+1)},\nabla_{\mathcal{D}}\varphi\rangle_{L^{2}}$
(3.2) $\displaystyle\quad=\langle
g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}\varphi\rangle_{L^{2}}+\delta
t_{\mathcal{D}}\langle
S(\Pi_{\mathcal{D}}v^{(n)}),\Pi_{\mathcal{D}}\varphi\rangle_{L^{2}},\quad\varphi\in
X_{{\mathcal{D}},0}.$

The gradient discretisation elements must enjoy some properties to build
convergent schemes, as stated below. Note that, in practice, different
numerical methods encompassed in the gradient discretisation framework verify
these properties. See [7, Chapter 7], for more details.

First, for $u_{1}\in H_{0}^{1}({\mathbb{D}})$, let

$S_{\mathcal{D}}(u_{1})=\min_{u_{2}\in
X_{{\mathcal{D}},0}}\left(\|\Pi_{\mathcal{D}}u_{2}-u_{1}\|_{L^{2}({\mathbb{D}})}+\|\nabla_{\mathcal{D}}u_{2}-\nabla
u_{1}\|_{L^{2}({\mathbb{D}})^{d}}\right).$ (3.3)

Next, for ${\boldsymbol{\theta}}\in H_{\rm div}:=\\{{\boldsymbol{\theta}}\in
L^{2}({\mathbb{D}})^{d}\;:\;{\rm div}{\boldsymbol{\psi}}\in
L^{2}({\mathbb{D}}),\;{\boldsymbol{\theta}}\cdot\mathbf{n}=0\mbox{ on
}\partial{\mathbb{D}}\\}$, let

$W_{\mathcal{D}}({\boldsymbol{\theta}})=\max_{u_{2}\in
X_{{\mathcal{D}},0}\setminus\\{0\\}}\frac{\Big{|}\langle\nabla_{\mathcal{D}}u_{2},{\boldsymbol{\theta}}\rangle_{L^{2}({\mathbb{D}})^{d}}+\langle\Pi_{\mathcal{D}}u_{2},\mathop{\rm
div}({\boldsymbol{\theta}})\rangle_{L^{2}({\mathbb{D}})}\Big{|}}{\|u_{2}\|_{\mathcal{D}}}.$
(3.4)

######  Definition 3.3.

A sequence $({\mathcal{D}}_{m})_{m\in{\mathbb{N}}}$ of gradient
discretisations is

  * •

consistent if

    1. (1)

$\displaystyle\lim_{m\to\infty}S_{{\mathcal{D}}_{m}}(u_{1})=0$, for all
$u_{1}\in H_{0}^{1}({\mathbb{D}})$,

    2. (2)

$\Pi_{{\mathcal{D}}_{m}}J_{{\mathcal{D}}_{m}}\varphi\to\varphi$ strongly in
$L^{2}({\mathbb{D}})$,, for all $u_{1}\in L^{2}({\mathbb{D}})$,

    3. (3)

$\delta t_{{\mathcal{D}}_{m}}\to 0$.

  * •

limit–conformity if
$\displaystyle\lim_{m\to\infty}W_{{\mathcal{D}}_{m}}({\boldsymbol{\theta}})=0$,
for all ${\boldsymbol{\theta}}\in H_{\rm div}$.

  * •

compact if for any sequence $(u_{m})_{m\in{\mathbb{N}}}\in
X_{{\mathcal{D}}_{m}}$, such that
$(\|u_{m}\|_{{\mathcal{D}}_{m}})_{m\in{\mathbb{N}}}$ is bounded, the sequence
$(\Pi_{{\mathcal{D}}_{m}}u_{m})_{m\in{\mathbb{N}}}$ is relatively compact in
$L^{2}({\mathbb{D}})$.

##  4\. Discrete Energy Estimates

Using the same reasonings as in [8], we establish here some estimates on
solutions and their gradients. Throughout this section, the letter $C$ denotes
different positive constants, which do not depend on the discretisation, and
$C_{0}:=|S(0)|$.

######  Lemma 4.1.

Let $v$ be a solution to the approximate problem (3.2). Then, for some
positive constant $C$, we have

$\displaystyle{\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{L^{2}({\mathbb{D}})}^{2}$
$\displaystyle+\|\nabla_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}}\times(0,T))^{d}}^{2}$
(4.1)
$\displaystyle+\sum_{n=0}^{N-1}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{L^{2}({\mathbb{D}})}^{2}\Big{]}$
$\displaystyle\leq C.$

###### Proof.

It is known that

$(a_{1}-a_{2})a_{1}=\frac{1}{2}(a_{1}^{2}-a_{2}^{2})+\frac{1}{2}(a_{1}-a_{2})^{2},\quad\forall
a_{1},a_{2}\in{\mathbb{R}}.$ (4.2)

Taking $\varphi=v^{(n+1)}$ in (3.2) as a test function and use the above
relation to get

$\displaystyle{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}+{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
(4.3) $\displaystyle\qquad+\delta
t_{\mathcal{D}}\|\nabla_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle={\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}+{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}(v^{(n+1)}-v^{(n)}){\rangle}$
$\displaystyle\qquad+{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}+\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}.$

Summing over $n=0,...,k$, with $k\in\\{0,...,N-1\\}$, and performing the
Cauchy–Schwarz inequality yield

$\displaystyle{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(k+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}+{\displaystyle\frac{1}{2}}\sum_{n=0}^{k}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\quad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\|\nabla_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}\|\Delta^{(n+1)}B\|_{\mathcal{H}}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}$
$\displaystyle\quad+\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\|S(\Pi_{\mathcal{D}}v^{(n+1)})\|_{{L^{2}({\mathbb{D}}\times(0,T))}}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}.$

Applying Young’s inequality $\alpha\beta\leq\alpha^{2}+\frac{\beta^{2}}{4}$ to
the second term on the RHS and the Lipschitz continuity assumption to the last
term on RHS, we have

$\displaystyle{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(k+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\frac{1}{4}\sum_{n=0}^{k}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
(4.4) $\displaystyle\quad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\|\nabla_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\Big{(}L\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+C_{0}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}\Big{)}.$

After applying Young’s inequality to the last term on the RHS, we have the
following inequality

$\displaystyle{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(k+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\frac{1}{4}\sum_{n=0}^{k}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
(4.5) $\displaystyle\qquad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\|\nabla_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\qquad+\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\qquad+\delta
t_{\mathcal{D}}\sum_{n=0}^{k}\Big{(}\frac{2L+1}{2}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+\frac{C_{0}^{2}}{2}\Big{)}.$

It is useful to see that the expectation of $\Delta^{(n+1)}B$ is zero, which
results in eliminating the third term on the RHS. Thus, taking the expectation
implies to

$\displaystyle{\displaystyle\frac{1}{2}}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(k+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\frac{1}{4}\sum_{n=0}^{k}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
(4.6) $\displaystyle\qquad+{\mathbb{E}}\Big{[}\sum_{n=0}^{k}\delta
t_{\mathcal{D}}\|\nabla_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}\Big{]}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\sum_{n=0}^{k}{\mathbb{E}}\Big{[}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{]}$
$\displaystyle\qquad+{\mathbb{E}}\Big{[}\sum_{n=0}^{k}\delta
t_{\mathcal{D}}\Big{(}\frac{2L+1}{2}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+\frac{C_{0}^{2}}{2}\Big{)}\Big{]}.$

The assumption on $g$ implies that [8, Equation 14] still holds. The second
term on the RHS is thus expressed as

${\mathbb{E}}[\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}]\leq\delta
t_{\mathcal{D}}({\rm
Tr}{\mathcal{Q}})(g_{1}{\mathbb{E}}[\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}]+g_{2}).$
(4.7)

Together with (4.6), this implies

$\displaystyle{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(k+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
(4.8)
$\displaystyle\leq\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+2({{\rm
Tr}}{\mathcal{Q}})g_{2}T+2({{\rm Tr}}{\mathcal{Q}})g_{1}\sum_{n=0}^{k}\delta
t_{\mathcal{D}}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
$\displaystyle\qquad+{\mathbb{E}}\Big{[}\sum_{n=0}^{k}\delta
t_{\mathcal{D}}(2L+1)\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+C_{0}^{2}\Big{]}.$

The consistency property gives, in which $S_{0}$ is a positive constant
depending on $v_{0}$

$\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}\leq S_{0}.$

Use this inequality and apply the discrete version of Gronwall’s lemma to
(4.8) to arrive at

$\max_{1\leq n\leq
N}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}\leq
C{\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}.$ (4.9)

Relations (4.6) and (4.9) give

$\displaystyle{\mathbb{E}}\Big{[}\|\nabla_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle+\sum_{n=0}^{N-1}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
$\displaystyle\leq C{\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}.$

Take the supremum of (4.5) on $k$ and the expectations to conclude

$\displaystyle{\mathbb{E}}\Big{[}\max_{1\leq n\leq
N-1}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$ (4.10)
$\displaystyle\leq\|\Pi_{\mathcal{D}}v^{(0)}\|_{{L^{2}({\mathbb{D}})}}^{2}+2{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{]}$
$\displaystyle\qquad+2{\mathbb{E}}\Big{[}\max_{0\leq k\leq
N-1}\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{]}$
$\displaystyle\qquad+\delta t_{\mathcal{D}}{\mathbb{E}}\Big{[}\max_{0\leq
k\leq
N-1}\sum_{n=0}^{k}\Big{(}(2L+1)\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+C_{0}^{2}\Big{)}\Big{]}.$

Now, the third term on the RHS can be written as, thanks to the stochastic
integral and the BurkHölder–Davis–Gundy inequality: [11, Theorem 2.4]

$\displaystyle{\mathbb{E}}\Big{[}\max_{0\leq k\leq
N-1}\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{]}$
(4.11) $\displaystyle\leq\frac{1}{4}{\mathbb{E}}\Big{[}\max_{0\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}+C^{2}g_{1}T\max_{0\leq
n\leq
N}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
$\displaystyle\qquad+C^{2}g_{2}T.$

With (4.7), the second term on the RHS of (4.10) is estimated as

$\displaystyle{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{]}$
(4.12) $\displaystyle\leq({\rm Tr}{\mathcal{Q}})g_{1}T\max_{1\leq n\leq
N}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}+({{\rm
Tr}}{\mathcal{Q}})g_{2}T.$

Substituting estimates (4.9), (4.11) and (4.12) in inequality (4.10) gives

${\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}\leq C.$

∎

######  Lemma 4.2.

Let $v$ be a solution to the approximate problem (3.2). Then, for some
positive constant $C$, we have

${\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{4}+\|\nabla
v^{(n)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{4}\Big{]}\leq C.$ (4.13)

###### Proof.

Introduce the following notation

$V^{(n)}:=\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}+\frac{\delta
t_{\mathcal{D}}}{2}\|\nabla_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}+A_{k},$

where $A_{K}$ is defined by

$A_{k}:=\displaystyle\sum_{i=0}^{k}\delta
t_{\mathcal{D}}\|\nabla_{\mathcal{D}}v^{(i)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}.$

We begin with deriving bound on the quantity
$\max_{n}{\mathbb{E}}\Big{[}(V^{(n)})^{2}\Big{]}$. Using the above formula,
Equality (4.3) becomes

$\displaystyle
V^{(n+1)}-V^{(n)}+\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}+\frac{\delta
t_{\mathcal{D}}}{2}\|\nabla_{\mathcal{D}}v^{(n+1)}-\nabla_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}$
$\displaystyle\leq
2{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}(v^{(n+1)}-v^{(n)}){\rangle}+{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+\delta
t_{\mathcal{D}}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}.$

Multiplying the above inequality by $V^{(n+1)}$ yields

$\displaystyle
V^{(n+1)}\Big{(}V^{(n+1)}-V^{(n)}\Big{)}+V^{(n+1)}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
(4.14) $\displaystyle\leq V^{(n+1)}\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}+{\mathbb{T}}_{1}+{\mathbb{T}}_{2},$

where

$\displaystyle{\mathbb{T}}_{1}:=2V^{(n+1)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}(v^{(n+1)}-v^{(n)}){\rangle},\mbox{
and }$
$\displaystyle{\mathbb{T}}_{2}:=V^{(n+1)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}.$

For the estimation of ${\mathbb{T}}_{1}$ and ${\mathbb{T}}_{2}$, we apply the
Cauchy–Schwarz and the Young inequalities. This gives

$\displaystyle{\mathbb{T}}_{1}$ $\displaystyle\leq
2V^{(n+1)}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\|\Pi_{\mathcal{D}}v^{(n+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
(4.15) $\displaystyle\leq
2V^{(n+1)}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+{\displaystyle\frac{1}{2}}V^{(n+1)}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\leq
2\Big{(}V^{(n+1)}-V^{(n)}\Big{)}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+2V^{(n)}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+{\displaystyle\frac{1}{2}}V^{(n+1)}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\leq\frac{1}{8}(V^{(n+1)}-V^{(n)}\Big{)}^{2}+8\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{4}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{4}$
$\displaystyle\quad+2V^{(n)}\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+{\displaystyle\frac{1}{2}}V^{(n+1)}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2},$

and

$\displaystyle{\mathbb{T}}_{2}$
$\displaystyle={\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
(4.16)
$\displaystyle=\Big{(}V^{(n+1)}-V^{(n)}\Big{)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+V^{(n)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\leq\frac{1}{16}\Big{(}V^{(n+1)}-V^{(n)}\Big{)}^{2}+4\|g(\Pi_{\mathcal{D}}v^{(n)})\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\quad+V^{(n)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}.$

We know that

$\displaystyle\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\leq
V^{(n)},\mbox{ and }\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\leq
g_{1}V^{(n)}+g_{2}.$ (4.17)

Using the moments’ bounds (4.15) and (4.16), we conclude from (4.14) that,
thanks to (4.17)

$\displaystyle{\displaystyle\frac{1}{2}}\Big{(}V^{(n+1)}\Big{)}^{2}-{\displaystyle\frac{1}{2}}\Big{(}V^{(n)}\Big{)}^{2}+\frac{5}{16}\Big{(}V^{(n+1)}-V^{(n)}\Big{)}^{2}$
(4.18)
$\displaystyle\quad+{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\leq
8\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{4}+10V^{(n)}\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+V^{(n)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+V^{(n+1)}\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}.$

We use the Cauchy–Shwarz and the Young inequalities, the Lipschitz continuity
condition, and (4.17) to deal with the last term on the RHS. It follows that

$\displaystyle{\displaystyle\frac{1}{2}}\Big{(}V^{(n+1)}\Big{)}^{2}-{\displaystyle\frac{1}{2}}\Big{(}V^{(n)}\Big{)}^{2}+\frac{5}{16}\Big{(}V^{(n+1)}-V^{(n)}\Big{)}^{2}$
(4.19)
$\displaystyle\quad+{\displaystyle\frac{1}{2}}\|\Pi_{\mathcal{D}}v^{(n+1)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\leq
8\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{4}+10V^{(n)}\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}$
$\displaystyle\quad+V^{(n)}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}$
$\displaystyle\quad+V^{(n+1)}\delta
t_{\mathcal{D}}\Big{(}\frac{2L+1}{2}V^{(n+1)}+\frac{C_{0}^{2}}{2}\Big{)}.$

The expectation of the third term on the RHS equals zero. With relation
(4.17), the first two terms on the RHS can be estimated as

$\displaystyle
8{\mathbb{E}}\Big{[}\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{4}\Big{]}+10{\mathbb{E}}\Big{[}V^{(n)}\Big{(}g_{1}V^{(n)}+g_{2}\Big{)}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{]}$
(4.20) $\displaystyle\leq 8\delta t_{\mathcal{D}}^{2}({{\rm
Tr}}({\mathcal{Q}}))^{2}{\mathbb{E}}\Big{[}(g_{1}V^{(n)}+g_{2})^{2}\Big{]}+10\delta
t_{\mathcal{D}}{{\rm
Tr}}({\mathcal{Q}}){\mathbb{E}}\Big{[}V^{(n)}(g_{1}V^{(n)}+g_{2})].$

Using the above expression, the inequality (4.1), and the discrete version of
Gronwall’s lemma, summing (4.19) over $n=0,...,k$, and taking the expectations
imply to

$\max_{1\leq n\leq N}{\mathbb{E}}\Big{[}\Big{(}V^{(n)}\Big{)}\Big{]}\leq C.$
(4.21)

We perform the formula (4.2) to (4.14) and sum over $n=0,...,k$ to conclude
that, thanks to the estimates (4.15) and (4.16)

$\displaystyle{\displaystyle\frac{1}{2}}\Big{(}v^{(k+1)}\Big{)}^{2}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}\Big{(}v^{(0)}\Big{)}^{2}+8\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{4}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{4}$
$\displaystyle\quad+2\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{(}V^{(n)}\Big{)}$
$\displaystyle\quad+4\sum_{n=0}^{k}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}$
$\displaystyle\quad+\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{(}V^{(n)}\Big{)}$
$\displaystyle\quad+\sum_{n=0}^{k}V^{(n+1)}\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}.$

Hence, we take a maximum over $k$ and then take expectation to attain

$\displaystyle{\mathbb{E}}\Big{[}\max_{1\leq n\leq
N-1}\Big{(}V^{(k+1)}\Big{)}^{2}\Big{]}$ (4.22)
$\displaystyle\leq\Big{(}V^{(0)}\Big{)}^{2}+16{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{4}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{4}\Big{]}$
$\displaystyle\quad+4{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\Big{(}V^{(n)}\Big{)}\Big{]}$
$\displaystyle\quad+8{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\|\Delta^{(n+1)}B\|_{\mathcal{H}}^{2}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
$\displaystyle\quad+2{\mathbb{E}}\Big{[}\max_{0\leq n\leq
N-1}\sum_{n=0}^{k}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{(}V^{(n)}\Big{)}\Big{]}$
$\displaystyle\quad+{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}V^{(n+1)}\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}\Big{]},$

Applying the BurkHölder–Davis–Gundy inequality (with constant $C$), one has

$\displaystyle
2{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}{\langle}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(n+1)}B,\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{(}V^{(n)}\Big{)}\Big{]}$
(4.23) $\displaystyle\leq 2C{\mathbb{E}}\Big{[}\Big{(}\sum_{n=0}^{N-1}\delta
t_{\mathcal{D}}\|g(\Pi_{\mathcal{D}})v^{(n)}\|_{\mathcal{H}}^{2}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{(}V^{(n)}\Big{)}^{2}\Big{)}^{\displaystyle\frac{1}{2}}\Big{]}$
$\displaystyle\leq 2C{\mathbb{E}}\Big{[}\max_{0\leq n\leq
N-1}V^{(n)}\Big{(}\sum_{n=0}^{N-1}\delta
t_{\mathcal{D}}(g_{1}V^{(n)}+g_{2})(V^{(n)})\Big{)}^{\displaystyle\frac{1}{2}}\Big{]}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}{\mathbb{E}}\Big{[}\max_{0\leq
n\leq N-1}(V^{(n)})^{2}\Big{]}+4C^{2}\sum_{n=0}^{N-1}\delta
t_{\mathcal{D}}{\mathbb{E}}\Big{[}g_{1}(V^{(n)})^{2}+g_{2}\Big{]}$
$\displaystyle\leq{\displaystyle\frac{1}{2}}{\mathbb{E}}\Big{[}\max_{0\leq
n\leq N-1}(V^{(n)})^{2}\Big{]}+4C^{2}T\max_{0\leq n\leq
N-1}\Big{(}{\mathbb{E}}\Big{[}(V^{(n)})^{2}\Big{]}+g_{2}\Big{)}.$

The last term on the RHS of (4.22) can be introduced as

$\displaystyle{\mathbb{E}}\Big{[}\sum_{n=0}^{N-1}V^{(n)}\delta
t_{\mathcal{D}}{\langle}S(\Pi_{\mathcal{D}}v^{(n+1)}),\Pi_{\mathcal{D}}v^{(n+1)}{\rangle}\Big{]}$
(4.24) $\displaystyle\leq{\mathbb{E}}\Big{[}\sum_{n=0}^{k}\delta
t_{\mathcal{D}}V^{(n)}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}\|S(\Pi_{\mathcal{D}}v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\max_{0\leq n\leq
N-1}(V^{(n)})\sum_{n=0}^{N-1}\delta
t_{\mathcal{D}}\|\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}\|S(\Pi_{\mathcal{D}}v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\max_{0\leq n\leq
N-1}(V^{(n)})^{2}+\Big{]}+C\max_{0\leq n\leq
N-1}{\mathbb{E}}\Big{[}(V^{(n)})^{2}\Big{]}+C\Big{]}.$

The estimates (4.19), (4.21), (4.22), (4.23), and (4.24) lead to

${\mathbb{E}}\Big{[}\max_{0\leq n\leq N-1}(V^{(n)})^{2}\Big{]}\leq C,$

which completes the proof of the assertion. ∎

######  Lemma 4.3.

If $v$ is a solution the approximate problem (3.2), then, for some positive
constant $C$, we have, for every $r\in\\{1,...,N-1\\}$

${\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}\displaystyle\sum_{n=0}^{N-r}\|\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}\leq
t^{(r)}C.$ (4.25)

###### Proof.

For an arbirtry $i\in\\{0,...,r\\}$, we sum (3.2) over $i$ to obtain

$\displaystyle\delta
t_{\mathcal{D}}\|\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}={\mathbb{T}}_{1}+{\mathbb{T}}_{2}+{\mathbb{T}}_{3},$
(4.26)

where

$\displaystyle{\mathbb{T}}_{1}:=-\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\sum_{i=0}^{r-1}{\langle}\nabla_{\mathcal{D}}v^{(n+i+1)},\nabla_{\mathcal{D}}v^{(n+r)}-\nabla_{\mathcal{D}}v^{(n)}{\rangle},$
(4.27) $\displaystyle{\mathbb{T}}_{2}:=\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}\sum_{i=0}^{r-1}{\langle}g(\Pi_{\mathcal{D}}v^{(n+i)})\Delta^{(n+i+1)}B,\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}{\rangle},\mbox{
and }$ $\displaystyle{\mathbb{T}}_{3}:=\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\sum_{i=0}^{r-1}{\langle}S(\Pi_{\mathcal{D}}v^{(n+i+1)}),\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}{\rangle}.$

We make use of Cauchy–Schwarz inequalities to get a bound on the expectation
of ${\mathbb{T}}_{1}$ as follows

$\displaystyle{\mathbb{E}}[{\mathbb{T}}_{1}]$
$\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}\|\nabla_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{L^{2}({\mathbb{D}})^{d}}\int_{t^{(n)}}^{t^{(n+r)}}\|\nabla_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})^{d}}\Big{]}$
(4.28)
$\displaystyle\leq\Big{(}t^{(r)}\Big{)}^{\frac{1}{2}}{\mathbb{E}}\Big{[}\Big{(}\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}\|\nabla_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{L^{2}({\mathbb{D}})^{d}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\quad\times\Big{(}\sum_{n=1}^{N-r}\delta
t_{\mathcal{D}}\int_{t^{(n)}}^{t^{(n+r)}}\|\nabla_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})^{d}}^{2}\Big{)}^{\frac{1}{2}}\Big{]}$
$\displaystyle\leq\Big{(}t^{(r)}\Big{)}^{\frac{1}{2}}{\mathbb{E}}\Big{[}\Big{(}2\int_{t^{(r)}}^{t^{(N)}}\|\nabla
v\|_{L^{2}({\mathbb{D}})^{d}}^{2}+2\int_{t^{(r)}}^{t^{(N)}}\|\nabla
v\|_{L^{2}({\mathbb{D}})^{d}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\quad+\Big{(}r\delta
t_{\mathcal{D}}\int_{t^{(1)}}^{t^{(N)}}\|\nabla
v\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{)}^{\frac{1}{2}}\Big{]}$
$\displaystyle\leq
2\Big{(}t^{(r)}\Big{)}^{\frac{1}{2}}{\mathbb{E}}\Big{[}\|\nabla_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}\Big{]}.$

We similarly use Cauchy–Schwarz inequalities to estimate the term
${\mathbb{T}}_{3}$, thanks to Lipschitz continuity assumption

$\displaystyle{\mathbb{E}}[{\mathbb{T}}_{3}]$
$\displaystyle={\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\sum_{i=0}^{r-1}{\langle}S(\Pi_{\mathcal{D}}v^{(n+i+1)}),\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{]}$
(4.29) $\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\sum_{i=0}^{r-1}\|S(\Pi_{\mathcal{D}}v^{(n+i+1)})\|_{{L^{2}({\mathbb{D}})}}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\sum_{i=0}^{r-1}L\|\Pi_{\mathcal{D}}v^{(n+i+1)}\|_{{L^{2}({\mathbb{D}})}}+C_{0}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}dsp\sum_{i=0}^{r-1}\|S(\Pi_{\mathcal{D}}v^{(n+i+1)})\|_{{L^{2}({\mathbb{D}})}}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}^{2}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\sum_{i=0}^{r-1}L\|\Pi_{\mathcal{D}}v^{(n+i+1)}\|_{{L^{2}({\mathbb{D}})}}+C_{0}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}\int_{t^{(n)}}^{t^{(n+r)}}L\|\Pi_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}})}}+C_{0}\Big{]}$
$\displaystyle\leq\Big{(}t^{(r)}\Big{)}^{\frac{1}{2}}{\mathbb{E}}\Big{[}\delta
t_{\mathcal{D}}\Big{(}\sum_{n=1}^{N-r}\|\Pi_{\mathcal{D}}(v^{(n+r)}-v^{(n)})\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\quad\times\Big{(}\sum_{n=1}^{N-r}\int_{t^{(n)}}^{t^{(n+r)}}L\|\Pi_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{)}^{\frac{1}{2}}+C_{0}\Big{]}$
$\displaystyle\leq\Big{(}t^{(r)}\Big{)}^{\frac{1}{2}}\Big{(}\delta
t_{\mathcal{D}}r\Big{)}^{\frac{1}{2}}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}u\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}.$

For the estimation of ${\mathbb{T}}_{2}$, we apply Young’s inequality to
attain

$\displaystyle{\mathbb{E}}[{\mathbb{T}}_{2}]$ (4.30) $\displaystyle=\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}{\mathbb{E}}\Big{[}{\langle}\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)g(\Pi_{\mathcal{D}}v(t))dB(t),\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}{\rangle}\Big{]}$
$\displaystyle\leq\frac{1}{4}\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
$\displaystyle\quad+\delta
t_{\mathcal{D}}\sum_{n=1}^{N-r}{\mathbb{E}}\Big{[}\|\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)g(\Pi_{\mathcal{D}}v(t))dB(t)\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}.$

Because of the Itô isometry, (2.2) and Lemma 4.1, the last term on the RHS is
estimated as

$\displaystyle{\mathbb{E}}\Big{[}\|\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)g(\Pi_{\mathcal{D}}v(t))dB(t)\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
(4.31) $\displaystyle\leq({{\rm
Tr}}{\mathcal{Q}}){\mathbb{E}}\Big{[}\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)\|g(\Pi_{\mathcal{D}}v(t))\|_{\mathcal{H}}^{2}\|\Big{]}$
$\displaystyle\leq({{\rm
Tr}}{\mathcal{Q}}){\mathbb{E}}\Big{[}\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)(g_{1}\|\Pi_{\mathcal{D}}v(t)\|_{{L^{2}({\mathbb{D}})}}^{2}+g_{2})\Big{]}$
$\displaystyle\leq({{\rm
Tr}}{\mathcal{Q}})t^{(r)}{\mathbb{E}}\Big{[}g_{1}\max_{1\leq n\leq
N}\|v^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{2}+g_{2}\Big{]}$ $\displaystyle\leq
C\|v^{(0)}\|_{{L^{2}({\mathbb{D}})}}t^{(r)}.$

Together with (4.28), (4.30), (4.31) and (4.29) complete the proof. ∎

######  Definition 4.4.

The dual norm $|\cdot|_{\star,{\mathcal{D}}}$ is defined by: for all
$v\in\Pi_{\mathcal{D}}(X_{{\mathcal{D}},0})$,

$|v|_{{\mathcal{D}},\star}:=\displaystyle\sup\Big{\\{}\displaystyle\int_{\mathbb{D}}u(x)\Pi_{\mathcal{D}}\psi(x)\,\mathrm{d}x\;:\;\psi\in
X_{{\mathcal{D}},0}\mbox{ and
}\|\nabla_{\mathcal{D}}\psi\|_{L^{2}({\mathbb{D}})^{d}}=1\Big{\\}}.$ (4.32)

######  Lemma 4.5.

Let $v$ be a solution of (3.2). Then, for all $r=1,...,N-1$, there exists a
positive constant $C$ satisfying

${\mathbb{E}}\Big{[}|\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}|_{{\mathcal{D}},\star}^{4}\Big{]}\leq
C\left(t^{(r)}\right)^{2}.$ (4.33)

Also, for any $t_{1},t_{2}\in[0,T]$, we have

${\mathbb{E}}\Big{[}|\Pi_{\mathcal{D}}v(t_{1})-\Pi_{\mathcal{D}}v(t_{2})|_{{\mathcal{D}},\star}^{4}\Big{]}\leq
C(|t_{1}-t_{2}|+\delta t_{\mathcal{D}})^{2}.$ (4.34)

###### Proof.

Taking an arbitrary $w$, such that $w\in
X_{{\mathcal{D}},0},\|\nabla_{\mathcal{D}}w\|_{L^{2}({\mathbb{D}})^{d}}=1$,
the relation (4.26) can be rewritten as

$\displaystyle{\mathbb{E}}\Big{[}|\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)}|_{{\mathcal{D}},\star}^{4}\Big{]}$
$\displaystyle={\mathbb{E}}\Big{[}|\sup_{w}{\langle}\Pi_{\mathcal{D}}v^{(n+r)}-\Pi_{\mathcal{D}}v^{(n)},\Pi_{\mathcal{D}}w{\rangle}|^{4}\Big{]}$
(4.35) $\displaystyle\leq{\mathbb{T}}_{1}+{\mathbb{T}}_{2}+{\mathbb{T}}_{3},$

where

$\displaystyle{\mathbb{T}}_{1}:=2^{3}(\delta
t_{\mathcal{D}})^{4}{\mathbb{E}}\Big{[}\Big{(}\sup_{w}\sum_{i=0}^{r-1}{\langle}\nabla_{\mathcal{D}}v^{(n+i+1)},\nabla_{\mathcal{D}}w{\rangle}\Big{)}^{4}\Big{]},$
$\displaystyle{\mathbb{T}}_{2}:=2^{3}{\mathbb{E}}\Big{[}\Big{(}\sup_{w}\sum_{i=0}^{r-1}{\langle}g(\Pi_{\mathcal{D}}v^{(n+i)})\Delta^{(n+i+1)}B,\Pi_{\mathcal{D}}w{\rangle}\Big{)}^{4}\Big{]},\mbox{
and }$ $\displaystyle{\mathbb{T}}_{3}:=2^{3}(\delta
t_{\mathcal{D}})^{4}{\mathbb{E}}\Big{[}\Big{(}\sup_{w}\sum_{i=0}^{r-1}{\langle}S(\Pi_{\mathcal{D}}v^{(n+i+1)}),\Pi_{\mathcal{D}}w{\rangle}\Big{)}^{4}\Big{]}$

To get a bound on ${\mathbb{T}}_{1}$, we apply Lemma 4.2 and the Hölder
inequality

$\displaystyle{\mathbb{T}}_{1}$ $\displaystyle\leq C(\delta
t_{\mathcal{D}})^{4}{\mathbb{E}}\Big{[}\Big{(}\sum_{i=0}^{r-1}\|\nabla_{\mathcal{D}}v^{(n+i+1)}\|_{L^{2}({\mathbb{D}})^{d}}\sup_{w}\|\nabla_{\mathcal{D}}w\|_{L^{2}({\mathbb{D}})^{d}}\Big{)}^{4}\Big{]}$
(4.36) $\displaystyle\leq
C{\mathbb{E}}\Big{[}\Big{(}\int_{t^{(n)}}^{t^{(n+r)}}\|\nabla_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})^{d}}\Big{)}^{4}\Big{]}$
$\displaystyle\leq
C(t^{(r)}){\mathbb{E}}\Big{[}\Big{(}\int_{t^{(n)}}^{t^{(n+r)}}\|\nabla_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})^{d}}^{2}\Big{)}^{2}\Big{]}$
$\displaystyle\leq
C(t^{(r)}){\mathbb{E}}\Big{[}\Big{\|}\nabla_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}\Big{]}\leq
c(t^{(r)}).$

For the estimation of ${\mathbb{T}}_{2}$, we employ the BurkHölder–Davis–Gundy
inequality, (2.2) and (4.1)

$\displaystyle{\mathbb{T}}_{2}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\|\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)g(\Pi_{\mathcal{D}}v(t))dB(t)\|_{{L^{2}({\mathbb{D}})}}^{4}\Big{]}$
(4.37)
$\displaystyle\leq{\mathbb{E}}\Big{[}\Big{(}\int_{0}^{T}\chi_{[t^{(n)},t^{(n+r)}]}(t)(g_{1}\|\Pi_{\mathcal{D}}v(t)\|_{{L^{2}({\mathbb{D}})}}^{2}+g_{2})\Big{)}^{2}\Big{]}$
$\displaystyle\leq C(t^{(r)})^{2}{\mathbb{E}}\Big{[}g_{1}\max_{1\leq n\leq
N}\|\Pi_{\mathcal{D}}u^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{4}+g_{2}\Big{]}\leq
Ct^{(r)}.$

We make use the Lipschitz continuity condition and Lemma (4.1) to estimate
${\mathbb{T}}_{3}$

$\displaystyle{\mathbb{T}}_{3}$ $\displaystyle\leq
C(t^{(r)})^{2}{\mathbb{E}}\Big{[}LC_{0}\sup_{w}\|\Pi_{\mathcal{D}}w\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{]}$
(4.38) $\displaystyle\quad+C\delta
t_{\mathcal{D}}{\mathbb{E}}\Big{[}L\sup_{w}\|\Pi_{\mathcal{D}}w\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{(}\sum_{i=0}^{r-1}\|\Pi_{\mathcal{D}}v^{(n+i+1)}\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{)}\Big{]}$
$\displaystyle\leq
C(t^{(r)})^{2}+C(t^{(r)}){\mathbb{E}}\Big{[}\Big{(}\int_{t^{(n)}}^{t^{(n+r)}}\|\Pi_{\mathcal{D}}v(t)\|_{{L^{2}({\mathbb{D}})}}^{2}\Big{)}^{4}\Big{]}$
$\displaystyle\leq
C(t^{(r)})^{2}+C(t^{(r)}){\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{2}\Big{]}$
$\displaystyle\leq
C(t^{(r)})^{2}+C(t^{(r)})\Big{(}{\mathbb{E}}\Big{[}\|\Pi_{\mathcal{D}}v\|_{{L^{2}({\mathbb{D}}\times(0,T))}}^{4}\Big{]}\Big{)}^{2}\leq
C(t^{(r)}).$

Altogether (4.35), (4.36), (4.37) and (4.38) yield the estimate (4.33). The
estimate (4.34) can be obtained by using the fact that when
$t_{1},t_{2}\in[0,T]$, $t_{1}<t_{2}$, $n\leq r$, and
$t_{1}\in(t^{(2)},t^{(3)}]$, then $t^{(2-n)}\leq|t_{2}-t_{1}|+\delta
t_{\mathcal{D}}$. ∎

Let $v$ be a solution to the approximate problem (3.2). For every $0\leq t\leq
T$, we have $n\in\\{0,....,N-1\\}$ such that $t^{(n)}\leq t\leq t^{(n+1)}$. We
introduce

$G_{\mathcal{D}}(t):=G_{\mathcal{D}}^{(n)}:=\displaystyle\sum_{i=0}^{n}g(\Pi_{\mathcal{D}}v^{(n)})\Delta^{(i+1)}B.$
(4.39)

As proved in [8], we can establish an estimate on $G_{\mathcal{D}}$.

######  Lemma 4.6.

For any $\beta\in(0,\frac{1}{2})$, the following inequalities hold

${\mathbb{E}}\Big{[}\|G_{\mathcal{D}}\|_{H^{\infty}(0,T,{L^{2}({\mathbb{D}})})}^{4}\Big{]}\leq
C\mbox{ and
}{\mathbb{E}}\Big{[}\|G_{\mathcal{D}}\|_{H^{\beta,4}(0,T,{L^{2}({\mathbb{D}})})}^{2}\Big{]}\leq
C,$ (4.40)

where $C$ is a positive constant depending on $\beta$.

###### Proof.

The BurkHölder–Davis–Gundy gives

$\displaystyle{\mathbb{E}}\Big{[}\|G_{\mathcal{D}}\|_{H^{\infty}(0,T,{L^{2}({\mathbb{D}})})}^{4}\Big{]}$
$\displaystyle={\mathbb{E}}\Big{[}\max_{1\leq n\leq
N}\Big{\|}\sum_{i=0}^{n}g(\Pi_{\mathcal{D}}v^{(i)})\Delta^{(i+1)}B\Big{\|}_{L^{2}({\mathbb{D}})}^{4}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\Big{(}\int_{o}^{T}\|g(\Pi_{\mathcal{D}}v)\|_{L(K,{L^{2}({\mathbb{D}})})}^{2}\Big{)}^{2}\Big{]}.$

∎

Hence, the first estimate is a consequence of (2.2) and Lemma 4.13. Similar to
(4.38), we have

${\mathbb{E}}\Big{[}\|G_{\mathcal{D}}^{(n+r)}-G_{\mathcal{D}}^{(n)}\|_{{L^{2}({\mathbb{D}})}}^{4}\Big{]}\leq
C(t^{(r)})^{2}.$ (4.41)

Reasoning as in [8, Lemma 7.2], the second estimate follows from the above
inequality and the first estimate.

##  5\. Main Results

In [8, Section 4], it is proved that the sequence
$(\Pi_{{\mathcal{D}}_{m}}v_{m},\nabla_{{\mathcal{D}}_{m}}v_{m},G_{{\mathcal{D}}_{m}},B)_{m\in{\mathbb{N}}}$
is tight. Now, using our estimates established in the previous section, we can
show that the sequence
$(\Pi_{{\mathcal{D}}_{m}}v_{m},\nabla_{{\mathcal{D}}_{m}}v_{m},G_{{\mathcal{D}}_{m}},B)_{m\in{\mathbb{N}}}$
converges almost sure up to a change of probability space. The proof of the
following lemma is identical to [8, Lemma 4.2] ((slightly adjusted to the fact
that $\bar{v}$ is a solution to the model considered here).

######  Lemma 5.1.

We have a probability space
$(\overline{\Omega},\overline{\mathcal{F}},\overline{\mathbb{F}},\overline{\mathbb{P}})$,
$(\widehat{v}_{m},\overline{Z}_{m},\overline{B}_{m})_{m\in{\mathbb{N}}}$, and
$(\bar{v},\overline{Z},\overline{B})$, in which the following items are
fulfilled:

  * •

$\widehat{v}_{m}\in X_{{\mathcal{D}}_{m},0},\;\forall m\in{\mathbb{N}}$,

  * •

the laws of
$(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m},\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}),\overline{Z}_{m},\overline{B}_{m})$
and
$(\Pi_{{\mathcal{D}}_{m}}v_{m},\nabla_{{\mathcal{D}}_{m}}v_{m}),Z_{m},{B}_{m})$
are the same, $\forall m\in{\mathbb{N}}$

  * •

$(\bar{v},\overline{Z},\overline{B})$ has values in
$L^{2}(0,T;H_{0}^{1}({\mathbb{D}}))\times L^{2}(0,T;L_{\rm
w}^{2}({\mathbb{D}}))\times C([0,T];L^{2}({\mathbb{D}}))$,

  * •

up to a subsequence, the following convergences hold $\mathbb{P}$-almost
surely,

$\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}\to\bar{v}\mbox{ strongly in
}L^{2}(0,T;L^{2}({\mathbb{D}})),$ (5.1)
$\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}\to\nabla\bar{v}\mbox{ weakly in
}L^{2}(0,T;L^{2}({\mathbb{D}})^{d}),$ (5.2)
$\overline{Z}_{m}\to\overline{Z}\mbox{ strongly in }L^{2}(0,T;L_{\rm
w}^{2}({\mathbb{D}})),$ (5.3) $\overline{B}_{m}\to\overline{B}\mbox{ strongly
in }C([0,T];L^{2}({\mathbb{D}})),$ (5.4)

  * •

$\widehat{v}_{m}$ satisfies the approximate problem (3.2), where
$B=\overline{B}_{m}$, and the following convergences up to a subsequence holds
for almost all $0<r,s<T$,

$\displaystyle\lim_{m\to\infty}\|(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(r)-\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s))-(\bar{v}(r)-\bar{v}(s))\|_{L^{2}}=0,$
(5.5)
$\displaystyle\lim_{m\to\infty}\|(\overline{Z}_{m}(s)-\overline{Z}_{m}(r))-(\overline{Z}(r)-\overline{Z}(s))\|_{L^{2}}=0.$
(5.6)

######  Theorem 5.2.

Assume that Hypotheses 2.1 hold. Let $({\mathcal{D}}_{m})_{m\in{\mathbb{N}}}$
be a sequence of gradient discretisations satisfying the consistency,
limit–conformity and compactness properties. Then, for any $m\geq 1$, the
approximate scheme (3.2) (in which ${\mathcal{D}}:={\mathcal{D}}_{m}$) has a
random process $v_{m}$ solution.

Also, we have a weak martingale solution
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{t})_{t\in[0,T]},\widetilde{\mathbb{P}},\widetilde{B},\widetilde{v})$
satisfying (2.3), and a sequence $\\{\widetilde{v}_{m}\\}$ of random processes
defined on $\widetilde{\Omega}$, which is of the same law as $v_{m}$, such
that up to a subsequence, the following convergences are satisfied
$\widetilde{\mathbb{P}}$-almost surely:

$\displaystyle\Pi_{{\mathcal{D}}_{m}}\widetilde{v}_{m}\to\widetilde{v},\quad\mbox{strongly
in }L^{2}({\mathbb{D}}\times(0,T)),$
$\displaystyle\nabla_{{\mathcal{D}}_{m}}\widetilde{v}_{m}\to\nabla\widetilde{v},\quad\mbox{weakly
in }L^{2}({\mathbb{D}}\times(0,T))^{d}.$

###### Proof.

Following the same reasoning as in [1, Corollary 5.2] and using the estimate
(4.1), we can prove the existence of at least one solution $v$ to the scheme
(3.2).

Now, consider the smooth countable set $\\{\psi_{i}\neq
0\;:\;i\in{\mathbb{N}}\\}$ to introduce

$w_{i}:=\displaystyle\frac{\psi_{i}}{(\|\psi_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla\psi_{i}\|_{L^{2}({\mathbb{D}})^{d}})}.$
(5.7)

Let $I_{\mathcal{D}}:H_{0}^{1}({\mathbb{D}})\cap L^{2}({\mathbb{D}})\to
X_{{\mathcal{D}},0}$ be an interpolator defined by

$I_{\mathcal{D}}w:=\arg\min_{\varphi\in
X_{{\mathcal{D}},0}}\Big{(}\|\Pi_{\mathcal{D}}\varphi-w\|_{{L^{2}({\mathbb{D}})}}+\|\nabla_{\mathcal{D}}\varphi-\nabla
w\|_{L^{2}({\mathbb{D}})^{d}}\Big{)}.$ (5.8)

Applying the consistency property with $\varphi=0$ together with
$\|w_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla
w_{i}\|_{L^{2}({\mathbb{D}})^{d}}=1$, one has, thanks to (5.7) and (5.8)

$\displaystyle\|\Pi_{\mathcal{D}}I_{\mathcal{D}}w_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla_{\mathcal{D}}I_{\mathcal{D}}w_{i}\|_{L^{2}({\mathbb{D}})^{d}}$
(5.9)
$\displaystyle\leq\|\Pi_{\mathcal{D}}I_{\mathcal{D}}w_{i}-w_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla_{\mathcal{D}}I_{\mathcal{D}}w_{i}-\nabla
w_{i}\|_{L^{2}({\mathbb{D}})^{d}}$
$\displaystyle\quad+\|w_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla
w_{i}\|_{L^{2}({\mathbb{D}})^{d}}$ $\displaystyle\leq
S_{\mathcal{D}}(w_{i})+\|w_{i}\|_{{L^{2}({\mathbb{D}})}}+\|\nabla
w_{i}\|_{L^{2}({\mathbb{D}})^{d}}$ $\displaystyle\leq 2.$

Thanks to Lemmas 4.1 and 4.5, and the convergence stated in (5.5), [8, Lemmas
4.3 and 5.1] asserts that the process $\overline{Z}$ is a square integrable
continuous martingale, and there exists a quadratic variation of
$\overline{Z}$ given by for all $\alpha,\beta\in{L^{2}({\mathbb{D}})}$,

$Z(t)(\alpha,\beta)=\displaystyle\int_{0}^{t}{\langle}(g(\bar{v}){\mathcal{Q}}^{\frac{1}{2}})^{\star}(\alpha),(g(\bar{v}){\mathcal{Q}}^{\frac{1}{2}})^{\star}(\beta){\rangle}_{K}\,\mathrm{d}s,\;\forall
t\geq 0.$

[6, Theorem 8.2] shows the existence of a probability space
$(\widehat{\mathbb{D}},\widehat{{\mathcal{F}}},\widehat{\mathbb{P}})$ and a
${\mathcal{Q}}-$ Brownian motion $\widehat{B}$, in which $\overline{Z}$ and
$\bar{v}$ are defined on this space and, for any $t\geq 0$,

$\overline{Z}(t,\cdot)=\displaystyle\int_{0}^{t}g(\bar{v})(s,\cdot)d\widehat{B}(s).$
(5.10)

For any $t\in[0,T]$ and for each $m\in{\mathbb{N}}$, we have a natural number
$0\leq k\leq N_{m-1}$, such that $t\in(t^{(k)},t^{(k+1)}]$. Recalling that
$\bar{v}_{m}$ solves the gradient scheme with $\overline{B}_{m}$ replaced by
$\overline{B}$, we have

$\displaystyle{\langle}d_{{\mathcal{D}}_{m}}^{(n+\frac{1}{2})}\widehat{v}_{m},\Pi_{{\mathcal{D}}_{m}}\varphi{\rangle}+\delta
t_{{\mathcal{D}}_{m}}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)},\nabla_{{\mathcal{D}}_{m}}\varphi{\rangle}$
$\displaystyle={\langle}g(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)})\Delta^{(n+1)}\overline{B}_{m},\Pi_{{\mathcal{D}}_{m}}\varphi{\rangle}\quad+\delta
t_{{\mathcal{D}}_{m}}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)}),\Pi_{{\mathcal{D}}_{m}}\varphi{\rangle}.$

Summing this relation from $n=0$ to $n=k$, and choosing
$\varphi:=I_{{\mathcal{D}}_{m}}\psi$, where $\psi\in H_{0}^{1}({\mathbb{D}})$,
we obtain,

$\displaystyle{\langle}\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(t),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}-{\langle}\Pi_{{\mathcal{D}}_{m}}v^{(0)},\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}$
(5.11) $\displaystyle\quad+\sum_{n=0}^{k}\delta
t_{{\mathcal{D}}_{m}}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)},\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}$
$\displaystyle={\langle}\widehat{M}_{m},\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}+\sum_{n=0}^{k}\delta
t_{{\mathcal{D}}_{m}}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)}),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}.$

Since $\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\to\psi$ in
$L^{2}({\mathbb{D}})$ and $({\mathcal{D}}_{m})_{m\in{\mathbb{N}}}$ is
consistent, we have for almost every $t$, thanks to (5.5) and (5.6)

$\displaystyle{\langle}\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(t),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\to{\langle}\widehat{v}(t),\psi{\rangle}\mbox{
in }L^{2}(\widehat{\mathbb{D}})$ (5.12)
$\displaystyle{\langle}\Pi_{{\mathcal{D}}_{m}}\widehat{v}^{(0)},\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\to{\langle}\widehat{v}_{0},\psi{\rangle}\mbox{
in }L^{2}(\widehat{\mathbb{D}})$
$\displaystyle{\langle}\widehat{Z}_{m}(t),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\to{\langle}\widehat{Z}(t),\psi{\rangle}\mbox{
in }L^{2}(\widehat{\mathbb{D}}).$

Now, we notice that

$\displaystyle\sum_{n=0}^{k}\delta
t_{{\mathcal{D}}_{m}}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)},\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}$
(5.13)
$\displaystyle=\int_{0}^{t}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s),\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s+\int_{t}^{\lceil
t/\delta t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s),\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s,$

and

$\displaystyle\sum_{n=0}^{k}\delta
t_{{\mathcal{D}}_{m}}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}^{(n+1)}),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}$
(5.14)
$\displaystyle=\int_{0}^{t}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s+\int_{t}^{\lceil
t/\delta t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s,$

We make use the convergence stated in (5.2),
$\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\to\nabla\psi$ in
$L^{2}({\mathbb{D}})^{d}$, and
$\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\to\psi$ in
$L^{2}({\mathbb{D}})^{d}$ to obtain, for any $t\in[0,T]$

$\displaystyle\int_{0}^{t}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s),\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s\to\int_{0}^{t}{\langle}\nabla\widehat{v}_{m}(s),\nabla\psi{\rangle}\,\mathrm{d}s,$
(5.15)

and

$\displaystyle\int_{0}^{t}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s\to\int_{0}^{t}{\langle}S(\widehat{v}_{m}(s)),\psi{\rangle}\,\mathrm{d}s.$
(5.16)

These prove the convergence of the first term of the RHS of (5.13) (resp.
(5.14)). Finally, as shown below, we observe that the expectation of the
absolute value of the last term in the RHS of (5.13) (resp. (5.14)) tends to
zero as $m\to\infty$, which concludes the proof.

$\displaystyle{\mathbb{E}}\Big{[}\Big{|}\int_{t}^{\lceil t/\delta
t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}{\langle}\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s),\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s\Big{|}\Big{]}$
(5.17) $\displaystyle\leq{\mathbb{E}}\Big{[}\int_{t}^{\lceil t/\delta
t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}\|\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}\|\nabla_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}\,\mathrm{d}s\Big{]}$
$\displaystyle\leq C\delta
t_{{\mathcal{D}}_{m}}^{1/2}{\mathbb{E}}\Big{[}\Big{(}\int_{0}^{T}\|\nabla_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)\|_{{L^{2}({\mathbb{D}}\times(0,T))^{d}}}^{2}\,\mathrm{d}s\Big{)}^{1/2}\Big{]}$
$\displaystyle\leq C\delta t_{{\mathcal{D}}_{m}}^{1/2},$

and

$\displaystyle{\mathbb{E}}\Big{[}\Big{|}\int_{t}^{\lceil t/\delta
t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}{\langle}S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)),\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi{\rangle}\,\mathrm{d}s\Big{|}\Big{]}$
(5.18) $\displaystyle\leq{\mathbb{E}}\Big{[}\int_{t}^{\lceil t/\delta
t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}\|S(\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s))\|_{{L^{2}({\mathbb{D}})}}\|\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\|_{{L^{2}({\mathbb{D}})}}\,\mathrm{d}s\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}\int_{t}^{\lceil t/\delta
t_{{\mathcal{D}}_{m}}\rceil\delta
t_{{\mathcal{D}}_{m}}}L\|\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)\|_{{L^{2}({\mathbb{D}})}}\|\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\|_{{L^{2}({\mathbb{D}})}}\,\mathrm{d}s\Big{]}$
$\displaystyle\quad+C_{0}\delta
t_{{\mathcal{D}}_{m}}\|\Pi_{{\mathcal{D}}_{m}}I_{{\mathcal{D}}_{m}}\psi\|_{{L^{2}({\mathbb{D}})}}$
$\displaystyle\leq C\delta
t_{{\mathcal{D}}_{m}}^{1/2}{\mathbb{E}}\Big{[}\Big{(}\int_{0}^{T}L\|\Pi_{{\mathcal{D}}_{m}}\widehat{v}_{m}(s)\|_{{L^{2}({\mathbb{D}})}}^{2}\,\mathrm{d}s\Big{)}^{1/2}\Big{]}+C_{0}\delta
t_{{\mathcal{D}}_{m}}$ $\displaystyle\leq C(\delta
t_{{\mathcal{D}}_{m}}+\delta t_{{\mathcal{D}}_{m}}^{1/2}).$

∎

##  6\. Numerical results

To evaluate the validity of the gradient scheme for the stochastic reaction-
diffusion model, we investigate a particular variant of a numerical method
that falls in the gradient discretisation method; it is called the hybrid
mimetic mixed method (HMM). We advise the reader to [1, Section 4.2] for the
full presentation of this method for deterministic reaction-diffusion
equations. As a test, we use the HMM method to solve the equation (1.1) with
non-homogeneous Dirichlet boundary conditions over the domain
${\mathbb{D}}=[-5,5]^{2}$, and with the reaction term
$S(\bar{v})=\bar{v}-{\bar{v}}^{3}$. The initial MATLAB code for the HMM scheme
is provided as a companion of the work in [10]. The model with a reaction
function of this nature is referred to as the stochastic Allen-Cahn equation,
and it has been the subject of extensive research in the literature. For
instance, [32, 15, 12, 21], and the references provided there illustrate this.
We initially select the noise term $g(\bar{v})=\rho\bar{v}$, where $\rho$
denotes the noise intensity. The exact travelling wave solution in such a case
is given as [3]

$\bar{v}(t,x,y)=(-\frac{c}{6}+2\sqrt{\frac{\rho^{2}+2}{8}-\frac{c^{2}}{48}}\tanh(\sqrt{\frac{\rho^{2}+2}{8}-\frac{c^{2}}{48}}(x+y-ct)))e^{\rho
B(t)-\frac{\rho^{2}}{2}t},$ (6.1)

where $c$ is the wave speed and is chosen as $c=0.3$ in our numerical
experiments. The analytical solution presented in (6.1) is used to extract the
initial and boundary conditions.

Simulations are conducted on a sequence of triangular and Kershowa meshes,
with a noise intensity of $\rho=0.2$, up to $T=1$. Samples of these meshes are
presented in Figure 6.1. The averages are calculated over $100$ realizations
of the Brownian motion, with time steps of $\delta t=0.0001,0.00025,0.001$,
and $0.00125$, respectively, where $\delta t=0.0001$ for the coarsest mesh and
$\delta t=0.00125$ for the finest one. The relative $L_{2}$-errors on the
solution $\bar{v}$ and on the gradient $\nabla\bar{v}$ are respectively
examined by the following norms:

$\frac{{\mathbb{E}}\Big{[}\|\bar{v}(\cdot)-\Pi_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})}\Big{]}}{{\mathbb{E}}\Big{[}\|\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})}\Big{]}}\mbox{
and
}\frac{{\mathbb{E}}\Big{[}\|\nabla\bar{v}(\cdot)-\nabla_{{\mathcal{D}}}v\|_{L^{2}({\mathbb{D}})^{2}}\Big{]}}{{\mathbb{E}}\Big{[}\|\nabla\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})^{2}}\Big{]}},$

where ${\mathbb{E}}[\cdot]$ denotes the expectation operator. The obtained
errors and the corresponding rate of convergence with regard to the mesh size
$h$ are given in Table 1 for the triangular mesh and, in Table 2 for the
Kershowa mesh. We have observed that the HMM’s convergence rate is of order
$1$, which is consistent with the expectation of convergence rate for the
first order methods.

In addition, we investigate the validity of the HMM method by investigating
the influences of multiplicative noise on the dynamics of the simulated waves
that the stochastic Allen-Cahn model demonstrates. Two examples of such noise-
induced waves dynamics are displayed in Figure 6.2 and Figure 6.3.

Figure 6.2 presents the HMM travelling wave solutions of the underlying model
at $T=1$ for various noise intensity levels, designated as $\rho=0$, $\rho=1$,
$\rho=5$, and $\rho=10$. The simulation was conducted on a triangular mesh
with the size of $h=0.15625$ and a time step of $0.001$. In the absence of
noise ($\rho=0$), the model exhibits a travelling wave with a maximal value of
approximately $1$. Nevertheless, the wave begins to lose its wave-form shape
when the noise is incorporated, and annihilates completely at high noise
levels, as illustrated in Figure 6.2 (c) for $\rho=5$ and Figure 6.2 (d) for
$\rho=10$. Consequently, the wave propagation is largely unaffected by weak
noise, whereas it is unable to propagate in the strong noise regime. This
phenomenon is known as  wave propagation failure in which no propagation wave
is observed.

Additionally, we further evaluate the HMM method in the presence of another
multiplicative noise source, denoted by

$g(\bar{v})=\rho\bar{v}(\bar{v}-1),$

where $\rho$ is the noise intensity. The intriguing aspect of this form of
noise is that the fluctuations do not influence the model in either the rest
state ($\bar{v}=0$) or the excited state ($\bar{v}=1$), as the noise vanishes
at both states. The initial and boundary conditions are determined using the
exact travelling wave solution (6.1) for the deterministic case ($\rho=0$).
The simulation results are depicted in Figure 6.3 for various levels of noise
intensity, $\rho=0$, $\rho=1$, $\rho=2$, and $\rho=3$, while maintaining the
other parameters, as in Figure 6.2. The results indicate that a single
travelling wave is produced in the deterministic state ($\rho=0$).
Nevertheless, the phenomenon of waves backfiring can occur when an appropriate
amount of noise is added. The transition from the excited state to the rest
can be induced in the midst of the wave for sufficiently large noise
($\rho=3$), resulting in the wave breaking into two halves that are moving in
opposite directions, as illustrated in Figure 6.3 (d).

|
---|---
(a) triangular mesh | (b) Kershowa mesh
Figure 6.1. Samples of the various 2D meshes |
---|---
(a) $\sigma=0$ | (b) $\sigma=1$
|
(c) $\sigma=5$ | (d) $\sigma=10$
Figure 6.2. Travelling wave solutions on the range $[-5,5]^{2}$, taking at levels of noise $\rho=0$, $\rho=1$, $\rho=5$ and $\rho=10$, with parameters chosen as $\delta t=0.001$, $T=1$, and $c=0.3$. |
---|---
(a) $\sigma=0$ | (b) $\sigma=1$
|
(c) $\sigma=2$ | (d) $\sigma=3$
Figure 6.3. Travelling wave solutions on the range $[-5,5]^{2}$, taking at levels of noise $\rho=0$, $\rho=1$, $\rho=2$ and $\rho=3$, with parameters chosen as $\delta t=0.001$, $T=1$, and $c=0.3$. $h$ | $\frac{{\mathbb{E}}[\|\bar{v}(\cdot)-\Pi_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})}]}{{\mathbb{E}}[\|\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})}]}$ | rate | $\frac{{\mathbb{E}}[\|\nabla\bar{v}(\cdot)-\nabla_{{\mathcal{D}}}v\|_{L^{2}({\mathbb{D}})^{2}}]}{{\mathbb{E}}[\|\nabla\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})^{2}}]}$ | rate
---|---|---|---|---
0.6250000 | 0.4708741 | – | 1.1986664 | –
0.3125000 | 0.2286052 | 1.1986664 | 0.6831530 | 1.0424831
0.1562500 | 0.1299039 | 0.8154136 | 0.3359330 | 1.0240353
0.0781250 | 0.0644115 | 1.0120541 | 0.1783213 | 0.9136945
Table 1. The relative errors on $\bar{v}$ and $\nabla\bar{v}$ and the convergence rates w.r.t. $h$ the size of triangular mesh with noise intensity $\rho=0.2$. $h$ | $\frac{{\mathbb{E}}[\|\bar{v}(\cdot)-\Pi_{\mathcal{D}}v\|_{L^{2}({\mathbb{D}})}]}{{\mathbb{E}}[\|\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})}]}$ | rate | $\frac{{\mathbb{E}}[\|\nabla\bar{v}(\cdot)-\nabla_{{\mathcal{D}}}v\|_{L^{2}({\mathbb{D}})^{2}}]}{{\mathbb{E}}[\|\nabla\bar{v}(\cdot)\|_{L^{2}({\mathbb{D}})^{2}}]}$ | rate
---|---|---|---|---
1.1155656 | 0.8482250 | – | 1.2973389 | –
0.8385224 | 0.4732923 | 2.0437227 | 1.0346060 | 0.7926934;
0.6717051 | 0.3026483 | 2.0157696 | 0.8655410 | 0.8043447
0.5602472 | 0.2117210 | 1.9692431 | 0.7127184 | 1.0706944
Table 2. The relative errors on $\bar{v}$ and $\nabla\bar{v}$ and the
convergence rates w.r.t. $h$ the size of Kershowa mesh with noise intensity
$\rho=0.2$.

##  7\. Conclusion

The GDM was used to develop a generic numerical analysis for a stochastic
reaction-diffusion model driven by a multiplicative noise. Based on natural
assumptions, we proved the existence of a weak martingale solution and
established convergence results that are applicable to various numerical
schemes. As a result of our analysis, we were able to develop an HMM method
for the underlying model and investigate the effects of multiplicative noise
on the behaviour of the solutions we obtained through numerical testing. We
observed that the HMM’s convergence rate is of order $1$, which matches the
expectation of convergence rate for the first order methods. We also
investigated the influences of this multiplicative noise on the dynamics of
the travelling waves in the model under investigation, particularly the
occurrence of wave propagation failure and wave backfiring as a result of
sufficiently large noise.

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