A finite element approximation for the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise

TL;DR

This paper introduces a new finite element method for solving the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise, ensuring convergence and enabling the proof of weak martingale solutions.

Contribution

It develops an unconditionally convergent linear finite element scheme using the Doss-Sussmann technique for the stochastic LLG equation with multi-dimensional noise.

Findings

The scheme is unconditionally convergent.

Existence of weak martingale solutions is established.

The method effectively handles multi-dimensional stochastic noise.

Abstract

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau--Lifshitz--Gilbert (LLG) equation with multi-dimensional noise. By using the Doss-Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

Equations

$$\{\begin{array}{ll}& \frac{\partial \mathbf{M}}{\partial t}={\lambda }_1\mathbf{M}\times \Delta \mathbf{M}-{\lambda }_2\mathbf{M}\times (\mathbf{M}\times \Delta \mathbf{M})\text{ in }(0,T)\times D,\\ & \\ & \frac{\partial \mathbf{M}}{\partial \mathbf{n}}=0\text{ in }(0,T)\times \partial D,\\ & \\ & \mathbf{M}(0,\cdot )={\mathbf{M}}_0(\cdot )\text{ in }D,\end{array}$$

$$|\mathbf{M}(t,x)|=1,t\in [0,T],x\in D$$

$$\displaystyle\begin{cases}&d\boldsymbol{M}=\big{(}\lambda_{1}\boldsymbol{M}\times\Delta\boldsymbol{M}-\lambda_{2}\boldsymbol{M}\times(\boldsymbol{M}\times\Delta\boldsymbol{M})\big{)}dt+\sum_{i=1}^{q}(\boldsymbol{M}\times\boldsymbol{g}_{i})\circ dW_{i}(t),\\ \\ &\dfrac{\partial\boldsymbol{M}}{\partial\boldsymbol{n}}=0\quad\text{ on }(0,T)\times\partial D,\\ \\ &\boldsymbol{M}(0,\cdot)=\boldsymbol{M}_{0}\quad\text{ in }D,\end{cases}$$

$${\mathbb{H}}^1(U)$$

$$\mathbf{u}\times \nabla \mathbf{v}$$

$$\nabla \mathbf{u}\times \nabla \mathbf{v}$$

$${⟨\mathbf{w}\times \nabla \mathbf{v},\nabla \mathbf{u}⟩}_{{\mathbb{L}}^2(D)}$$

$${⟨\mathbf{M}(t),\mathbf{\psi }⟩}_{{\mathbb{L}}^2(D)}-{⟨{\mathbf{M}}_0,\mathbf{\psi }⟩}_{{\mathbb{L}}^2(D)}$$

$$-{\lambda }_2{\int }_0^t{⟨\mathbf{M}\times \nabla \mathbf{M},\nabla (\mathbf{M}\times \mathbf{\psi })⟩}_{{\mathbb{L}}^2(D)}ds$$

$$+\sum _{i=1}^q{\int }_0^t{⟨\mathbf{M}\times {\mathbf{g}}_i,\mathbf{\psi }⟩}_{{\mathbb{L}}^2(D)}\circ d{W}_i(s).$$

$$G_i^{\star }=-G_i,$$

$$\text{and}{\left(G_i^2\right)}^{\star }=G_i^2.$$

$$Z_t=I+\sum _{i=1}^q{\int }_0^t{G}_i{Z}_s\circ d{W}_i(s),t\ge 0.$$

$$\left|Z_t\mathbf{u}\right|=|\mathbf{u}|\mathbb{P}-a.s\mathrm{for}\mathrm{every}\mathbf{u}\in {\mathbb{R}}^3.$$

$$Z_t=I+\frac{1}{2}\sum _{i=1}^q{\int }_0^t{G}_i^2{Z}_{s}ds+\sum _{i=1}^q{\int }_0^t{G}_i{Z}_{s}d{W}_i(s),t\ge 0.$$

$$Z_t^{\mathbf{u}}=\mathbf{u}+\frac{1}{2}\sum _{i=1}^q\underset{0}{\overset{t}{\int }}{G}_i^2{Z}_s^{\mathbf{u}}ds+\sum _{i=1}^q\underset{0}{\overset{t}{\int }}{G}_i{Z}_s^{\mathbf{u}}d{W}_i(s).$$

$$d|Z_t^{\mathbf{u}}{|}^2$$

$$=\sum _{i=1}^q⟨Z_t^{\mathbf{u}},G_i^2{Z}_t^{\mathbf{u}}⟩dt+2\sum _{i=1}^q⟨Z_t^{\mathbf{u}},G_i{Z}_t^{\mathbf{u}}⟩d{W}_i(t)+\sum _{i=1}^q|G_i{Z}_t^{\mathbf{u}}{|}^{2}dt$$

$$=0,$$

$$|Z_t^{\mathbf{u}}{|}^2=|\mathbf{u}{|}^2,\mathrm{for}\mathrm{all}t\ge 0,\mathbb{P}-a.s..$$

$$\mathbb{E}{\left|Z_t(y)-Z_s(x)\right|}^p\le 2^{p-1}\mathbb{E}{\left|Z_t(y)-Z_s(y)\right|}^p+2^{p-1}\mathbb{E}{\left|Z_s(y)-Z_s(x)\right|}^p.$$

$$\mathbb{E}{\left|Z_t(y)-Z_s(y)\right|}^p\le C_1|t-s{|}^{\frac{p}{2}}.$$

$$|g_i(x)-g_i(y)|\le c_i|x-y{|}^{\alpha },x,y\in \overline{D},i=1,\dots ,q$$

$$|G_i(x)h|$$

$$|(G_i(x)-G_i(y))h|$$

$$\left|\left(G_i^2(x)-G_i^2(y)\right)h\right|$$

$$Z_s(y)-Z_s(x)$$

$$-\frac{1}{2}\sum _{i=1}^q{\int }_0^s{G}_i^2(x)Z_r(x)dr-\sum _{i=1}^q{\int }_0^s{G}_i(x)Z_r(x)d{W}_i(r)$$

$$=\frac{1}{2}\sum _{i=1}^q{\int }_0^s\left(G_i^2(y)-G_i^2(x)\right)Z_r(y)dr+\frac{1}{2}\sum _{i=1}^q{\int }_0^s{G}_i^2(x)\left(Z_r(y)-Z_r(x)\right)dr$$

$$+\sum _{i=1}^q{\int }_0^s\left(G_i(y)-G_i(x)\right)Z_r(y)d{W}_i(r)+\sum _{i=1}^q{\int }_0^s{G}_i(x)\left(Z_r(y)-Z_r(x)\right)d{W}_i(r).$$

$$\mathbb{E}{\left|Z_s(y)-Z_s(x)\right|}^p$$

$$\mathbb{E}{\left|Z_s(y)-Z_s(x)\right|}^p\le \tilde{C}{e}^{\tilde{C}T}|x-y{|}^{\alpha p}.$$

$$\mathbb{E}{\left|Z_t(y)-Z_s(x)\right|}^p\le c_1|t-s{|}^{\frac{p}{2}}+c_2|x-y{|}^{\alpha p}.$$

$$\frac{p}{2}\ge r\mathrm{and}p\alpha \ge r.$$

$$\mathbb{E}{\left|Z_t(y)-Z_s(x)\right|}^p\le c\left(|t-s{|}^r+|x-y{|}^r\right),$$

$$d\xi_{t}(x)=\frac{1}{2}\sum_{i=1}^{q}\bigl{(}G_{i}^{2}\xi_{t}(x)+H_{i}Z_{t}(x)\bigr{)}\,dt+\sum_{i=1}^{q}\bigl{(}G_{i}\xi_{t}(x)+I_{i}Z_{t}(x)\bigr{)}\,dW_{i}(t),$$

Full text

1. 1 Introduction
2. 2 Definition of a weak solution and the main result
3. 3 The auxiliary equation for the diffusion part
4. 4 Equivalence of weak solutions
5. 5 The finite element scheme
6. 6 The main result
7. 7 Appendix

1. Introduction

The deterministic Landau-Lifschitz-Gilbert (LLG) equation provides a basis for the theory and applications of ferromagnetic materials and fabrication of magnetic memories in particular, see for example [15, 9, 12, 17]. Let us recall, that in this theory we consider a ferromagnetic material filling the domain $D$ and a function $\boldsymbol{u}\in H^{1,2}\left(D,\mathbb{S}^{2}\right)$, where $\mathbb{S}^{2}$ stands for the unit sphere in $\mathbb{R}^{3}$, represents a configuration of magnetic moments across the domain $D$, that is $\boldsymbol{u}(x)$ is the magnetisation vector at the point $x\in D$. According to the Landau and Lifschitz theory of ferrormagnetizm [17], modified later by Gilbert [12], the time evolution of magnetic moments $\boldsymbol{M}(t,x)$ is described, in the simplest case, by the Landau-Lifschitz-Gilbert (LLG) equation

[TABLE]

where $\lambda_{1}\not=0$ and $\lambda_{2}>0$ are constants, and $\boldsymbol{n}$ stands for the outward normal vector on $\partial D$; see e.g. [9]. We assume that $\boldsymbol{M}_{0}\in H^{1,2}\left(D,\mathbb{S}^{2}\right)$, and then one can show that

[TABLE]

In this paper we are concerned with a stochastic version of the LLG equation. Randomly fluctuating fields were originally introduced in physics by Néel in [neel] as formal quantities responsible for magnetization fluctuations. The necessity of being able to describe deviations from the average magnetization trajectory in an ensemble of noninteracting nanoparticles was later emphasised by Brown in  [6, 7]. According to a non-rigorous arguments of Brown the magnetisation $\boldsymbol{M}$ evolves randomly according to a stochastic version of (1.1) that takes the form, (see [8] for more details about the physical background and derivation of this equation)

[TABLE]

where $\boldsymbol{g}_{i}\in\mathbb{W}^{2,\infty}(D)$, $i=1,\cdots,q$, satisfy the homogeneous Neumann boundary conditions and $\left(W_{i}\right)_{i=1}^{q}$ is a $q$-dimensional Wiener process. In view of the property (1.2) for the deterministic system, we require that $\boldsymbol{M}$ also satisfies (1.2). To this end we are forced to use the Stratonovich differential $\circ\>dW_{i}(t)$ in equation (1.3). Mathematical theory of equation (1.3) has been initiated only recently, in  [8], where the existence of weak martingale solutions to (1.3) was proved for the case $q=1$ using the Galerkin-Faedo approximations. Let us note, that usually the Galerkin-Faedo approximations do not provided a useful computational tool for solving an equation.

The aim of this paper is two-fold. We will prove the existence of solutions to the stochastic LLG equation (1.3) and at the same time will provide an efficient and flexible algorithm for solving numerically this equation. To this end we will use the finite element method and a new transformation of the Stratonovich type equation (1.3) to a deterministic PDE (4.1) with coefficients determined by a stochastic ODE (3.6) that can be solved separately. The deterministic PDE we obtain, has solutions absolutely continuous with respect to time, hence convenient for the construction of a convergent finite elemetn scheme. Our approach is based on the Doss-Sussmann technique [11, 18]. This transformation was introduced in [13] to study the stochastic LLG equation with a single Wiener process ($q=1$), in which case the auxiliary ODE is deterministic. Since the vector fields $\boldsymbol{u}\times\boldsymbol{g}_{i}$ are non-commuting, the case of $q>1$ is more difficult and requires new arguments.

We apply the finite element method to the PDE resulting from this transformation and prove the convergence of linear finite element scheme to a weak martingale solution to  (1.3) (after taking an inverse transformation). Our proof is simpler than the proof in  [8] and covers the case of $q>1$. We note here that under appropriate assumptions even the case of infinite-dimensional noise ($q=\infty$) can be handled in exactly the same way.

Let us recall that the first convergent finite element scheme for the stochastic LLG equation was studied in [5] and is based on a Crank–Nicolson type time-marching evolution, relying on a nonlinear iteration solved by a fixed point method. On the other hand, there has been an intensive development of a new class of numerical methods for the LLG equation (1.1) based on a linear iterations, yielding unconditional convergence and stability [1, 3]. The ideas developed there are extended and generalized in [13, 2] in order to take into account the stochastic term. A fully linear discrete scheme for (1.3) is studied in [13] but with one-dimensional noise. The method is based on the so–called Doss-Sussmann technique [11, 18], which allows one to replace the stochastic partial differential equation (PDE) by an equivalent PDE with random coefficients. In contrast,  [2] considers, for a more general noise, a projection scheme applied directly to the original stochastic equation (1.3). However, this approach requires a quite specific and complicated treatment of the stochastic term. In this paper, we propose a convergent $\theta$-linear scheme for the numerical solution of the tranformed equation and prove unconditional stability and convergence for the scheme when $\theta>1/2$. To the best of our knowledge this is a new result for this problem.

The paper is organised as follows. In Section 2 we define the notion of weak martingale solutions to (1.3) and state our main result. In Section 3, we introduce an auxiliary stochastic ODE and prove some properties of solution necessary for the transformation of equation (1.3) to a deterministic PDE with random coefficients. Details of this transformation are presented in Section 4. We also show in this section how a weak solution to (1.3) can be obtained from a weak solution of the reformulated form. In Section 5 we introduce our finite element scheme and present a proof for the stability of approximate solutions. Section 6 is devoted to the proof of the main theorem, namely the convergence of finite element solutions to a weak solution of the reformulated equation. Finally, in the Appendix we collect, for the reader’s convenience, a number of facts that are used in the course of the proof.

Throughout this paper, $c$ denotes a generic constant that may take different values at different occurrences. In what follows we will also use the notation $D_{T}=(0,T)\times D$.

2. Definition of a weak solution and the main

result

In this section we state the definition of a weak solution to (1.3) and present our main result. Before doing so, we introduce some suitable Sobolev spaces, and fix some notation. The standing assumption for the rest of the paper is that $D$ is a bounded open domain in ${\mathbb{R}}^{3}$ with a smooth boundary.

For any $U\subset{\mathbb{R}}^{d}$, $d\geq 1$, we denote by $\mathbb{L}^{2}(U)$ the space of Lebesgue square-integrable functions defined on $U$ and taking values in ${\mathbb{R}}^{3}$. The function space $\mathbb{H}^{1}(U)$ is defined as:

[TABLE]

Remark 2.1**.**

For $\boldsymbol{u},\boldsymbol{v}\in\mathbb{H}^{1}(D)$ we denote

[TABLE]

Definition 2.2**.**

Given $T\in(0,\infty)$ and a family of functions $\left\{g_{i}:\,i=1,\ldots,q\right\}\subset\mathbb{L}^{\infty}(D)$, a weak martingale solution $(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\in[0,T]},\mathbb{P},W,\boldsymbol{M})$ to (1.3), for the time interval $[0,T]$, consists of

1. (a)

a filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\in[0,T]},\mathbb{P})$ with the filtration satisfying the usual conditions, 2. (b)

a $q$-dimensional $({\mathcal{F}}_{t})$-adapted Wiener process $W=(W_{t})_{t\in[0,T]}$, 3. (c)

a progressively measurable process $\boldsymbol{M}:[0,T]\times\Omega\rightarrow\mathbb{L}^{2}(D)$

such that

1. (1)

$\boldsymbol{M}(\cdot,\omega)\in C([0,T];\mathbb{H}^{-1}(D))$* for $\mathbb{P}$-a.e. $\omega\in\Omega$;* 2. (2)

$\mathbb{E}\left(\operatorname*{ess\,sup\/}_{t\in[0,T]}\|\nabla\boldsymbol{M}(t)\|^{2}_{\mathbb{L}^{2}(D)}\right)<\infty$; 3. (3)

$|\boldsymbol{M}(t,x)|=1$* for each $t\in[0,T]$, a.e. $x\in D$, and $\mathbb{P}$-a.s.;* 4. (4)

for every $t\in[0,T]$, for all $\boldsymbol{\psi}\in{\mathbb{C}}_{0}^{\infty}(D)$, $\mathbb{P}$-a.s.:

[TABLE]

As the main result of this paper, we will establish a finite element scheme defined via a sequence of functions which are piecewise linear in both the space and time variables. We also prove that this sequence contains a subsequence converging to a weak martingale solution in the sense of Definition 2.2. A precise statement will be given in Theorem 6.9.

3. The auxiliary equation for the diffusion part

In this section we introduce the auxiliary equation (3.12) that will be used in the next section to define a new variable from $\boldsymbol{M}$, and establish some properties of its solution.

Let $\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{q}\in C\left(\overline{D},\mathbb{R}^{3}\right)$, be fixed. For $i=1,\ldots,q$, and $x\in\overline{D}$ we define linear operators $G_{i}(x):\mathbb{R}^{3}\to\mathbb{R}^{3}$ by $G_{i}(x)\boldsymbol{u}=\boldsymbol{u}\times\boldsymbol{g}_{i}(x)$. In what follows we suppress the argument $x$. It is easy to check that

[TABLE]

We will consider a stochastic Stratonovitch equation on the algebra $\mathcal{L}\left(\mathbb{R}^{3}\right)$ of linear operators in $\mathbb{R}^{3}$:

[TABLE]

Lemma 3.1**.**

*Let $\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{q}\in C\left(\overline{D},\mathbb{R}^{3}\right)$. Then the following holds.

(a) For every $x\in\overline{D}$ equation (3.3) has a unique strong solution, which has a $t$-continuous version in $\mathcal{L}\left(\mathbb{R}^{3}\right)$.

(b) For every $t\geq 0$ and $x\in\overline{D}$*

[TABLE]

*In particular, for every $t\geq 0$ the operator $Z_{t}$ is invertible and $Z_{t}^{-1}=Z_{t}^{\star}$.

(c) If moreover $\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{q}\in C^{\alpha}\left(\overline{D},\mathbb{R}^{3}\right)$ for a certain $\alpha\in(0,1)$ then the mapping $(t,x)\to Z_{t}(x)$ has a continuous version in $\mathcal{L}\left({\mathbb{R}}^{3}\right)$.*

Proof.

Equation (3.3) can be equivalently written as an Itô equation

[TABLE]

Since the coefficients of equation (3.5) are Lipschitz, the existence and uniqueness of strong solutions to equation (3.5), and the existence of its continuous version is standard, see for example Theorem 18.3 in [16]. Hence, the same result holds for (3.3).

To prove (b) we fix $x\in\overline{D}$, $t\geq 0$ and $\boldsymbol{u}\in\mathbb{R}^{3}$ and put $Z^{\boldsymbol{u}}=Z\boldsymbol{u}$. Then equation (3.5) yields

[TABLE]

Applying the Itô formula to the process $|Z_{t}^{\boldsymbol{u}}|^{2}$ and invoking (3.1) we obtain

[TABLE]

or equivalently

[TABLE]

To prove (c), we begin by letting $0\leq s<t\leq T$ and $x,y\in\overline{D}$. For any $p\geq 1$ we have

[TABLE]

It is well known that there exists $C_{1}>0$ such that

[TABLE]

If there exists $\alpha\in(0,1]$ such that

[TABLE]

then for a certain $C>0$, for any $h\in{\mathbb{R}}^{3}$ there holds

[TABLE]

Then

[TABLE]

Using (3.9) we obtain

[TABLE]

Therefore, invoking the Gronwall Lemma we obtain

[TABLE]

Combining (3.7), (3.8) and (3.10) we obtain

[TABLE]

Let $\beta>0$ and $r=d+1+\beta$. Let $p$ be chosen in such a way that

[TABLE]

The set $[0,T]\times D$ can be covered by a finite number of open sets $B_{k}$ with the property $|t-s|^{r}+|x-y|^{r}<1$ on every set $B_{k}$. In each $B_{k}$, (3.11) then yields

[TABLE]

and the result then follows by the Kolmogorv-Chentsov theorem, see p. 57 of [16]. ∎

Lemma 3.2**.**

*Assume that $\boldsymbol{g}_{i}\in C^{1+\alpha}_{b}(D,\mathbb{R}^{3})$. Then the following holds.

(a) For every $t\geq 0$ we have $Z_{t}\in C^{1}_{b}(D,\mathcal{L}(\mathbb{R}^{3}))$ $\mathbb{P}$-a.s.

(b) For every $x\in D$ the process $\xi_{t}(x)=\nabla Z_{t}(x)$ is the unique solution of the linear Itô equation*

[TABLE]

with $\xi_{0}(x)=0$ and the operators $H_{i},I_{i}\in\mathcal{L}\left(\mathbb{R}^{3}\right)$ defined as

[TABLE]

*(c) For every $\gamma<\min\left(\alpha,\frac{1}{2}\right)$ the mapping $(t,x)\to\nabla Z_{t}(x)$ is $\gamma$-Hölder continuous.

(d) We have*

[TABLE]

Proof.

(a) Let $E$ denote the Banach space of continuous and adapted processes $Z$ taking values in the space of linear operators $\mathcal{L}\left(\mathbb{R}^{3}\right)$ and endowed with the norm

[TABLE]

For every $x\in D$ we define a mapping

[TABLE]

It is easy to check that the assumptions of Lemma 9.2, p. 238 in [10] are satisfied and therefore (a) holds.

(b) The proof is completely analogous to the proof of Theorem 9.8 in [10], and is hence omitted.

(c) The proof is analogous to the proof of part (c) of Lemma 3.1.

(d) The estimate follows easily from (c). ∎

For every $\boldsymbol{u}\in\mathbb{L}^{2}(D)$ we will consider the $L^{2}(D)$-valued process $Z_{t}(u)$ defined by

[TABLE]

Clearly,

[TABLE]

where the equality holds in $\mathbb{L}^{2}(D)$. The process $Z_{t}$ is now an operator-valued process taking values $\mathcal{L}\left(\mathbb{L}^{2}(D)\right)$ and it will still be denoted by $Z_{t}$. The next lemmas follow immediately from the properties of the matrix-valued process considered above.

Lemma 3.3**.**

*Assume that $\left\{g_{i}:\,i=1,\ldots,q\right\}\subset C_{b}^{1+\alpha}(D)$. Then for every $\boldsymbol{u}\in\mathbb{L}^{2}(D)$ the stochastic differential equation (3.12) has a unique strong continuous solution in $\mathbb{L}^{2}(D)$. Moreover, there exists $\Omega_{0}\subset\Omega$ such that $\mathbb{P}\left(\Omega_{0}\right)=1$ and for every $\omega\in\Omega_{0}$ the following holds.

(a) For all $t\geq 0$ and every $\boldsymbol{u}\in\mathbb{L}^{2}(D)$,*

[TABLE]

(b) For every $t\geq 0$ the mapping $\boldsymbol{u}\to Z_{t}(\omega,\boldsymbol{u})$ defines a linear bounded operator $Z_{t}(\omega)$ on $\mathbb{L}^{2}(D)$. In particular,

[TABLE]

Moreover, for every $T>0$ there exists a constant $C_{T}>0$ such that

[TABLE]

(c) For every $t\geq 0$ the operator $Z_{t}(\omega)$ is invertible and the inverse operator is the unique solution of the stochastic differential equation on $\mathbb{L}^{2}(D)$:

[TABLE]

Finally,

[TABLE]

Lemma 3.4**.**

Assume that $\boldsymbol{g}_{i}\in C_{b}^{1+\alpha}\left(D,\mathbb{R}^{3}\right)$ for $i=1,\cdots,q$. Then, for every $\boldsymbol{u}\in\mathbb{H}^{1}(D)$ $Z_{t}(\boldsymbol{u})\in\mathbb{H}^{1}$ $\mathbb{P}$-a.s. Furthermore, the process $\xi_{t}(\boldsymbol{u}):=\nabla Z_{t}(\boldsymbol{u})$, is the unique solution of the linear equation

[TABLE]

with $\xi_{0}(\boldsymbol{u})=\nabla\boldsymbol{u}$.

Lemma 3.5**.**

For any $\boldsymbol{u},\boldsymbol{v}\in\mathbb{L}^{2}(D)$, there holds for all $t\geq 0$ and $\mathbb{P}$-a.s.:

[TABLE]

Proof.

Let $Z_{t}^{\boldsymbol{u}}:=Z_{t}(\boldsymbol{u})$ and $Z_{t}^{\boldsymbol{v}}:=Z_{t}(\boldsymbol{v})$ for all $t\geq 0$. We now prove (3.17); the property (3.13) can be obtained in the same manner. Using the Itô formula for $Z_{t}^{\boldsymbol{u}}\times Z_{t}^{\boldsymbol{v}}$ and (3.6), we obtain

[TABLE]

Using an elementary identity

[TABLE]

we find that

[TABLE]

and

[TABLE]

Invoking (3.20) and (3.21), equation (3.18) we obtain

[TABLE]

Therefore, the process $V_{t}:=Z_{t}^{\boldsymbol{u}}\times Z_{t}^{\boldsymbol{v}}$ is a solution of the following stochastic differential equation:

[TABLE]

On the other hand, it follows from (3.6) that the process $Z_{t}(\boldsymbol{u}\times\boldsymbol{v})$ satisfies the same equation. Hence,  (3.17) follows from the uniqueness of solutions to (3.18).

∎

Lemma 3.6**.**

For any $\boldsymbol{u},\boldsymbol{v}\in\mathbb{H}^{1}(D)$, there holds for all $t\geq 0$ and $\mathbb{P}$-a.s.:

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Let $\xi_{t}^{\boldsymbol{u}}:=\xi_{t}(\boldsymbol{u})$ and $\xi_{t}^{\boldsymbol{v}}:=\xi_{t}(\boldsymbol{v})$ for all $t\geq 0$. In addition, we consider a $C^{\infty}$ function \phi:\bigl{(}\mathbb{L}^{2}(D)\bigr{)}^{2}\rightarrow{\mathbb{R}} defined by $\phi(\boldsymbol{x},\boldsymbol{y})=\left\langle{\boldsymbol{x}},{\boldsymbol{y}}\right\rangle_{\mathbb{L}^{2}(D)}$. By using the Itô Lemma we obtain

[TABLE]

Using (3.1) and (3.2), we deduce from (3.23) that

[TABLE]

Integrating by parts for the first and the last term in the right hand side of the above equation and noting the homogeneous Neumann boundary condition of $\boldsymbol{g}_{i}$, we obtain

[TABLE]

Hence, the resutl follows from replacing $t$ by $s$ and intergrating (3) over $[0,t]$. ∎

Remark 3.7**.**

By using integration by parts and the homogeneous Neumann boundary conditions of $\boldsymbol{g}_{i}$ for $i=1,\cdots,q$ we obtain some symmetry properties of functions $F_{1,i}$, $F_{2,i}$ and $F$: for any $\boldsymbol{u},\boldsymbol{v},\boldsymbol{v}_{1},\boldsymbol{v}_{2}\in\mathbb{H}^{1}(D)$,

[TABLE]

and hence, $F(t,\boldsymbol{u},\boldsymbol{v})=F(t,\boldsymbol{v},\boldsymbol{u})$. Furthermore, it follows from (3.13) that

[TABLE]

The following lemmas state some important properties of $F$ used throughout this paper.

Lemma 3.8**.**

Assume that $\boldsymbol{g}_{i}\in\mathbb{W}^{2,\infty}(D)$ for $i=1,\cdots,q$. Then for any $\boldsymbol{u},\boldsymbol{v}\in L^{2}(\Omega;\mathbb{H}^{1}(D))$ there exists a constant $c$ depending on $T$ and $\{\boldsymbol{g}_{i}\}_{i=1,\cdots,q}$ such that

[TABLE]

and for any $\epsilon>0$,

[TABLE]

Proof.

It follows from (3.22) that

[TABLE]

For convenience, we next estimate $|F(\tau,\boldsymbol{u},\boldsymbol{v})|$, which is a slightly more general version of $|F(\tau,\boldsymbol{u},\boldsymbol{u})|$. By using the elementary inequality

[TABLE]

the assumption $\boldsymbol{g}_{i}\in\mathbb{W}^{2,\infty}(D)$ and (3.3), there holds

[TABLE]

This implies that

[TABLE]

Then, by using the Burkholder-Davis-Gundy inequality, Hölder inequality, (3.3) and (3.29), we estimate

[TABLE]

We use (3.30) and (3.32) with $\boldsymbol{v}=\boldsymbol{u}$ and $\epsilon=1$ together with (3.27) to deduce

[TABLE]

Hence, the result (3.25) follows immediately by using Gronwall’s inequality.

To prove (3.26) we note that

[TABLE]

Hence, it follows from (3.30), (3.32) and (3.25) that

[TABLE]

which completed the proof of the lemma. ∎

Lemma 3.9**.**

For any \boldsymbol{u}\in L^{2}\bigl{(}\Omega;\mathbb{L}^{2}(D)\bigr{)}, \boldsymbol{v}\in L^{2}\bigl{(}\Omega;\mathbb{H}^{1}(D)\bigr{)} and $0\leq s\leq T$, there exists a constant $c$ depending on $\{\boldsymbol{g}_{i}\}_{i=1,\cdots,q}$ such that

[TABLE]

Proof.

From the definition of function $F$ in Lemma 3.6 and the triangle inequality, there holds

[TABLE]

From Remark 3.7, we note that

[TABLE]

and therefore, by using (3), the last term of (3.34) can be estimated as follows:

[TABLE]

We now estimate \bigl{|}F_{1,i}(\tau,\boldsymbol{u},\boldsymbol{v})\bigr{|} by integrating by parts and then using Hölder’s inequality, the assumption $\boldsymbol{g}_{i}\in\mathbb{W}^{2,\infty}(D)$ and (3.3) as follows:

[TABLE]

and therefore,

[TABLE]

Hence, by using Hölder inequality we obtain from (3.34)–(3) that there holds:

[TABLE]

Via the Minkowski inequality and (3.25), we observe that

[TABLE]

The required result follows from (3) and (3.38), which completes the proof of this lemma. ∎

4. Equivalence of weak solutions

In this section we use the process $(Z_{t})_{t\geq 0}$ defined in the preceding section to define a new process $\boldsymbol{m}$ from $\boldsymbol{M}$. Let

[TABLE]

We will show that this new variable $\boldsymbol{m}$ is differentiable with respect to $t$.

In the next lemma, we introduce the equation satisfied by $\boldsymbol{m}$ so that $\boldsymbol{M}$ is a solution to (1.3) in the sense of (4).

Lemma 4.1**.**

If $\boldsymbol{m}(\cdot,\omega)\in H^{1}(0,T;\mathbb{L}^{2}(D))\cap L^{2}(0,T;\mathbb{H}^{1}(D))$, for $\mathbb{P}$-a.s. $\omega\in\Omega$, satisfies

[TABLE]

and for any $\boldsymbol{\psi}\in L^{2}(0,T;\mathbb{W}^{1,\infty}(D))$

[TABLE]

Then $\boldsymbol{M}=Z_{t}\boldsymbol{m}$ satisfies (4) $\mathbb{P}$-a.s..

Proof.

Using Itô’s formula for $\boldsymbol{M}=Z_{t}\boldsymbol{m}$, we deduce

[TABLE]

Multiplying both sides by a test function $\boldsymbol{\psi}\in{\mathbb{C}}_{0}^{\infty}(D)$ and integrating over $D$ we obtain

[TABLE]

where in the last step we used (3.16). On the other hand, it follows from (4.1) that, for all $\boldsymbol{\xi}\in L^{2}(0,t;\mathbb{W}^{1,\infty}(D))$, there holds:

[TABLE]

Using (4) with $\boldsymbol{\xi}=Z_{s}^{-1}\boldsymbol{\psi}$ for the last term on the right hand side of (4) we deduce

[TABLE]

It follows from (3.17) that

[TABLE]

which complete the proof. ∎

The following lemma shows that the constraint on $|\boldsymbol{m}|$ is inherited by $|\boldsymbol{M}|$.

Lemma 4.2**.**

The process $\boldsymbol{M}$ satisfies

[TABLE]

if and only if $\boldsymbol{m}$ defined in (4.1) satisfies

[TABLE]

Proof.

The proof follows by using (3.16):

[TABLE]

∎

In the next lemma we provide a relationship between equation (4.1) and its Gilbert form.

Lemma 4.3**.**

Let $\boldsymbol{m}(\cdot,\omega)\in H^{1}(0,T;\mathbb{L}^{2}(D))\cap L^{2}(0,T;\mathbb{H}^{1}(D))$ for $\mathbb{P}$-a.s. $\omega\in\Omega$ satisfy

[TABLE]

and

[TABLE]

where $\mu=\lambda_{1}^{2}+\lambda_{2}^{2}$. Then $\boldsymbol{m}$ satisfies (4.1).

Proof.

For each $\boldsymbol{\psi}\in L^{2}(0,T;\mathbb{W}^{1,\infty}(D))$, using Lemma 7.1 in the Appendix, there exists $\boldsymbol{\varphi}\in L^{2}(0,T;\mathbb{H}^{1}(D))$ such that

[TABLE]

We can write (4.6) as

[TABLE]

From  (4.5) and (3.3), we obtain that

[TABLE]

On the other hand, by using (4.9), (3.19) and a standard identity

[TABLE]

we obtain

[TABLE]

Moreover, we have

[TABLE]

Summing (4), (4) and (4.12) gives

[TABLE]

The desired equation (4.1) follows by noting (4.7) and using (3.19), (4.10) and (4.9). ∎

Remark 4.4**.**

By using (4.10) and (4.5) we can rewrite (4.6) as

[TABLE]

or equivalently, thanks to Lemma 3.6,

[TABLE]

where $\boldsymbol{w}=\boldsymbol{m}\times\boldsymbol{\varphi}$ for $\boldsymbol{\varphi}\in L^{2}(0,T;\mathbb{H}^{1}(D))$. We note in particular that $\boldsymbol{w}\cdot\boldsymbol{m}=0$. This property will be exploited later in the design of the finite element scheme.

We state the following lemma as a consequence of Lemmas 4.3, 4.2 and 4.1.

Lemma 4.5**.**

Let $\boldsymbol{m}(\cdot,\omega)\in H^{1}(0,T;\mathbb{L}^{2}(D))\cap L^{2}(0,T;\mathbb{H}^{1}(D))$ for $\mathbb{P}$-a.s. $\omega\in\Omega$. If $\boldsymbol{m}$ is a solution of (4.5)– (4.6), then $\boldsymbol{M}=Z_{t}\boldsymbol{m}$ is a weak martingale solution of (1.3) in the sense of Definition 2.2.

Proof.

By using Lemmas 4.1, 4.2 and 4.3 together with the imbedding $H^{1}(0,T;\mathbb{L}^{2}(D))\hookrightarrow C(0,T;\mathbb{H}^{-1}(D)$, we deduce that $\boldsymbol{M}$ satisfies $(1)$, $(2)$, $(3)$, $(4)$ in Definition 2.2, which completes the proof. ∎

Thanks to the above lemma, we now can now restrict our attention to solving equation (4.6) rather than  (4).

5. The finite element scheme

In this section we design a finite element scheme to find approximate solutions to (4.6). In the next section, we prove that the finite element solutions converge to a solution of (4.6). Then, thanks to Lemma 4.5, we obtain a weak solution of (4).

Let $\mathbb{T}_{h}$ be a regular tetrahedrization of the domain $D$ into tetrahedra of maximal mesh-size $h$. We denote by ${\mathcal{N}}_{h}:=\{\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{N}\}$ the set of vertices and introduce the finite-element space $\mathbb{V}_{h}\subset\mathbb{H}^{1}(D)$, which is the space of all continuous piecewise linear functions on $\mathbb{T}_{h}$. A basis for $\mathbb{V}_{h}$ can be chosen to be $\{\phi_{n}\boldsymbol{\xi}_{1},\phi_{n}\boldsymbol{\xi}_{2},\phi_{n}\boldsymbol{\xi}_{3}\}_{1\leq n\leq N}$, where $\{\boldsymbol{\xi}_{i}\}_{i=1,\cdots,3}$ is the canonical basis for ${\mathbb{R}}^{3}$ and $\phi_{n}(\boldsymbol{x}_{m})=\delta_{n,m}.$ Here $\delta_{n,m}$ denotes the Kronecker delta symbol. The interpolation operator from ${\mathbb{C}}^{0}(D)$ onto $\mathbb{V}_{h}$, denoted by $I_{\mathbb{V}_{h}}$, is defined by

[TABLE]

Before introducing the finite element scheme, we state the following result proved by Bartels [4], which will be used in the subsequent analysis.

Lemma 5.1**.**

Assume that

[TABLE]

Then for all $\boldsymbol{u}\in\mathbb{V}_{h}$ satisfying $|\boldsymbol{u}(\boldsymbol{x}_{l})|\geq 1$, $l=1,2,\cdots,J$, there holds

[TABLE]

When $d=2$, we note that condition (5.1) holds for Delaunay triangulation. Roughly speaking, a Delaunay triangulation is one in which no vertex is contained inside the perimeter of any triangle. When $d=3$, condition (5.1) holds if all dihedral angles of the tetrahedra in $\mathbb{T}_{h}|_{\mathbb{L}^{2}(D)}$ are less than or equal to $\pi/2$; see [4]. In what follows we assume that (5.1) holds.

To discretize the equation (4.6), we fix a positive integer $J$, choose the time step $k$ to be $k=T/J$ and define $t_{j}=jk$, $j=0,\cdots,J$. For $j=1,2,\ldots,J$, the solution $\boldsymbol{m}(t_{j},\cdot)$ is approximated by $\boldsymbol{m}^{(j)}_{h}\in\mathbb{V}_{h}$, which is computed as follows.

Since

[TABLE]

we can define $\boldsymbol{m}_{h}^{(j+1)}$ from $\boldsymbol{m}_{h}^{(j)}$ by

[TABLE]

where $\boldsymbol{v}_{h}^{(j)}$ is an approximation of $\boldsymbol{m}_{t}(t_{j},\cdot)$. Hence, it suffices to propose a scheme to compute $\boldsymbol{v}_{h}^{(j)}$.

Motivated by the property $\partial_{t}\boldsymbol{m}\cdot\boldsymbol{m}=0$, we will find $\boldsymbol{v}_{h}^{(j)}$ in the space $\mathbb{W}^{(j)}_{h}$ defined by

[TABLE]

Given $\boldsymbol{m}_{h}^{(j)}\in\mathbb{V}_{h}$, we use (4.4) to define $\boldsymbol{v}_{h}^{(j)}$ instead of using (4.6) so that the same test and trial functions can be used (see Remark 4.4). Hence, we define by $\boldsymbol{v}_{h}^{(j)}\in\mathbb{W}_{h}^{(j)}$ satisfying the following equation

[TABLE]

We summarise the algorithm as follows.

Algorithm 5.1**.**

**

**Step 1: **

Set $j=0$. Choose $\boldsymbol{m}^{(0)}_{h}=I_{\mathbb{V}_{h}}\boldsymbol{m}_{0}$.

**Step 2: **

Find $\boldsymbol{v}_{h}^{(j)}\in\mathbb{W}_{h}^{(j)}$ satisfying (5).

**Step 3: **

Define

[TABLE]

**Step 4: **

Set $j=j+1$, and return to Step 2 if $j<J$. Stop if $j=J$.

Since $\left|\boldsymbol{m}_{h}^{(0)}(x_{n})\right|=1$ and $\boldsymbol{v}_{h}^{(j)}(x_{n})\cdot\boldsymbol{m}_{h}^{(j)}(x_{n})=0$ for all $n=1,\ldots,N$ and $j=0,\ldots,J$, we obtain (by induction)

[TABLE]

In particular,  (5.6) shows that the algorithm is well defined.

We finish this section by proving the following three lemmas concerning some properties of $\boldsymbol{m}_{h}^{(j)}$ and $R_{h,k}$.

Lemma 5.2**.**

For any $j=0,\ldots,J$,

[TABLE]

where $|D|$ denotes the measure of $D$.

Proof.

The first inequality follows from (5.6) and the second can be obtained by integrating $\boldsymbol{m}_{h}^{(j)}(\boldsymbol{x})$ over $D$. ∎

Lemma 5.3**.**

There exist a deterministic constant $c$ depending on $\boldsymbol{m}_{0}$, $\{\boldsymbol{g}_{i}\}_{i=1}^{q}$, $\lambda_{1}$ and $\lambda_{2}$ such that for any $\theta\in[1/2,1]$ and for $j=1,\ldots,J$ there holds

[TABLE]

Proof.

Taking $\boldsymbol{w}_{h}^{(j)}=\boldsymbol{v}_{h}^{(j)}$ in equation (5) yields to the following identity:

[TABLE]

or equivalently

[TABLE]

From Lemma 5.1 it follows that

[TABLE]

or equivalently,

[TABLE]

Therefore, together with (5.7), we deduce

[TABLE]

Thus, it follows from (3.26) that

[TABLE]

By choosing $\epsilon=\frac{\mu^{-1}\lambda_{2}}{c(T+T^{1/2}+1)}$ in the right hand side of this inequality and using Lemma 5.2 we deduce

[TABLE]

Replacing $j$ by $i$ in the above inequality and summing for $i$ from [math] to $j-1$ yeilds

[TABLE]

Since $\boldsymbol{m}_{0}\in\mathbb{H}^{2}(D)$ it can be shown that there exists a deterministic constant $c$ depending only on $\boldsymbol{m}_{0}$ such that

[TABLE]

Hence, inequality (5) implies

[TABLE]

By using induction and (5.9) we can show that

[TABLE]

Summing over $i$ from 0 to $j-1$ and using $1+x\leq e^{x}$ we obtain

[TABLE]

This together with (5) gives the desired result. ∎

6. The main result

In this section, we will use the finite element function $\boldsymbol{m}_{h}^{(j)}$ to construct a sequence of functions that converges in an appropriate sense to a function that is a weak martingale solution of (1.3) in the sense of Definition 2.2.

The discrete solutions $\boldsymbol{m}_{h}^{(j)}$ and $\boldsymbol{v}_{h}^{(j)}$ constructed via Algorithm 5.1 are interpolated in time in the following definition.

Definition 6.1**.**

For all $x\in D$, $\boldsymbol{u},\boldsymbol{v}\in\mathbb{V}_{h}$ and all $t\in[0,T]$, let $j\in\{0,...,J\}$ be such that $t\in[t_{j},t_{j+1})$. We then define

[TABLE]

We note that $\boldsymbol{m}_{h,k}(t)$ is an ${\mathcal{F}}_{t_{j}}$ adapted process for $t\in[t_{j},t_{j+1})$. The above sequences have the following obvious bounds.

Lemma 6.2**.**

There exist a deterministic constant $c$ depending on $\boldsymbol{m}_{0}$, $\boldsymbol{g}$, $\mu_{1}$, $\mu_{2}$ and $T$ such that for all $\theta\in[1/2,1]$,

[TABLE]

where $\boldsymbol{m}_{h,k}^{*}=\boldsymbol{m}_{h,k}$ or $\boldsymbol{m}_{h,k}^{-}$. In particular, when $\theta\in[0,\frac{1}{2})$,

[TABLE]

Proof.

It is easy to see that

[TABLE]

Both inequalities are direct consequences of Definition 6.1, Lemmas 5.2, and 5.3, noting that the second inequality requires the use of the inverse estimate (see e.g. [14])

[TABLE]

∎

The next lemma provides a bound of $\boldsymbol{m}_{h,k}$ in the $\mathbb{H}^{1}$-norm and establishes relationships between $\boldsymbol{m}_{h,k}^{-}$, $\boldsymbol{m}_{h,k}$ and $\boldsymbol{v}_{h,k}$.

Lemma 6.3**.**

Assume that $h$ and $k$ approach [math], with the further condition $k=o(h^{2})$ when $\theta\in[0,\frac{1}{2})$. The sequences $\{\boldsymbol{m}_{h,k}\}$, $\{\boldsymbol{m}_{h,k}^{-}\}$, and $\{\boldsymbol{v}_{h,k}\}$ defined in Definition 6.1 satisfy the following properties:

[TABLE]

Proof.

The results can be obtained by using Lemma 6.2 and the arguments in the proof of [13, Lemma 6.3]. ∎

We now prove some properties of $F$ and $F_{k}$, which will be used in the next two lemmas.

Lemma 6.4**.**

For any \boldsymbol{u},\boldsymbol{v}\in L^{2}\bigl{(}\Omega;L^{2}(0,T;\mathbb{H}^{1}(D))\bigr{)}, there exists a constant $c$ depending on $T$ and $\{\boldsymbol{g}_{i}\}_{i=1,\cdots,q}$ such that

[TABLE]

here, $F^{*}=F$ or $F_{k}$. Furthermore,

[TABLE]

Proof.

Proof of (6.5): The first result of the lemma for $F^{*}=F$ can be deduced from Lemma 3.9 by replacing $s\equiv t$, $\boldsymbol{u}\equiv\boldsymbol{u}(t,\cdot)$, $\boldsymbol{v}\equiv\boldsymbol{v}(t,\cdot)$ and using Hölder’s inequality as follows:

[TABLE]

We first note that

[TABLE]

then apply Lemma 3.9 for $s\equiv t_{j}$, $\boldsymbol{u}\equiv\boldsymbol{u}(t,\cdot)$ and $\boldsymbol{v}\equiv\boldsymbol{v}(t,\cdot)$ to deduce

[TABLE]

Hence,  (6.5) with function $F^{*}=F_{k}$ follows by using Hölder’s inequality.

Proof of (6.4): Noting that

[TABLE]

Here

[TABLE]

in which $\tilde{F}^{j}_{1,i}(s,\boldsymbol{x},\boldsymbol{y})=F_{1,i}(s+t_{j},\boldsymbol{x},\boldsymbol{y})$, $\tilde{F}^{j}_{2,i}(s,\boldsymbol{x},\boldsymbol{y})=F_{2,i}(s+t_{j},\boldsymbol{x},\boldsymbol{y})$ and $\tilde{W}_{i}(s)=W_{i}(s+t_{j})-W_{i}(t_{j})$. By using the same arguments as in the proof of Lemma 3.9 we obtain the same result for the upper bound of $\tilde{F}^{j}$, namely

[TABLE]

Hence, there holds

[TABLE]

Therefore, it follows from (6) and (6) that

[TABLE]

The result follows immediately by using Hölder’s inequality, which completes the proof of the lemma. ∎

The following two Lemmas 6.5 and  6.6 show that  $\boldsymbol{m}_{h,k}^{-}$ and $\boldsymbol{m}_{h,k}$, respectively, satisfy a discrete form of (4.6).

Lemma 6.5**.**

Assume that $h$ and $k$ approach [math] with the following conditions

[TABLE]

Then for any \boldsymbol{\psi}\in C_{0}^{\infty}\big{(}(0,T);{\mathbb{C}}^{\infty}(D)\big{)}, there holds $\mathbb{P}$-a.s.

[TABLE]

where

[TABLE]

Furthermore, $\mathbb{E}|I_{i}|=O(h)$ for $i=1,2,3$.

Proof.

For $t\in[t_{j},t_{j+1})$, we use equation (5) with \boldsymbol{w}_{h}^{(j)}=I_{\mathbb{V}_{h}}\big{(}\boldsymbol{m}_{h,k}^{-}(t,\cdot)\times\boldsymbol{\psi}(t,\cdot)\big{)} to see

[TABLE]

Integrating both sides of the above equation over $(t_{j},t_{j+1})$ and summing over $j=0,\ldots,J-1$ we deduce

[TABLE]

This implies

[TABLE]

Hence it suffices to prove that $\mathbb{E}|I_{i}|=O(h)$ for $i=1,2,3$. First, by using Lemma 5.2 we obtain

[TABLE]

and

[TABLE]

Lemma 6.2 and (6.11) together with Hölder’s inequality and Lemma 7.2 yield

[TABLE]

The bound for $\mathbb{E}|I_{2}|$ can be obtained similarly, using Lemma 6.2 and noting that when $\theta\in[0,\frac{1}{2}]$, a suitable bound on $k\left\|\nabla\boldsymbol{v}_{h,k}\right\|_{\mathbb{L}^{2}(D_{T})}$ can be deduced from the inverse estimate as follows:

[TABLE]

The bound for $\mathbb{E}|I_{3}|$ can be obtained by noting the linearity of $F$ in Remark 3.7 and using Lemmas 6.4 and 7.2. Indeed,

[TABLE]

This completes the proof of the lemma. ∎

Lemma 6.6**.**

Assume that $h$ and $k$ approach 0 satisfying (6.10). Then for any \boldsymbol{\psi}\in C_{0}^{\infty}\big{(}(0,T);{\mathbb{C}}^{\infty}(D)\big{)}, there holds $\mathbb{P}$-a.s.

[TABLE]

where

[TABLE]

Furthermore, $\mathbb{E}|I_{i}|=O(h)$ for $i=1,\cdots,6$ and $\mathbb{E}|I_{7}|=O(h+k^{1/2})$.

Proof.

From Lemma 6.5 it follows that

[TABLE]

Hence it suffices to prove that $\mathbb{E}|I_{i}|=O(h)$ for $i=4,\cdots,6$. First, by using the triangle inequality and Hölder’s inequality, we obtain

[TABLE]

Therefore, the required bound on $\mathbb{E}|I_{4}|$ can be obtained by using (6.11), (6.12) and Lemmas 6.2, 6.3. The bounds on $\mathbb{E}|I_{5}|$ and $\mathbb{E}|I_{6}|$ can be obtaineded similarly.

In order to prove the bound for $\mathbb{E}|I_{7}|$, we first use the triangle inequality then Remark 3.7 and Lemma 6.4 to obtain

[TABLE]

in which (6.1) and (6.2) are used to obtain the last inequality. This completes the proof of the lemma. ∎

In order to prove the convergence of random variables $\boldsymbol{m}_{h,k}$, we first state a result of tightness for the family ${\mathcal{L}}(\boldsymbol{m}_{h,k})$. We then use the Skorohod theorem to define another probability space and an almost surely convergent sequence defined in this space whose limit is a weak martingale solution of equation (4.4). The proof of the following results are omitted since they are relatively simple modification of the proof of the corresponding results from [13].

Lemma 6.7**.**

Assume that $h$ and $k$ approach [math], and further that  (6.10) holds. Then the set of laws $\{{\mathcal{L}}(\boldsymbol{m}_{h,k})\}$ on the Banach space C\big{(}[0,T];\mathbb{H}^{-1}(D)\big{)}\cap L^{2}(0,T;\mathbb{L}^{2}(D)) is tight.

Proposition 6.8**.**

Assume that $h$ and $k$ approach [math], and further that  (6.10) holds. Then there exist:

1. (a)

a probability space $(\Omega^{\prime},{\mathcal{F}}^{\prime},\mathbb{P}^{\prime})$; 2. (b)

a sequence $\{\boldsymbol{m}^{\prime}_{h,k}\}$ of random variables defined on $(\Omega^{\prime},{\mathcal{F}}^{\prime},\mathbb{P}^{\prime})$ and taking values in C\big{(}[0,T];\mathbb{H}^{-1}(D)\big{)}\cap L^{2}(0,T;\mathbb{L}^{2}(D)); and 3. (c)

a random variable $\boldsymbol{m}^{\prime}$ defined on $(\Omega^{\prime},{\mathcal{F}}^{\prime},\mathbb{P}^{\prime})$ and taking values in C\big{(}[0,T];\mathbb{H}^{-1}(D)\big{)}\cap L^{2}(0,T;\mathbb{L}^{2}(D)),

satisfying

1. (1)

${\mathcal{L}}(\boldsymbol{m}_{h,k})={\mathcal{L}}(\boldsymbol{m}_{h,k}^{\prime})$, 2. (2)

$\boldsymbol{m}_{h,k}^{\prime}\rightarrow\boldsymbol{m}^{\prime}$* in C\big{(}[0,T];\mathbb{H}^{-1}(D)\big{)}\cap L^{2}(0,T;\mathbb{L}^{2}(D)) strongly, $\mathbb{P}^{\prime}$-a.s..*

Moreover, the sequence $\{\boldsymbol{m}^{\prime}_{h,k}\}$ satisfies

[TABLE]

here, $c$ is a positive constant only depending on $\{\boldsymbol{g}_{i}\}_{i=1,\cdots,q}$.

We are now ready to state and prove our main theorem.

Theorem 6.9**.**

Assume that $T>0$, $\boldsymbol{M}_{0}\in\mathbb{H}^{1}(D)$ satisfies (LABEL:equ:m0) and $\boldsymbol{g}_{i}\in\mathbb{W}^{2,\infty}(D)$ for $i=1,\cdots,q$ satisfy the homogeneous Neumann boundary condition. Then $\boldsymbol{m}^{\prime}$, the sequence $\{\boldsymbol{m}^{\prime}_{h,k}\}$ and the probability space $(\Omega^{\prime},{\mathcal{F}}^{\prime},\mathbb{P}^{\prime})$ given by Proposition 6.8 satisfy:

1. (1)

the sequence of $\{\boldsymbol{m}^{\prime}_{h,k}\}$ converges to $\boldsymbol{m}^{\prime}$ weakly in $L^{2}(\Omega^{\prime};\mathbb{H}^{1}(D_{T}))$; and 2. (2)

\big{(}\Omega^{\prime},{\mathcal{F}}^{\prime},({\mathcal{F}}^{\prime}_{t})_{t\in[0,T]},\mathbb{P}^{\prime},\boldsymbol{M}^{\prime}\big{)}* is a weak martingale solution of (1.3), where*

[TABLE]

Proof.

From (6.16) and property (2) of Proposition 6.8, there exists a set $V\subset\Omega^{\prime}$ such that $\mathbb{P}^{\prime}(V)=1$ and for all $\omega^{\prime}\in V$ there hold

[TABLE]

Hence, by using Lebesgue’s dominated convergence theorem, we deduce

[TABLE]

which implies from (6.14) that

[TABLE]

In order to prove Part (2), by noting Lemma 4.5 and Remark 4.4 we only need to prove that $\boldsymbol{m}^{\prime}$ satisfies (4.5) and (4.4), namely

[TABLE]

and

[TABLE]

where

[TABLE]

By using (6.15) and (6.17), we obtain (6.19) immediately.

In order to prove (6.20), we first find the equation satisfied by $\boldsymbol{m}_{h,k}^{\prime}$ and then pass to the limit when $h$ and $k$ approach [math].

By using Lemmas 6.6 and property (1) of Proposition 6.8, it follows that for any $\boldsymbol{\psi}\in C^{\infty}_{0}(0,T;\mathbb{C}^{\infty}(D))$ that there holds

[TABLE]

To pass to the limit in (6.21), we first using (6.17)–(6.19) and the same arguments as in [13, Theorem 6.8] to obtain that as $h$ and $k$ tend to [math],

[TABLE]

Then, by using Remark 3.7 and (6.5) with $F^{*}=F$, we estimate

[TABLE]

Since $\boldsymbol{m}^{\prime}\in L^{2}(\Omega^{\prime};\mathbb{H}^{1}(D_{T}))$, it follows from (6.14) and (6.17) that

[TABLE]

as $h$ and $k$ tend to [math]. From (6.22)–(6.25) we deduce that

[TABLE]

and hence, together with (6.21) $\mathbb{E}|{\mathcal{I}}(\boldsymbol{m}^{\prime},\boldsymbol{\psi})|=0$. This implies (6.20) which completes the proof of our main theorem. ∎

7. Appendix

For the reader’s convenience we will recall the following results, which are proved in [13].

Lemma 7.1**.**

For any real constants $\lambda_{1}$ and $\lambda_{2}$ with $\lambda_{1}\not=0$, if $\boldsymbol{\psi},\boldsymbol{\zeta}\in{\mathbb{R}}^{3}$ satisfy $|\boldsymbol{\zeta}|=1$, then there exists $\boldsymbol{\varphi}\in{\mathbb{R}}^{3}$ satisfying

[TABLE]

As a consequence, if $\boldsymbol{\zeta}\in\mathbb{H}^{1}(D_{T})$ with $|\boldsymbol{\zeta}(t,x)|=1$ a.e. in $D_{T}$ and $\boldsymbol{\psi}\in L^{2}(0,T;\mathbb{W}^{1,\infty}(D))$, then $\boldsymbol{\varphi}\in L^{2}(0,T;\mathbb{H}^{1}(D))$.

Lemma 7.2**.**

For any $\boldsymbol{v}\in{\mathbb{C}}(D)$, $\boldsymbol{v}_{h}\in\mathbb{V}_{h}$ and $\boldsymbol{\psi}\in{\mathbb{C}}_{0}^{\infty}(D_{T})$,

[TABLE]

where $\boldsymbol{m}_{h,k}^{-}$ is defined in Defintion 6.1

The next lemma defines a discrete $\mathbb{L}^{p}$-norm in $\mathbb{V}_{h}$, equivalent to the usual $\mathbb{L}^{p}$-norm.

Lemma 7.3**.**

There exist $h$-independent positive constants $C_{1}$ and $C_{2}$ such that for all $p\in[1,\infty]$ and $\boldsymbol{u}\in\mathbb{V}_{h}$,

[TABLE]

where $\Omega\subset{\mathbb{R}}^{d}$, d=1,2,3.

Acknowledgements

The authors acknowledge financial support through the ARC Discovery projects DP140101193 and DP120101886. They are grateful to Vivien Challis for a number of helpful conversations.