Weak convergence of the backward Euler method for stochastic Cahn–Hilliard equation with additive noise

Meng Cai mcai1993@126.com Siqing Gan sqgan@csu.edu.cn Yaozhong Hu yaozhong@ualberta.ca School of Mathematics and Statistics, HNP-LAMA, Central South University, 410083, Hunan, China Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Canada

Abstract

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen–Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn–Hilliard equation setting for the first time.

keywords:

stochastic Cahn–Hilliard equation , weak convergence rate , backward Euler method

MSC:

60H35 , 60H15 , 65C30

^†^†volume: 00

\journalname\runauth

M. Cai, S.Gan and Y. Hu \jidanms \jnltitlelogo \CopyrightLine2011Published by Elsevier Ltd.

\dochead

1 Introduction

During the last decades, there have been overwhelming activities on the analysis of numerical stochastic partial differential equation (SPDE) under globally Lipschitz condition and a fast growing number of studies on Allen–Cahn type SPDE with non-globally Lipschitz coefficients. However, numerical analysis of stochastic Cahn–Hilliard equation, which is another prominent SPDE model with non-globally Lipschitz coefficients, is at its beginning and is far from being well understood. The Cahn–Hilliard equation is of fundamental importance in various applications to, such as, the complicated phase separation and coarsening phenomena in a melted alloy [6, 8], spinodal decomposition for binary mixture [7], the diffusive process of populations and oil film spreading over a solid surface [12]. Our motivating example arises from a simplified mesoscopic physical model for phase separation. The aim of this article is to investigate the weak convergence rate of a full discretization for stochastic Cahn–Hilliard equation driven by additive noise,

\left\{\begin{array}[]{lll}\mathrm{d}X(t)+A(AX(t)+F(X(t)))\,\mathrm{d}t=\mathrm{d}W(t),\quad t\in(0,T],\\ X(0)=X_{0}.\end{array}\right.

(1)

Let $\mathbf{D}$ be a bounded connected open domain of $\mathbb{R}^{d},d=1,2,3$ with smooth boundary and let $H:=L^{2}(\mathbf{D},\mathbb{R})$ be the Hilbert space with the usual scalar product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$ . The space $\dot{H}:=\{v\in H:\int_{\mathbf{D}}v\mathrm{d}x=0\}$ is a subspace of $H$ . We make the following assumptions.

Assumption 1.1.

$-A:\mathrm{dom}(A)\subseteq\dot{H}\to\dot{H}$ is the Neumann Laplacian defined by $-Au=\Delta u,u\in\mathrm{dom}(A)=\{u\in H^{2}(\mathbf{D})\cap\dot{H}:\frac{\partial u}{\partial n}=0\,\,\mathrm{on}\,\,\partial\mathbf{D}\}$ .

Assumption 1.2.

$F:L^{6}(\mathbf{D},\mathbb{R})\rightarrow H$ is the Nemytskii operator given by

F(v)(x)=f(v(x))=v^{3}(x)-v(x),\quad x\in\mathbf{D},v\in L^{6}(\mathbf{D},\mathbb{R}).

(2)

Assumption 1.3.

The noise process $\{W(t)\}_{t\in[0,T]}$ is an $\dot{H}$ -valued $Q$ -Wiener process with the covariance operator $Q$ satisfying

\big{\|}A^{\frac{1}{2}}Q^{\frac{1}{2}}\big{\|}_{\mathcal{L}_{2}}<\infty.

(3)

Assumption 1.4.

The initial value $X_{0}$ is deterministic and satisfies

|X_{0}|_{4}<\infty,

(4)

where the norm $|\cdot|_{4}$ is defined in (14) below.

We point out that Assumption 1.3 is the same as that in [21, 24, 28]. The assumption on the initial datum can be relaxed, but at the expense of having the constant $C$ depending on $t^{-1}$ , by exploiting the smoothing effect of the semigroup $E(t),t\in[0,T]$ and standard non–smooth data error estimates.

Based on the above assumptions and following the semigroup framework in [19], we see that the model (1) admits a unique mild solution

\displaystyle X(t)=E(t)X_{0}-\int_{0}^{t}E(t-s)APF(X(s))\,\mathrm{d}s+\int_{0}^{t}E(t-s)\,\mathrm{d}W(s),\quad t\in[0,T],

where $E(t)$ denotes the analytic semigroup generated by $-A^{2}$ . We refer the readers to [3, 10, 14, 15, 18, 20, 27] for the existence and uniqueness of the mild solution for such equation. Since the exact solutions are rarely known explicitly, numerical simulations are often used to investigate the behavior of the solutions. We choose the spatial semi-discretization by the spectral Galerkin method, i.e., projecting the equation to vector space $H_{N}$ , spanned by the first $N$ eigenvectors of $A$ . The approximated equation of (1) is in the form

\mathrm{d}X^{N}(t)+A(AX^{N}(t)+P_{N}F(X^{N}(t)))\mathrm{d}t=P_{N}\mathrm{d}W(t),\,t\in(0,T];\,\,X^{N}(0)=P_{N}X_{0}\,,

where $P_{N}$ is the spectral Galerkin projection operator onto the space $H_{N}$ . In the temporal direction, we apply the backward Euler method to the above equation. The fully discrete scheme is then given by

X_{t_{m}}^{M,N}-X_{t_{m-1}}^{M,N}+\tau A^{2}X_{t_{m}}^{M,N}+\tau P_{N}AF(X_{t_{m}}^{M,N})=P_{N}\Delta W_{m},\quad m\in\{1,2,\cdots,M\}.

Here $\Delta W_{m}:=W(t_{m})-W(t_{m-1})$ , $\tau=\tfrac{T}{M}$ is the time stepsize and $t_{m}=m\tau$ . The main result, concerning the weak convergence rates of the full discretization, reads

\big{|}\mathbb{E}[\Phi(X(T))]-\mathbb{E}[\Phi(X_{T}^{M,N})]\big{|}\leq C\big{(}\lambda_{N}^{-2}+\tau\big{)},\,\forall\,\Phi\in C_{b}^{2}(\dot{H},\mathbb{R}).

(5)

Here and throughout this article, $C$ denotes a generic positive constant that is independent of the discretization parameters $M,N$ and may change from line to line and $C_{b}^{2}(\dot{H},\mathbb{R})$ (or $C_{b}^{2}$ ) represents the space of not necessarily bounded mappings from $\dot{H}$ to $\mathbb{R}$ that have continuous and bounded Fréchet derivatives up to order 2. We split the weak error into two terms, both the spatial error and the temporal error, which are analyzed in Section 3 and Section 4, respectively. The result given by the above inequality (5) is on the weak rate of convergence. It is strictly greater than the strong ones (see Corollary 4.1) as expected. It is seen that the weak rate (which is 1.0 in time) is not twice as the strong one, contrary to the common belief. Indeed, the order is limited to $1$ since an implicit Euler scheme is used.

The idea for error analysis to obtain (5) goes as follows. At first, the weak error is separated into two parts, the spatial error and the temporal error,

\mathbb{E}\big{[}\Phi(X(T))\big{]}-\mathbb{E}\big{[}\Phi(X_{T}^{M,N})\big{]}\\ =\big{(}\mathbb{E}\big{[}\Phi(X(T))\big{]}-\mathbb{E}\big{[}\Phi(X^{N}(T))\big{]}\big{)}+\big{(}\mathbb{E}\big{[}\Phi(X^{N}(T))\big{]}-\mathbb{E}\big{[}\Phi(X_{T}^{M,N})\big{]}\big{)}.

(6)

To simplify the notation, we often write $\mathcal{O}_{t}$ for $\int_{0}^{t}E(t-r)\mathrm{d}W(r)$ and $\mathcal{O}^{N}_{t}=P_{N}\mathcal{O}_{t}$ . By introducing two processes $\bar{X}(t):=X(t)-\mathcal{O}_{t}$ and $\bar{X}^{N}(t):=X^{N}(t)-\mathcal{O}_{t}^{N}$ , we can further split the spatial error as

\begin{split}\mathbb{E}\big{[}\Phi(X(T))\big{]}-\mathbb{E}\big{[}\Phi(X^{N}(T))\big{]}&=\big{(}\mathbb{E}\big{[}\Phi(\bar{X}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}\big{)}\\ &\quad+\big{(}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{N})\big{]}\big{)}.\end{split}

(7)

To proceed, one relies on the Taylor expansion of the test function $\Phi$ . The key argument to estimate the first term on the right hand of (7) is to bound the error between $\bar{X}^{N}(T)$ and $\bar{X}(T)$ by that in a strong sense,

\displaystyle\begin{split}&\big{|}\mathbb{E}\big{[}\Phi(\bar{X}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}\big{|}\\ &\quad\leq C\Big{|}\mathbb{E}\int_{0}^{1}\Phi^{\prime}\big{(}X(T)+\lambda(\bar{X}^{N}(T)-\bar{X}(T))\big{)}\big{(}\bar{X}^{N}(T)-\bar{X}(T)\big{)}\mathrm{d}\lambda\Big{|}\\ &\quad\leq C\,\|\bar{X}(T)-P_{N}\bar{X}(T)\|_{L^{2}(\Omega,\dot{H})}+C\,\|P_{N}\bar{X}(T)-\bar{X}^{N}(T)\|_{L^{2}(\Omega,\dot{H})}.\end{split}

(8)

The error term $\|\bar{X}(T)-P_{N}\bar{X}(T)\|_{L^{2}(\Omega,\dot{H})}$ can be easily controlled owing to the higher spatial regularity of the stochastic process $\bar{X}(T)$ , in the absence of the stochastic convolution. The remaining term $e^{N}(t):=P_{N}\bar{X}(t)-\bar{X}^{N}(t)$ , satisfying the following random PDE,

\tfrac{\mathrm{d}}{\mathrm{d}t}e^{N}(t)+A^{2}e^{N}(t)+P_{N}A\big{[}F(X(t))-F(X^{N}(t))\big{]}=0,\quad e^{N}(0)=0,

(9)

must be carefully treated due to the presence of the unbounded operator $A$ before the nonlinear term $F$ . We use the monotonicity of the nonlinearity of $F$ and the regularities of $X(T)$ , $X^{N}(T)$ and $\mathcal{O}_{t}$ to derive $\Big{\|}\int_{0}^{T}|e^{N}(t)|_{1}^{2}\mathrm{d}t\Big{\|}_{L^{p}(\Omega,\mathbb{R})}\leq C\,\lambda_{N}^{-4}$ . Then, combining it with the mild solution of (9) leads to the desired weak orders (c.f. (69)-(73) below). Subsequently, we turn our attention to the second term in (7). Applying the Taylor expansion gives

\displaystyle\begin{split}&\big{|}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{N})\big{]}\big{|}\leq\Big{|}\mathbb{E}\big{[}\Phi^{{}^{\prime}}(X^{N}(T))(\mathcal{O}_{T}-\mathcal{O}_{T}^{N})\big{]}\Big{|}\\ &\quad+\Big{|}\mathbb{E}\Big{[}\int_{0}^{1}\Phi^{{}^{\prime\prime}}(X^{N}(T)+\lambda(\mathcal{O}_{T}-\mathcal{O}_{T}^{N}))(\mathcal{O}_{T}-\mathcal{O}_{T}^{N},\mathcal{O}_{T}-\mathcal{O}_{T}^{N})(1-\lambda)\mathrm{d}\lambda\Big{]}\Big{|}.\end{split}

(10)

The Malliavin integration by parts formula is the key ingredient to deal with the first term (c.f. (77)) and the second term can be easily estimated due to the boundedness of $\Phi^{\prime\prime}$ . It is now easy to explain why the weak rate of convergence is expected to be higher than strong convergence rate. As a byproduct of the weak error analysis, one can easily obtain the rate of the strong error,

\|X(t)-X^{N}(t)\|_{L^{2}(\Omega,\dot{H})}\leq\|\bar{X}(t)-\bar{X}^{N}(t)\|_{L^{2}(\Omega,\dot{H})}+\|\mathcal{O}_{t}-\mathcal{O}^{N}_{t}\|_{L^{2}(\Omega,\dot{H})}\leq C\lambda_{N}^{-\frac{3}{2}},

(11)

which is consistent with the results in [16, 27] and is lower than the weak convergence rate in (5), due to the presence of the second error. The basic idea to estimate temporal error is the same as that of the spatial error by essentially exploiting the discrete analogue of the arguments. The main point is that error must be uniform on the spatially discrete parameter $N$ .

Having sketched the central ideas of the weak error analysis, we review some relevant results in the literature. For the linearized stochastic Cahn–Hilliard equations, we refer to [11, 23, 25] for some strong convergence results of the finite element method. The authors in [21, 24] studied the strong convergence of the fully discrete finite element approximation for Cahn–Hilliard–Cook equation under spatial regular noise, but with no rates obtained. Later, the authors in [28] derives strong convergence rates of the mixed finite element method by using a priori strong moment bounds of the numerical approximations. For unbounded noise diffusion, the existence and regularity of solution have been investigated in [3, 14] and the absolute continuity has been studied in [2, 15]. Recently, the strong convergence rates of the spatial spectral Galerkin method and the temporal accelerated implicit Euler method for the stochastic Cahn–Hilliard equation were obtained in [16]. For weak convergence analysis in the non–globally Lipschitz setting, we are only aware of the papers [5, 9, 13, 17] concerning the stochastic Allen–Cahn equation. To the best of our knowledge, the weak convergence rates of a fully discrete method for the stochastic Cahn–Hilliard equation are absent in the literature. It is worthwhile to point out that issues from the presence of the unbounded operator in front of the nonlinear term make the weak error analysis much more challenging. To be more specific, in addition to the aforementioned difficulty in the weak analysis, the estimate of the Malliavin derivative for the spatial approximation process is also completely different, much more efforts are needed (c.f. Proposition 3.2). More recently, while this work was under review, we were aware of the preprint [4] posted in arXiv, concerning with numerical approximations of similar SPDEs, where Bréhier, Cui and Wang provide weak error estimates for another class of numerical schemes, whose weak order is twice as the strong order, for less regular problems. It is worth mentioning that the approach in the two works are substantially different. Different methods and different regularity regimes are dealt with.

The outline of the article is as follows. In the next section, we present some preliminaries, including the well-posedness and regularity of the mild solution and give a brief introduction to Malliavin calculus. Section 3 is devoted to the weak analysis of the spectral Galerkin method in space and Section 4 is concerned with the weak convergence rates of the backward Euler method in time.

2 Preliminaries

In this section, the mathematical setting, well-posedness and regularity of the model and a brief introduction to Malliavin calculus are given.

2.1 Mathematical setting

Given two real separable Hilbert spaces $(H,\langle\cdot,\cdot\rangle,\|\cdot\|)$ and $(U,\langle\cdot,\cdot\rangle_{U},\|\cdot\|_{U})$ , $\mathcal{L}(U,H)$ stands for the space of all bounded linear operators from $U$ to $H$ with the operator norm $\|\cdot\|_{\mathcal{L}(U,H)}$ and $\mathcal{L}_{2}(U,H)(\subset\mathcal{L}(U,H)$ ) denotes the space of all Hilbert-Schmidt operators from $U$ to $H$ . For simplicity, we write $\mathcal{L}(H)$ and $\mathcal{L}_{2}(H)$ (or $\mathcal{L}_{2}$ for short) instead of $\mathcal{L}(H,H)$ and $\mathcal{L}_{2}(H,H)$ , respectively. It is known, see e.g., [19], that $\mathcal{L}_{2}(U,H)$ is a Hilbert space equipped with the inner product and norm,

\displaystyle\left<T_{1},T_{2}\right>_{\mathcal{L}_{2}(U,H)}=\sum_{i\in\mathbb{N}^{+}}\left<T_{1}\phi_{i},T_{2}\phi_{i}\right>,\;\|T\|_{\mathcal{L}_{2}(U,H)}=\Big{(}\sum_{i\in\mathbb{N}^{+}}\|T\phi_{i}\|^{2}\Big{)}^{\frac{1}{2}},

(12)

where $\{\phi_{i}\}$ is an arbitrary orthonormal basis of $U$ . Let $H=L^{2}(\mathbf{D},\mathbb{R})$ and $\dot{H}=\{v\in H:\langle v,1\rangle=0\}$ . $V:=C(\mathbf{D},\mathbb{R})$ denotes the Banach space of all continuous functions with supremum norm $\|\cdot\|_{V}$ and $L^{r}(\mathbf{D},\mathbb{R}):=\{f:\mathbf{D}\to\mathbb{R},\int_{\mathbf{D}}|f(x)|^{r}dx<\infty\}$ . We define $P:H\rightarrow\dot{H}$ the generalized orthogonal projection by $Pv=v-|\mathbf{D}|^{-1}\int_{\mathbf{D}}v\mathrm{d}x$ , then $(I-P)v=|\mathbf{D}|^{-1}\int_{\mathbf{D}}v\mathrm{d}x$ is the average of $v$ over $\mathbf{D}$ .

It is easy to check that $A$ is a positive definite, self-adjoint and unbounded linear operator on $\dot{H}$ with compact inverse. For any $v\in H$ , we define $Av=APv$ , then there exists a family of eigenpairs $\{e_{j},\lambda_{j}\}_{j\in\mathbb{N}}$ such that

\displaystyle Ae_{j}=\lambda_{j}e_{j}\quad\text{and}\quad 0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{j}\leq\cdots\quad\text{with}\quad\lambda_{j}\rightarrow\infty,

(13)

where $e_{0}=|\mathbf{D}|^{-\frac{1}{2}}$ and $\{e_{j},j=1,\cdots\}$ forms an orthonormal basis of $\dot{H}$ . Straightforward applications of the spectral theory yield the fractional powers of $A$ on $\dot{H}$ , e.g., $A^{\alpha}v=\sum_{j=1}^{\infty}\lambda_{j}^{\alpha}\langle v,e_{j}\rangle e_{j}$ , $\alpha\in\mathbb{R}$ , $v\in\dot{H}$ . The space $\dot{H}^{\alpha}=\mathrm{dom}(A^{\frac{\alpha}{2}}),\alpha\in\mathbb{R}$ is a Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{\alpha}$ and the associated norm $|\cdot|_{\alpha}$ given by

\langle v,w\rangle_{\alpha}=\sum_{j=1}^{\infty}\lambda_{j}^{\alpha}\langle v,e_{j}\rangle\langle w,e_{j}\rangle,\quad|v|_{\alpha}=\|A^{\frac{\alpha}{2}}v\|=\Big{(}\sum_{j=1}^{\infty}\lambda_{j}^{\alpha}|\langle v,e_{j}\rangle|^{2}\Big{)}^{\frac{1}{2}}.

(14)

We also define $\|u\|_{\alpha}=\big{(}|u|_{\alpha}^{2}+|\langle u,e_{0}\rangle|^{2}\big{)}^{\frac{1}{2}}$ for $u\in H$ and the corresponding space is $H^{\alpha}:=\{u\in H:\|u\|_{\alpha}<\infty\}.$ A basic fact shows that for $\alpha=1,2$ , the norm $|\cdot|_{\alpha}$ on $\dot{H}^{\alpha}$ is equivalent to the standard Sobolev norm $\|\cdot\|_{H^{\alpha}(\mathbf{D})}$ (see [22, Theorems 2.9, 2.12] and [30, Theorem 16.9]). Since $H^{2}(\mathbf{D})$ is an algebra, there is a constant $C>0$ such that, for any $f,g\in\dot{H}^{2}$ ,

\|fg\|_{H^{2}(\mathbf{D})}\leq C\|f\|_{H^{2}(\mathbf{D})}\|g\|_{H^{2}(\mathbf{D})}\leq C|f|_{2}|g|_{2}.

(15)

We recall that the operator $-A^{2}$ generates an analytic semigroup $E(t)=e^{-tA^{2}}$ on $H$ due to (13) and we have

\displaystyle\begin{split}E(t)v=e^{-tA^{2}}v=Pe^{-tA^{2}}v+(I-P)v,\quad v\in H.\end{split}

(16)

With the aid of the eigenbasis of $A$ and Parseval’s identity, we have

$\displaystyle\\|A^{\mu}E(t)\\|_{\mathcal{L}(\dot{H})}$	$\displaystyle\leq Ct^{-\frac{\mu}{2}},\;t>0,\;\mu\geq 0,$	(17)
$\displaystyle\\|A^{-\nu}(I-E(t))\\|_{\mathcal{L}(\dot{H})}$	$\displaystyle\leq Ct^{\frac{\nu}{2}},\quad t\geq 0,\;\nu\in[0,2],$	(18)
$\displaystyle\int_{t_{1}}^{t_{2}}\\|A^{\varrho}E(s)v\\|^{2}\,\mathrm{d}s$	$\displaystyle\leq C\|t_{2}-t_{1}\|^{1-\varrho}\\|v\\|^{2},\;\forall v\in\dot{H},\varrho\in[0,1],$	(19)
$\displaystyle\Big{\\|}A^{2\rho}\int_{t_{1}}^{t_{2}}E(t_{2}-\sigma)v\,\mathrm{d}\sigma\Big{\\|}$	$\displaystyle\leq C\|t_{2}-t_{1}\|^{1-\rho}\\|v\\|,\;\forall v\in\dot{H},\rho\in[0,1].$	(20)

By Assumption 1.2, there exists a constant $C>0$ such that

	$\displaystyle-\langle F(u)-F(v),u-v\rangle$	$\displaystyle\leq\\|u-v\\|^{2},\quad u,v\in L^{6}(\mathbf{D},\mathbb{R}),$		(21)
	$\displaystyle\\|F(u)-F(v)\\|$	$\displaystyle\leq C(1+\\|u\\|_{V}^{2}+\\|v\\|_{V}^{2})\\|u-v\\|,\quad u,v\in V.$		(22)

2.2 Well-posedness and regularity results of the model

First at all, similar to [16, (2.5) $\&\,\,(2.7)$ ], we give the following lemma concerning the spatio-temporal regularity result of stochastic convolution $\mathcal{O}_{t}:=\int_{0}^{t}E(t-s)\mathrm{d}W(s)$ .

Lemma 2.1.

Suppose Assumptions 1.1 and 1.3 hold. Then for all $p\geq 1$ , the stochastic convolution $\mathcal{O}_{t}$ satisfies

\mathbb{E}\Big{[}\sup_{t\in[0,T]}|\mathcal{O}_{t}|_{V}^{p}\Big{]}+\sup_{t\in[0,T]}\mathbb{E}\Big{[}|\mathcal{O}_{t}|_{3}^{p}\Big{]}<\infty,

(23)

and for $\alpha\in[0,3]$ ,

\|\mathcal{O}_{t}-\mathcal{O}_{s}\|_{L^{p}(\Omega,\dot{H}^{\alpha})}\leq C|t-s|^{\text{min}\{\frac{1}{2},\frac{3-\alpha}{4}\}}.

(24)

The following theorem states the well-posedness and spatio-temporal regularity of the mild solution for stochastic Cahn-Hilliard equation (1), whose proofs can be found for example in [28, Theorem 2.1 & Theorem 2.2].

Theorem 2.1 (Well-posedness and regularity of the mild solution).

Under Assumptions 1.1-1.4, there is a unique mild solution of (1) satisfying

\displaystyle X(t)=E(t)X_{0}-\int_{0}^{t}\!\!E(t-s)AF(X(s))\,\mathrm{d}s+\int_{0}^{t}\!\!E(t-s)\mathrm{d}W(s),\,t\in[0,T].

(25)

Furthermore, for $p\geq 1$ ,

\displaystyle\sup_{t\in[0,T]}\|X(t)\|_{L^{p}(\Omega,\dot{H}^{3})}<\infty,

(26)

and for any $\alpha\in[0,3]$ ,

\displaystyle\|X(t)-X(s)\|_{L^{p}(\Omega,\dot{H}^{\alpha})}\leq C(t-s)^{\text{min}\{\frac{1}{2},\frac{3-\alpha}{4}\}},\,0\leq s<t\leq T.

(27)

Combining (26) and (15) yields the following result.

Corollary 2.1.

If Assumptions 1.1-1.4 are valid, then for all $p\geq 1$ ,

\sup_{t\in[0,T]}\|F(X(t))\|_{L^{p}(\Omega,\dot{H}^{2})}<\infty.

(28)

2.3 Introduction to Malliavin calculus

A brief introduction to Malliavin calculus is given in this subsection. For more details, one can consult the classical monograph [26]. Define a Hilbert space $U_{0}=Q^{\frac{1}{2}}(\dot{H})$ with inner product $\langle u,v\rangle_{U_{0}}=\langle Q^{-\frac{1}{2}}u,Q^{-\frac{1}{2}}v\rangle$ . Let $\mathcal{G}:L^{2}([0,T],U_{0})\rightarrow L^{2}(\Omega)$ be an isonormal Gaussian process. More precisely, for any deterministic mapping $\phi\in L^{2}([0,T],U_{0})$ , $\mathcal{G}(\phi)$ is centered Gaussian with the covariance structure

\mathbb{E}\big{[}\mathcal{G}(\phi_{1})\mathcal{G}(\phi_{2})\big{]}=\langle\phi_{1},\phi_{2}\rangle_{L^{2}([0,T],U_{0})},\,\,\phi_{1},\phi_{2}\in L^{2}([0,T],U_{0}).

(29)

For example (see e.g., [1]), we define the cylindrical $Q$ -Wiener process

W(t)u=\mathcal{G}(\chi_{[0,t]}\otimes u),\,u\in U_{0},\,t\in[0,T].

(30)

Given $u\in U_{0}$ , the process $W(t)u,t\in[0,T]$ , is a Brownian motion and we have

\mathbb{E}[W(t)uW(s)v]=\text{min}\{s,t\}\langle u,v\rangle_{U_{0}},\,\,u,v\in U_{0}.

(31)

Let $C_{p}^{\infty}(\mathbb{R}^{M},\mathbb{R})$ be the space of all $C^{\infty}$ -mappings with polynomial growth. We define the family of smooth $\dot{H}$ -valued cylindrical random variables as

\mathcal{S}(H)=\Big{\{}G=\sum_{i=1}^{N}g_{i}\big{(}\mathcal{G}(\phi_{1}),\ldots,\mathcal{G}(\phi_{M})\big{)}h_{i}:\phi_{1},\cdots,\phi_{M}\in L^{2}([0,T],U_{0}),\,g_{i}\in C_{p}^{\infty}(\mathbb{R}^{M},\mathbb{R}),\,h_{i}\in\dot{H},\,i\in\{1,\cdots,N\}\Big{\}}.

(32)

The Malliavin derivative of $G\in\mathcal{S}(\dot{H})$ is an element of $\mathcal{L}_{2}(U_{0},\dot{H})$ and given by

\mathcal{D}_{t}G:=\sum_{i=1}^{N}\sum_{j=1}^{M}\partial_{j}g_{i}\big{(}\mathcal{G}(\phi_{1}),\ldots,\mathcal{G}(\phi_{M})\big{)}h_{i}\otimes\phi_{j}(t),

(33)

where $h_{i}\otimes\phi_{j}(t)$ denotes the tensor product, that is, for $1\leq j\leq M$ and $1\leq i\leq N$ ,

\big{(}h_{i}\otimes\phi_{j}(t)\big{)}(u)=\langle\phi_{j}(t),u\rangle_{U_{0}}h_{i}\in\dot{H},\quad\forall\,\,u\in U_{0},~{}h_{i}\in\dot{H},~{}t\in[0,T].

(34)

If $G$ is $\mathcal{F}_{t}$ -measurable, then ${\mathcal{D}}_{s}G=0$ for $s>t$ . The derivative operator ${\mathcal{D}}$ is known to be closable and we define $\mathbb{D}^{1,2}(\dot{H})$ as the closure of $\mathcal{S}(\dot{H})$ with respect to the norm

\|G\|_{\mathbb{D}^{1,2}(\dot{H})}=\Bigl{(}\mathbb{E}\big{[}\|G\|^{2}\big{]}+\mathbb{E}\int_{0}^{T}\|{\mathcal{D}}_{t}G\|_{\mathcal{L}_{2}(U_{0},\dot{H})}^{2}\mathrm{d}t\Bigr{)}^{\frac{1}{2}}.

(35)

We are now ready to give the Malliavin integration by parts formula. For any $G\in\mathbb{D}^{1,2}(\dot{H})$ and an adapted process $\Psi\in L^{2}([0,T]\times\Omega,\mathcal{L}_{2}(U_{0},\dot{H}))$ ,

\mathbb{E}\left[\left\langle\int_{0}^{T}\Psi(t)\mathrm{d}W(t),G\right\rangle\right]=\mathbb{E}\left[\int_{0}^{T}\left\langle\Psi(t),\mathcal{D}_{t}G\right\rangle_{\mathcal{L}_{2}(U_{0},\dot{H})}\mathrm{d}t\right],

(36)

where the stochastic integral is Itô integral. To simplify the notation, we often write $\mathcal{L}_{2}^{0}$ instead of $\mathcal{L}_{2}(U_{0},\dot{H})$ . Next, we define ${\mathcal{D}}_{s}^{u}G=\langle{\mathcal{D}}_{s}G,u\rangle$ the derivative in the direction $u\in U_{0}$ . Then the Malliavin derivative acting on the Itô integral $\int_{0}^{t}\Psi(r)\mathrm{d}W(r)$ satisfies for all $u\in U_{0}$ ,

\mathcal{D}_{s}^{u}\int_{0}^{t}\Psi(r)\mathrm{d}W(r)=\int_{0}^{t}\mathcal{D}_{s}^{u}\Psi(r)\mathrm{d}W(r)+\Psi(s)u,\quad 0\leq s\leq t\leq T.

(37)

Given another separable Hilbert space $\mathcal{H}$ , if $\sigma\in C_{b}^{1}(\dot{H},\mathcal{H})$ and $G\in\mathbb{D}^{1,2}(\dot{H})$ , then $\sigma(G)\in\mathbb{D}^{1,2}(\mathcal{H})$ and the chain rule holds as ${\mathcal{D}}_{t}(\sigma(G))=\sigma^{\prime}(G)\mathcal{D}_{t}G$ .

3 Weak convergence rate of the spectral Galerkin method

This section is devoted to the weak error analysis of the spatial spectral Galerkin semi-discretization. In the beginning, we define a finite dimension space $H_{N}=\mathrm{span}\{e_{1},\cdots,e_{N}\}$ and the projection operator $P_{N}:\dot{H}^{\beta}\to H_{N}$ by $P_{N}x=\sum_{i=1}^{N}\langle x,e_{i}\rangle e_{i}$ for all $x\in\dot{H}^{\beta},\beta\in\mathbb{R}$ . As a result, $A$ commutes with $P_{N}$ and

\big{\|}\big{(}P_{N}-I\big{)}x\big{\|}\leq C\lambda_{N}^{-\frac{\beta}{2}}|x|_{\beta},\quad\forall~{}\beta\geq 0.

(38)

Applying the spectral Galerkin approximation to (1) results in the finite-dimensional stochastic differential equation, given by

\mathrm{d}X^{N}(t)+A^{2}X^{N}(t)\mathrm{d}t+AP_{N}F(X^{N}(t))\mathrm{d}t=P_{N}\mathrm{d}W(t),\,t\in(0,T];\,\,X^{N}(0)=P_{N}X_{0},

(39)

whose unique solution, in the mild form, is written as

X^{N}(t)=E(t)P_{N}X_{0}-\int_{0}^{t}E(t-s)AP_{N}F(X^{N}(s))\mathrm{d}s+\int_{0}^{t}E(t-s)P_{N}\mathrm{d}W(s).

(40)

Similarly to Lemma 2.1, the spatio-temporal regularity of the discrete stochastic convolution $\int_{0}^{t}E(t-s)P_{N}\mathrm{d}W(s)$ ( $\mathcal{O}_{t}^{N}$ for short) (see e.g., [27]) enjoys

\sup_{N\in\mathbb{N}}\sup_{t\in[0,T]}\mathbb{E}\Big{[}|\mathcal{O}_{t}^{N}|_{3}^{p}\Big{]}<\infty,\,\forall p\geq 1,

(41)

and for $\alpha\in[0,3]$ ,

\sup_{N\in\mathbb{N}}\big{\|}\mathcal{O}_{t}^{N}-\mathcal{O}_{s}^{N}\|_{L^{p}(\Omega,\dot{H}^{\alpha})}\leq C|t-s|^{\text{min}\{\frac{1}{2},\frac{3-\alpha}{4}\}},\,\forall p\geq 1.

(42)

It has to be noted that essential difficulties exist for analyzing a finite element method for the considered SPDE. Indeed, the orthogonal projection $P_{h}$ can not commute with operator $A$ , although $P_{N}$ commutes with $A$ . Moreover, compared with finite difference method, the spectral Galerkin method admits a simpler analysis, whose approximated solution is smooth and allows better control of the Lipschitz constant. The proof of the following regularity results is given in [27, Lemma 3.4].

Proposition 3.1 (Spatio-temporal regularity of spatial semi-discretization).

If Assumptions 1.1-1.4 are satisfied, then the mild solution of the spatial approximation process (40) admits for all $p\geq 1$ ,

\sup_{N\in\mathbb{N}}\mathbb{E}\big{[}\sup_{t\in[0,T]}\|X^{N}(t)\|_{L^{6}(\mathbf{D},\mathbb{R})}^{p}\big{]}<\infty.

(43)

Moreover, we have

\sup_{N\in\mathbb{N}}\sup_{t\in[0,T]}\|X^{N}(t)\|_{L^{p}(\Omega,\dot{H}^{3})}<\infty,

(44)

and for any $\alpha\in[0,3]$ ,

\sup_{N\in\mathbb{N}}\|X^{N}(t)-X^{N}(s)\|_{L^{p}(\Omega,\dot{H}^{\alpha})}\leq C(t-s)^{\text{min}\{\frac{1}{2},\frac{3-\alpha}{4}\}},\,0\leq s<t\leq T.

(45)

Combining (44) and (15) gives the next result.

Corollary 3.1.

Under Assumptions 1.1-1.4,

\sup_{t\in[0,T]}\|F(X^{N}(t))\|_{L^{p}(\Omega,\dot{H}^{2})}<\infty,\,\forall p\geq 1.

(46)

The next result shows that $X^{N}(t)$ is differentiable in Malliavin sense.

Proposition 3.2 (Boundedness of the Malliavin derivative).

Let Assumptions 1.1-1.4 hold. Then the Malliavin derivative of $X^{N}(t)$ satisfies

\displaystyle\mathbb{E}\big{[}\|\mathcal{D}_{s}X^{N}(t)\|_{\mathcal{L}_{2}(U_{0},\dot{H})}^{2}\big{]}<\infty,\quad 0\leq s\leq t\leq T.

(47)

Proof.

The existence of the Malliavin derivative $\mathcal{D}_{s}^{y}X^{N}(t)$ can be obtained by the standard argument such as the Picard iteration. Here, we will focus on the bound (47). Taking the Malliavin derivative on the equation (40) in the direction $y\in U_{0}$ and using the chain rule yield that for $0\leq s\leq t\leq T$ ,

\displaystyle\mathcal{D}_{s}^{y}X^{N}(t)=E(t-s)P_{N}y-\int_{s}^{t}E(t-r)AP_{N}F^{\prime}(X^{N}(r))\mathcal{D}_{s}^{y}X^{N}(r)\mathrm{d}r.

(48)

Following a standard strategy for the analysis of the Cahn–Hilliard equations, the proof of the upper bounds for $\mathcal{D}_{s}^{y}X^{N}(t)$ requires to exploit two energy estimates, in the $|\cdot|_{-1}$ and $|\cdot|$ norms. First, observe that for all $t\geq s$ , $\mathcal{D}_{s}^{y}X^{N}(t)$ is differentiable and satisfies

\displaystyle\frac{\mathrm{d}\mathcal{D}_{s}^{y}X^{N}(t)}{\mathrm{d}t}+A^{2}\mathcal{D}_{s}^{y}X^{N}(t)+AP_{N}F^{\prime}(X^{N}(t))\mathcal{D}_{s}^{y}X^{N}(t)=0.

(49)

Multiplying $A^{-1}\mathcal{D}_{s}^{y}X^{N}(t)$ on both sides of the above equation yields

\displaystyle\begin{split}\Big{\langle}\frac{\mathrm{d}\mathcal{D}_{s}^{y}X^{N}(t)}{\mathrm{d}t},A^{-1}\mathcal{D}_{s}^{y}X^{N}(t)\Big{\rangle}+\langle A^{2}\mathcal{D}_{s}^{y}X^{N}(t),A^{-1}\mathcal{D}_{s}^{y}X^{N}(t)\rangle+\langle AP_{N}F^{\prime}(X^{N}(t))\mathcal{D}_{s}^{y}X^{N}(t),A^{-1}\mathcal{D}_{s}^{y}X^{N}(t)\rangle=0.\end{split}

(50)

Next, integrating (50) over $[s,t]$ one obtains

\displaystyle\begin{split}|\mathcal{D}_{s}^{y}X^{N}(t)|_{-1}^{2}&=|y|_{-1}^{2}\!-\!2\!\!\int_{s}^{t}\!\!\!|\mathcal{D}_{s}^{y}X^{N}(r)|_{1}^{2}\mathrm{d}r\!-\!2\!\!\int_{s}^{t}\!\!\!\left\langle F^{\prime}(X^{N}(r))\mathcal{D}_{s}^{y}X^{N}(r),\mathcal{D}_{s}^{y}X^{N}(r)\right\rangle\mathrm{d}r\\ &=|y|_{-1}^{2}\!-\!2\int_{s}^{t}\!\!\!|\mathcal{D}_{s}^{y}X^{N}(r)|_{1}^{2}\mathrm{d}r+2\int_{s}^{t}\!\!\!\left\langle A^{\frac{1}{2}}\mathcal{D}_{s}^{y}X^{N}(r),A^{-\frac{1}{2}}\mathcal{D}_{s}^{y}X^{N}(r)\right\rangle\mathrm{d}r\\ &\quad-6\int_{s}^{t}\left\langle(X^{N}(r))^{2}\mathcal{D}_{s}^{y}X^{N}(r),\mathcal{D}_{s}^{y}X^{N}(r)\right\rangle\mathrm{d}r\\ &\leq|y|_{-1}^{2}-\int_{s}^{t}|\mathcal{D}_{s}^{y}X^{N}(r)|_{1}^{2}\mathrm{d}r+\int_{s}^{t}|\mathcal{D}_{s}^{y}X^{N}(r)|_{-1}^{2}\mathrm{d}r,\end{split}

(51)

where in the last step the elementary inequality $2ab\leq a^{2}+b^{2}$ was used. Hence, by Gronwall’s inequality we have

|\mathcal{D}_{s}^{y}X^{N}(t)|_{-1}^{2}\leq C|y|_{-1}^{2}.

(52)

Therefore, one has

\int_{s}^{t}|\mathcal{D}_{s}^{y}X^{N}(r)|_{1}^{2}\mathrm{d}r\leq C|y|_{-1}^{2}.

(53)

Next, we may multiply by $\mathcal{D}_{s}^{y}X^{N}(t)$ both sides of (49) to get

\displaystyle\begin{split}\left\langle\frac{\mathrm{d}\mathcal{D}_{s}^{y}X^{N}(t)}{\mathrm{d}t},\mathcal{D}_{s}^{y}X^{N}(t)\right\rangle+\left\langle A^{2}\mathcal{D}_{s}^{y}X^{N}(t),\mathcal{D}_{s}^{y}X^{N}(t)\right\rangle+\left\langle AP_{N}F^{\prime}(X^{N}(t))\mathcal{D}_{s}^{y}X^{N}(t),\mathcal{D}_{s}^{y}X^{N}(t)\right\rangle=0.\end{split}

(54)

Similarly, the energy estimate in the $|\cdot|$ norm is treated as follows:

\displaystyle\begin{split}|\mathcal{D}_{s}^{y}X^{N}(t)|^{2}&=|y|^{2}-2\int_{s}^{t}\|A\mathcal{D}_{s}^{y}X^{N}(r)\|^{2}\mathrm{d}r-2\int_{s}^{t}\langle F^{\prime}(X^{N}(r))\mathcal{D}_{s}^{y}X^{N}(r),A\mathcal{D}_{s}^{y}X^{N}(r)\rangle\mathrm{d}r\\ &\leq|y|^{2}-2\int_{s}^{t}\|A\mathcal{D}_{s}^{y}X^{N}(r)\|^{2}\mathrm{d}r+2\int_{s}^{t}\|A\mathcal{D}_{s}^{y}X^{N}(r)\|^{2}\mathrm{d}r\\ &\quad+\tfrac{1}{2}\int_{s}^{t}\|F^{\prime}(X^{N}(r))\mathcal{D}_{s}^{y}X^{N}(r)\|^{2}\mathrm{d}r\\ &\leq|y|^{2}+C\big{(}\sup_{r\in[s,t]}\|X^{N}(r)\|_{L^{6}}^{4}+1\big{)}\int_{s}^{t}|\mathcal{D}_{s}^{y}X^{N}(r)|_{1}^{2}\mathrm{d}r\\ &\leq|y|^{2}+C\,|y|_{-1}^{2}\big{(}\sup_{r\in[s,t]}\|X^{N}(r)\|_{L^{6}}^{4}+1\big{)}\\ &\leq|y|^{2}+C\,|y|^{2}\big{(}\sup_{r\in[s,t]}\|X^{N}(r)\|_{L^{6}}^{4}+1\big{)},\end{split}

(55)

where in the first inequality the elementary inequality $2ab\leq 2a^{2}+\tfrac{1}{2}b^{2}$ was used. What’s more, Hölder’s inequality $\|fg\|\leq C\|f\|_{L^{3}}\|g\|_{L^{6}}$ and Sobolev embedding inequality $\dot{H}^{\frac{d}{3}}\subset L^{6},d=1,2,3$ were used in the above second inequality. Choosing $y=Q^{\frac{1}{2}}e_{i},i=\{1,2,\cdots\}$ and taking expectation yield

\mathbb{E}\big{[}\|\mathcal{D}_{s}X^{N}(t)\|_{\mathcal{L}_{2}(U_{0},\dot{H})}^{2}\big{]}\leq C\,\|Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}^{2}\leq C\,\|A^{\frac{1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}^{2}<\infty,

(56)

where (3) and (43) were used. ∎

Remark 3.1.

The trace-class noise (i.e., $\|Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}<\infty$ ) is sufficient to obtain (56) and thus Proposition 3.2.

Let us now turn to some useful results on the nonlinear term $F$ .

Lemma 3.1.

Let $F$ be the Nemytskii operator defined in Assumption 1.2, then for $d=1,2,3$ ,

|F^{\prime}(x)y|_{1}\leq C\big{(}1+|x|_{2}^{2}\big{)}|y|_{1},\,x\in\dot{H}^{2},\,y\in\dot{H}^{1},

(57)

and

\displaystyle\begin{split}|F^{\prime}(\varsigma)\psi|_{-1}\leq C\big{(}1+|\varsigma|_{2}^{2}\big{)}|\psi|_{-1},\quad\forall\varsigma\in\dot{H}^{2},\psi\in\dot{H}.\end{split}

(58)

Proof.

The estimate (57) is an immediate consequence of [27, Lemma 3.2]. To see (58), with the aid of the self-adjointness of $A^{-\frac{1}{2}}$ and $F^{\prime}(\varsigma)$ , we have

\begin{split}\|A^{-\frac{1}{2}}F^{\prime}(\varsigma)\psi\|&=\sup_{\|\xi\|\leq 1}\langle A^{-\frac{1}{2}}F^{\prime}(\varsigma)\psi,\xi\rangle=\sup_{\|\xi\|\leq 1}\langle\psi,F^{\prime}(\varsigma)A^{-\frac{1}{2}}\xi\rangle=\sup_{\|\xi\|\leq 1}\langle A^{-\frac{1}{2}}\psi,A^{\frac{1}{2}}F^{\prime}(\varsigma)A^{-\frac{1}{2}}\xi\rangle\\ &\leq|\psi|_{-1}\sup_{\|\xi\|\leq 1}|F^{\prime}(\varsigma)A^{-\frac{1}{2}}\xi|_{1}\leq C\big{(}1+|\varsigma|_{2}^{2}\big{)}|\psi|_{-1}.\end{split}

(59)

The condition $\psi\in\dot{H}$ is used in the above first and second identities. This finishes the proof. ∎

Now, we are well prepared to carry out the weak error analysis of the spatial semi-discretization.

Theorem 3.1 (Weak convergence rate of the spatial approximation).

Let $X(T)$ and $X^{N}(T)$ , given by (25) and (40), be the solution of problems (1) and (39) respectively. Let Assumptions 1.1-1.4 hold. Then for $\Phi\in C_{b}^{2}$ , there exists a constant $C>0$ such that

\big{|}\mathbb{E}[\Phi(X(T))]-\mathbb{E}[\Phi(X^{N}(T))]\big{|}\leq C\,\lambda_{N}^{-2}.

(60)

Proof.

By introducing two processes $\bar{X}(t)=X(t)-\mathcal{O}_{t}$ and $\bar{X}^{N}(t)=X^{N}(t)-\mathcal{O}^{N}_{t}$ , we can separate the error $\mathbb{E}\big{[}\Phi(X(T))\big{]}-\mathbb{E}\big{[}\Phi(X^{N}(T))\big{]}$ into two terms as follows

\begin{split}\mathbb{E}\big{[}\Phi(X(T))\big{]}-\mathbb{E}\big{[}\Phi(X^{N}(T))\big{]}&=\Big{(}\mathbb{E}\big{[}\Phi(\bar{X}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}\Big{)}\\ &\quad+\Big{(}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{N})\big{]}\Big{)}\\ &=:I_{1}+I_{2}.\end{split}

(61)

To estimate $I_{1}$ , it suffices to consider the strong convergence between $\bar{X}(T)$ and $\bar{X}^{N}(T)$ . To be specific, by the Taylor expansion and triangle inequality we have

\begin{split}|I_{1}|&=\Big{|}\mathbb{E}\big{[}\Phi(\bar{X}(T)+\mathcal{O}_{T})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T})\big{]}\Big{|}\leq C\big{|}\mathbb{E}[\|\bar{X}(T)-\bar{X}^{N}(T)\|]\big{|}\\ &\leq C\|\bar{X}(T)-P_{N}\bar{X}(T)\|_{L^{2}(\Omega,\dot{H})}+C\|P_{N}\bar{X}(T)-\bar{X}^{N}(T)\|_{L^{2}(\Omega,\dot{H})}.\end{split}

(62)

To bound the first error term $\|\bar{X}(T)-P_{N}\bar{X}(T)\|_{L^{2}(\Omega,\dot{H})}$ , we need an estimate on $\bar{X}(t),t\in[0,T]$ , that is,

\begin{split}\|\bar{X}(t)&\|_{L^{p}(\Omega,\dot{H}^{4})}=\|A^{2}\bar{X}(t)\|_{L^{p}(\Omega,\dot{H})}\\ &\leq\|A^{2}E(t)X_{0}\|_{L^{p}(\Omega,\dot{H})}+\left\|\int_{0}^{t}A^{2}E(t-s)APF(X(t))\,\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\quad+\left\|\int_{0}^{t}A^{2}E(t-s)AP(F(X(t))-F(X(s)))\,\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\leq C\Big{(}|X_{0}|_{4}+\|F(X(t))\|_{L^{p}(\Omega,\dot{H}^{2})}+\int_{0}^{t}(t-s)^{-1}\|P(F(X(t))-F(X(s)))\|_{L^{p}(\Omega,\dot{H}^{2})}\mathrm{d}s\Big{)}\\ &\leq C\Big{(}|X_{0}|_{4}+\|F(X(t))\|_{L^{p}(\Omega,\dot{H}^{2})}\Big{)}\\ &\quad+C\Big{(}1+\sup_{r\in[0,t]}\|X(r)\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}\Big{)}\cdot\int_{0}^{t}(t-s)^{-1}\|X(t)-X(s)\|_{L^{2p}(\Omega,\dot{H}^{2})}\mathrm{d}s\\ &\leq C\Big{(}1+\int_{0}^{t}(t-s)^{-1}\cdot(t-s)^{\frac{1}{4}}\mathrm{d}s\Big{)}\\ &<\infty,\end{split}

(63)

where (17) and (20) were used in the above second inequality, (15) was used in the above third inequality, (26)-(28) were used in the above fourth inequality. As a result, by using (38), we get

\|\bar{X}(T)-P_{N}\bar{X}(T)\|_{L^{p}(\Omega,\dot{H})}=\|(I-P_{N})A^{-2}A^{2}\bar{X}(T)\|_{L^{p}(\Omega,\dot{H})}\leq C\lambda_{N}^{-2}.

(64)

In the next step, we consider the second term of (62) in the treatment of $I_{1}$ . For convenience, we denote $e^{N}(t)=P_{N}\bar{X}(t)-\bar{X}^{N}(t)$ , which satisfies

\tfrac{\mathrm{d}}{\mathrm{d}t}e^{N}(t)+A^{2}e^{N}(t)+AP_{N}\big{[}F(\bar{X}(t)+\mathcal{O}_{t})-F(\bar{X}^{N}(t)+\mathcal{O}^{N}_{t})\big{]}=0.

(65)

We multiply the above identity by $A^{-1}{e^{N}(t)}$ to get

\begin{split}\tfrac{1}{2}\tfrac{\mathrm{d}}{\mathrm{d}t}|e^{N}(t)|_{-1}^{2}+|e^{N}(t)|_{1}^{2}&=-\langle e^{N}(t),F(\bar{X}(t)+\mathcal{O}_{t})-F(P_{N}\bar{X}(t)+\mathcal{O}_{t})\rangle\\ &\quad-\langle{e^{N}(t)},F(P_{N}\bar{X}(t)+\mathcal{O}_{t})-F(\bar{X}^{N}(t)+\mathcal{O}_{t})\rangle\\ &\quad-\langle{e^{N}(t)},F(\bar{X}^{N}({t})+\mathcal{O}_{t})-F(\bar{X}^{N}(t)+\mathcal{O}^{N}_{t})\rangle\\ &\leq\tfrac{1}{2}\|e^{N}(t)\|^{2}+\tfrac{1}{2}\|F(\bar{X}(t)+\mathcal{O}_{t})-F(P_{N}\bar{X}(t)+\mathcal{O}_{t})\|^{2}\\ &\quad+\|e^{N}(t)\|^{2}+|e^{N}(t)|_{1}\cdot|F(\bar{X}^{N}(t)+\mathcal{O}_{t})-F(\bar{X}^{N}(t)+\mathcal{O}^{N}_{t})|_{-1}\\ &\leq\tfrac{3}{2}\|e^{N}(t)\|^{2}+\tfrac{1}{2}\|F(\bar{X}(t)+\mathcal{O}_{t})-F(P_{N}\bar{X}(t)+\mathcal{O}_{t})\|^{2}\\ &\quad+\tfrac{1}{4}|e^{N}(t)|_{1}^{2}+|F(\bar{X}^{N}(t)+\mathcal{O}_{t})-F(\bar{X}^{N}(t)+\mathcal{O}^{N}_{t}|_{-1}^{2}\\ &\leq\tfrac{3}{4}|e^{N}(t)|_{1}^{2}+\tfrac{9}{8}|e^{N}(t)|_{-1}^{2}+C\|\bar{X}(t)-P_{N}\bar{X}(t)\|^{2}(1+|\bar{X}(t)|_{2}^{4}+|\mathcal{O}_{t}|_{2}^{4})\\ &\quad+C|\mathcal{O}_{t}-\mathcal{O}^{N}_{t}|_{-1}^{2}(1+|\bar{X}^{N}({t})|_{2}^{4}+|\mathcal{O}_{t}|_{2}^{4}),\end{split}

(66)

where in the above first inequality we used Young’s inequality $ab\leq\frac{1}{2}a^{2}+\frac{1}{2}b^{2}$ , (21) and Cauchy-Schwartz inequality. Also, (22), Sobolev embedding inequality $\dot{H}^{2}\subset V$ , Young’s inequality $\frac{3}{2}ab\leq\frac{1}{2}a^{2}+\frac{9}{8}b^{2}$ , Taylor’s expansion and Lemma 3.1 were used in the above last inequality. By Gronwall’s inequality, we further deduce

\begin{split}|e^{N}(T)|_{-1}^{2}+\int_{0}^{T}|e^{N}(t)|_{1}^{2}\mathrm{d}t&\leq C\int_{0}^{T}\|\bar{X}(t)-P_{N}\bar{X}(t)\|^{2}(1+|\bar{X}(t)|_{2}^{4}+|\mathcal{O}_{t}|_{2}^{4})\mathrm{d}t\\ &\quad+C\int_{0}^{T}|\mathcal{O}_{t}-\mathcal{O}^{N}_{t}|_{-1}^{2}(1+|\bar{X}^{N}(t)|_{2}^{4}+|\mathcal{O}_{t}|_{2}^{4})\mathrm{d}t.\end{split}

Applying (23) and (38) gives

\|\mathcal{O}_{t}-\mathcal{O}^{N}_{t}\|_{L^{p}(\Omega,\dot{H}^{-1})}=\|(I-P_{N})A^{-2}A^{\frac{3}{2}}\mathcal{O}_{\color[rgb]{1,0,0}t}\|_{L^{p}(\Omega,\dot{H})}\leq C\lambda_{N}^{-2}~{}\|\mathcal{O}_{t}\|_{L^{p}(\Omega,\dot{H}^{3})}\leq C\lambda_{N}^{-2}.

(67)

With the aid of the regularity of $X(T)$ and $X^{N}(T)$ , (64), Hölder’s inequality and (67), one can find that

\begin{split}\Big{\|}\int_{0}^{T}|e^{N}(t)|_{1}^{2}\mathrm{d}t\Big{\|}_{L^{p}(\Omega,\mathbb{R})}&\leq C\int_{0}^{T}\|\bar{X}(t)-P_{N}\bar{X}(t)\|_{L^{4p}(\Omega,\dot{H})}^{2}\mathrm{d}t+C\int_{0}^{T}\|\mathcal{O}_{t}-\mathcal{O}^{N}_{t}\|_{L^{4p}(\Omega,\dot{H}^{-1})}^{2}\mathrm{d}t\\ &\leq C\lambda_{N}^{-4}.\end{split}

(68)

We are now ready to estimate

\begin{split}\|e^{N}(T)\|_{L^{p}(\Omega,\dot{H})}&=\Big{\|}P_{N}\Big{(}E(T)X_{0}-\int_{0}^{T}E(T-s)AF(X(s))\mathrm{d}s\Big{)}\\ &\quad-\Big{(}E(T)P_{N}X_{0}-\int_{0}^{T}E(T-s)AP_{N}F(X^{N}(s))\mathrm{d}s\Big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &=\left\|\int_{0}^{T}E(T-s)AP_{N}(F(X(s))-F(X^{N}(s)))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\leq\left\|\int_{0}^{T}E(T-s)AP_{N}(F(\bar{X}(s)+\mathcal{O}_{s})-F(P_{N}\bar{X}(s)+\mathcal{O}_{s}))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\quad+\left\|\int_{0}^{T}E(T-s)AP_{N}(F(P_{N}\bar{X}(s)+\mathcal{O}_{s})-F(\bar{X}^{N}(s)+\mathcal{O}_{s}))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\quad+\left\|\int_{0}^{T}E(T-s)AP_{N}(F(\bar{X}^{N}(s)+\mathcal{O}_{s})-F(\bar{X}^{N}(s)+\mathcal{O}^{N}_{s}))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &=:e^{N}_{1}(T)+e^{N}_{2}(T)+e^{N}_{3}(T).\end{split}

(69)

Again, by (17), (22), (23),(38), (63) and Sobolev embedding inequality $\dot{H}^{2}\subset V$ , we have

\begin{split}e^{N}_{1}(T)&=\left\|\int_{0}^{T}E(T-s)AP_{N}(F(\bar{X}(s)+\mathcal{O}_{s})-F(P_{N}\bar{X}(s)+\mathcal{O}_{s}))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\leq C\int_{0}^{T}(T-s)^{-\frac{1}{2}}\|\bar{X}(s)-P_{N}\bar{X}(s)\|_{L^{2p}(\Omega,\dot{H})}\mathrm{d}s\times\big{(}1+\sup_{s\in[0,T]}\|\bar{X}(s)\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}+\sup_{s\in[0,T]}\|\mathcal{O}_{s}\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}\big{)}\\ &\leq C\lambda_{N}^{-2}.\end{split}

(70)

From (17), (57) in Lemma 3.1, Hölder’s inequality, (68) and regularity of $X(t)$ and $X^{N}(t)$ , it follows that

\begin{split}e^{N}_{2}(T)&\leq C\left\|\int_{0}^{T}(T-s)^{{-\frac{1}{4}}}\big{|}F(P_{N}\bar{X}(s)+\mathcal{O}_{s})-F(\bar{X}^{N}(s)+\mathcal{O}_{s})\big{|}_{1}\mathrm{d}s\right\|_{L^{p}(\Omega,\mathbb{R})}\\ &\leq C\left\|\int_{0}^{T}(T-s)^{-\frac{1}{4}}|e^{N}(s)|_{1}\big{(}1+|\bar{X}(s)|_{2}^{2}+|\bar{X}^{N}(s)|_{2}^{2}+|\mathcal{O}_{s}|_{2}^{2}\big{)}\mathrm{d}s\right\|_{L^{p}(\Omega,\mathbb{R})}\\ &\leq C\left\|\int_{0}^{T}|e^{N}(s)|_{1}^{2}\mathrm{d}s\right\|_{L^{p}(\Omega,\mathbb{R})}^{\frac{1}{2}}\left(\int_{0}^{T}(T-s)^{-\frac{1}{2}}\mathrm{d}s\right)^{\frac{1}{2}}\\ &\leq C\lambda_{N}^{-2}.\end{split}

(71)

Similarly to the estimate of (70) with (58) and (67) instead, we obtain

\begin{split}e^{N}_{3}(T)&=\left\|\int_{0}^{T}E(T-s)A^{\frac{3}{2}}A^{-\frac{1}{2}}P_{N}(F(\bar{X}^{N}(s)+\mathcal{O}_{s})-F(\bar{X}^{N}(s)+\mathcal{O}^{N}_{s}))\mathrm{d}s\right\|_{L^{p}(\Omega,\dot{H})}\\ &\leq C\int_{0}^{T}(T-s)^{-\frac{3}{4}}\|\mathcal{O}_{s}-\mathcal{O}^{N}_{s}\|_{L^{2p}(\Omega,\dot{H}^{-1})}\mathrm{d}s\big{(}1+\sup_{s\in[0,T]}\|\bar{X}^{N}(s)\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}+\sup_{s\in[0,T]}\|\mathcal{O}_{s}\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}\big{)}\\ &\leq C\lambda_{N}^{-2}.\end{split}

(72)

Therefore, gathering estimates of $e^{N}_{1}(T)$ , $e^{N}_{2}(T)$ and $e^{N}_{3}(T)$ together yields

\|P_{N}\bar{X}(T)-\bar{X}^{N}(T)\|_{L^{2}(\Omega,\dot{H})}\leq C\lambda_{N}^{-2}.

(73)

Combining it with (64) yields

\|\bar{X}(T)-\bar{X}^{N}(T)\|_{L^{2}(\Omega,\dot{H})}\leq C\lambda_{N}^{-2},

(74)

and thus $|I_{1}|\leq C\lambda_{N}^{-2}$ . Next, we turn to the estimate of $|I_{2}|$ . Using Taylor’s expansion and the triangle inequality, we get

\displaystyle\begin{split}|I_{2}|&=\Big{|}\mathbb{E}\Big{[}\Phi^{{}^{\prime}}(X^{N}(T))(\mathcal{O}_{T}-\mathcal{O}_{T}^{N})+\int_{0}^{1}\Phi^{{}^{\prime\prime}}(X^{N}(T)+\lambda(\mathcal{O}_{T}-\mathcal{O}_{T}^{N}))(\mathcal{O}_{T}-\mathcal{O}_{T}^{N},\mathcal{O}_{T}-\mathcal{O}_{T}^{N})(1-\lambda)\mathrm{d}\lambda\Big{]}\Big{|}\\ &\leq\Big{|}\mathbb{E}\big{[}\Phi^{\prime}(X^{N}(T))(I-P_{N})\mathcal{O}_{T}\big{]}\Big{|}+C\,\mathbb{E}\big{[}\|\mathcal{O}_{T}-\mathcal{O}^{N}_{T}\|^{2}\big{]}.\end{split}

(75)

The second term can be easily bounded by utilizing (38) and the moment bound for $|\mathcal{O}_{T}|_{3}$ in Lemma 2.1, that is

\displaystyle\mathbb{E}\Big{[}\Big{\|}\mathcal{O}_{T}-\mathcal{O}^{N}_{T}\Big{\|}^{2}\Big{]}=\mathbb{E}\Big{[}\Big{\|}(I-P_{N})\mathcal{O}_{T}\Big{\|}^{2}\Big{]}\leq C\lambda_{N}^{-3}.

(76)

For the first term, (47) in Proposition 3.2, the Malliavin integration by parts formula (36), the chain rule of the Malliavin derivative, (17), (38) and (3) enable us to obtain

\displaystyle\begin{split}\Big{|}\mathbb{E}\big{[}\Phi^{\prime}(X^{N}(T))(I-P_{N})\mathcal{O}_{T}\big{]}\Big{|}&=\Big{|}\mathbb{E}\Big{[}\Big{\langle}\int_{0}^{T}(I-P_{N})E(T-s)\mathrm{d}W(s),\Phi^{{}^{\prime}}(X^{N}(T))\Big{\rangle}\Big{]}\Big{|}\\ &=\Big{|}\mathbb{E}\int_{0}^{T}\left<(I-P_{N})E(T-s),\mathcal{D}_{s}\Phi^{{}^{\prime}}(X^{N}(T))\right>_{\mathcal{L}_{2}^{0}}\mathrm{d}s\Big{|}\\ &\leq C\,\mathbb{E}\int_{0}^{T}\big{\|}(I-P_{N})E(T-s)\big{\|}_{\mathcal{L}_{2}^{0}}\|\Phi^{{}^{\prime\prime}}(X^{N}(T))\|_{\mathcal{L}}\|\mathcal{D}_{s}X^{N}(T)\|_{\mathcal{L}_{2}^{0}}\,\mathrm{d}s\\ &\leq C\,\int_{0}^{T}\|(I-P_{N})E(T-s)A^{-\frac{1}{2}}\|_{\mathcal{L}}\|A^{\frac{1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}\mathrm{d}s\\ &\leq C\,\lambda_{N}^{-2}\int_{0}^{T}(T-s)^{-\frac{3}{4}}\mathrm{d}s\leq C\,\lambda_{N}^{-2}.\end{split}

(77)

Hence, we obtain $|I_{2}|\leq C\lambda_{N}^{-2}$ . Gathering it with $|I_{1}|\leq C\lambda_{N}^{-2}$ then concludes the proof. ∎

4 Weak convergence rate of the backward Euler method

Based on the spatial spectral Galerkin approximation (39), this section concerns the weak error analysis of a backward Euler method in the temporal direction. We divide the interval $[0,T]$ into $M$ equidistant subintervals with the time step-size $\tau=\tfrac{T}{M}$ and denote the nodes $t_{m}=m\tau$ for $m\in\{0,1,\cdots,M\},\,M\in\mathbb{N}^{+}$ . Then, the fully discrete scheme reads

X_{t_{m}}^{M,N}-X_{t_{m-1}}^{M,N}+\tau A^{2}X_{t_{m}}^{M,N}+\tau P_{N}AF(X_{t_{m}}^{M,N})=P_{N}\Delta W_{m},\quad X_{0}^{M,N}=P_{N}X_{0},

(78)

where $\Delta W_{m}:=W(t_{m})-W(t_{m-1})$ for short. By introducing a family of operators $\{E_{\tau,N}^{m}\}_{m=1}^{M}$ : $E_{\tau,N}^{m}v=(I+\tau A^{2})^{-m}P_{N}v=\sum_{j=1}^{N}(1+\tau\lambda_{j}^{2})^{-m}\langle v,e_{j}\rangle e_{j}$ , $\forall\ v\in\dot{H}$ , we have

X_{t_{m}}^{M,N}=E_{\tau,N}^{m}X_{0}-\tau\sum_{j=1}^{m}E_{\tau,N}^{m-j+1}AF(X_{t_{j}}^{M,N})+\sum_{j=1}^{m}E_{\tau,N}^{m-j+1}\Delta W_{j}.

(79)

Thanks to [29, Theorem C.2], the implicit scheme (78) is well-defined. More details can be found in [27]. Following the proof of [21, (2.10)], it is easy to check that the operator $E_{\tau,N}^{m}$ satisfies

\|A^{\mu}E_{\tau,N}^{m}v\|\leq Ct_{m}^{-\frac{\mu}{2}}\|v\|,\quad\mu\in[0,2],\,v\in\dot{H},\,m\in\{1,2,\cdots,M\}

(80)

and there exists a constant $C$ such that for all $v\in\dot{H}$ ,

\Big{(}\tau\sum_{j=1}^{m}\|AE_{\tau,N}^{j}v\|^{2}\Big{)}^{\frac{1}{2}}\leq C\|v\|.

(81)

The regularity of the fully discrete approximation is derived in the following result.

Proposition 4.1.

Let Assumptions 1.1-1.4 be satisfied, then we have for all $p\geq 1$ ,

\sup_{N\in\mathbb{N}}\sup_{m\in\{0,1,\cdots,M\}}\|X^{M,N}_{t_{m}}\|_{L^{p}(\Omega,\dot{H}^{2})}<\infty.

(82)

Proof.

Firstly, by the proof in [27, Theorem 4.1], we have for $\eta\in(\frac{3}{2},2)$ and all $p\geq 1$ ,

\sup_{N\in\mathbb{N}}\sup_{m\in\{0,1,\cdots,M\}}\|X^{M,N}_{t_{m}}\|_{L^{p}(\Omega,\dot{H}^{\eta})}<\infty.

(83)

Next, from (80), the Burkholder-Davis-Gundy-type inequality, (81), (3), (4) and Sobolev embedding inequality $\dot{H}^{\eta}\subset V$ , it follows that

\begin{split}\sup_{N\in\mathbb{N}}&\sup_{m\in\{0,1,\cdots,M\}}\|X^{M,N}_{t_{m}}\|_{L^{p}(\Omega,\dot{H}^{2})}\\ &\leq\|X_{0}\|_{L^{p}(\Omega,\dot{H}^{2})}+\tau\sup_{N\in\mathbb{N}}\sup_{m\in\{0,1,\cdots,M\}}\sum_{j=1}^{m}t_{m-j+1}^{-\frac{3}{4}}\|PF(X_{t_{j}}^{M,N})\|_{L^{p}(\Omega,\dot{H}^{1})}\\ &\quad+\sup_{N\in\mathbb{N}}\sup_{m\in\{0,1,\cdots,M\}}\Big{(}\tau\sum_{j=1}^{m}\big{\|}A^{\frac{3}{2}}E_{\tau,N}^{m-j+1}Q^{\frac{1}{2}}\big{\|}_{\mathcal{L}_{2}}^{2}\Big{)}^{\frac{1}{2}}\\ &\leq C\Big{(}1+\|A^{\frac{1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}^{2}+\tau\sup_{N\in\mathbb{N}}\sup_{m\in\{0,1,\cdots,M\}}\sum_{j=1}^{m}t_{m-j+1}^{-\frac{3}{4}}\Big{\|}|X_{t_{j}}^{M,N}|_{1}~{}\|X_{t_{j}}^{M,N}\|_{V}^{2}\Big{\|}_{L^{p}(\Omega,\mathbb{R})}\Big{)}\\ &\leq C\Big{(}1+\sup_{m\in\{0,1,\cdots,M\}}\tau\sum_{j=1}^{m}t_{m-j+1}^{-\frac{3}{4}}\sup_{N\in\mathbb{N}}\sup_{j\in\{0,1,\cdots,M\}}\|X_{t_{j}}^{M,N}\|_{L^{3p}(\Omega,\dot{H}^{\eta})}\Big{)}\\ &<\infty.\end{split}

(84)

This completes the proof. ∎

Before presenting the main theorem, we introduce the notation $\lfloor s\rfloor:=\max\{0,\tau,\cdots,m\tau,\cdots\}\cap[0,s]$ , $\lceil s\rceil:=\min\{0,\tau,\cdots,m\tau,\cdots\}\cap[s,T]$ and $[s]:=\frac{\lfloor s\rfloor}{\tau}$ . The fully discrete approximation operator is then defined by

\Psi_{\tau}^{M,N}(t):=E(t)P_{N}-E_{\tau,N}^{k},\quad t\in[t_{k-1},t_{k}),\quad k\in\{1,2,\cdots,M\}.

(85)

The following lemma of the fully discrete approximation operator plays a pivotal role in the weak convergence analysis.

Lemma 4.1.

Under Assumption 1.1, we have the following statements.

(i)

Let $\rho\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

$\|\Psi_{\tau}^{M,N}(t)u\|\leq C\,t^{-\frac{\rho}{4}}\,|u|_{-\rho},\,u\in\dot{H}^{-\rho}.$ (86)
(ii)

Let $\beta\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

$\|\Psi_{\tau}^{M,N}(t)u\|\leq C\,\tau^{\frac{\beta}{4}}\,|u|_{\beta},\,u\in\dot{H}^{\beta}.$ (87)
(iii)

Let $\alpha\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

$\|\Psi_{\tau}^{M,N}(t)u\|\leq C\,\tau^{\frac{4-\alpha}{4}}\,t^{-1}\,|u|_{-\alpha},\,u\in\dot{H}^{-\alpha}.$ (88)
(iv)

Let $\mu\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

$\|\Psi_{\tau}^{M,N}(t)u\|\leq C\,\tau\cdot t^{-\frac{4-\mu}{4}}\,|u|_{\mu},\,u\in\dot{H}^{\mu}.$ (89)

(v)

Let $\nu\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

\Big{(}\int_{0}^{t}\|\Psi_{\tau}^{M,N}(s)u\|^{2}\mathrm{d}s\Big{)}^{\frac{1}{2}}\leq C\,\tau^{\frac{\nu}{4}}|u|_{\nu-2},\,u\in\dot{H}^{\nu-2}.

(90)

(vi)

Let $\delta\in[0,4]$ , there exists a constant $C$ such that for $t>0$ ,

\Big{\|}\int_{0}^{t}\Psi_{\tau}^{M,N}(s)u\mathrm{d}s\Big{\|}\leq C\,\tau^{\frac{4-\delta}{4}}|u|_{-\delta},\,u\in\dot{H}^{-\delta}.

(91)

Proof.

Elementary fact in [27, Lemma 5.3] yields (i), (ii), (iii), (v) and (vi). We then use the standard interpolation argument to prove (iv). For $\mu=0$ , it is a consequence of (iii) with $\alpha=0$ and for $\mu=4$ , it is a consequence of (ii) with $\beta=4$ . ∎

For clarity of exposition, we denote $\mathcal{O}_{T}^{M,N}:=\sum_{j=1}^{M}E_{\tau,N}^{M-j+1}\Delta W_{j}=\int_{0}^{T}E_{\tau,N}^{M-[s]}\mathrm{d}W(s).$ The next lemma gives the estimate between $\mathcal{O}_{t_{m}}^{N}$ and $\mathcal{O}_{t_{m}}^{M,N}$ .

Lemma 4.2.

Under Assumptions 1.1 and 1.3, we have for $p\geq 1$ ,

\sup_{m\in\{1,2,\cdots,M\}}\big{\|}\mathcal{O}_{t_{m}}^{M,N}-\mathcal{O}_{t_{m}}^{N}\big{\|}_{L^{p}(\Omega,\dot{H}^{-\beta})}\leq C\,\tau^{\frac{3+\beta}{4}},\,\beta\in[-3,1].

(92)

Proof.

The Burkholder-Davis-Gundy inequality and (v) in Lemma 4.1 with $\nu=3+\beta$ yield

\begin{split}\big{\|}\mathcal{O}_{t_{m}}^{M,N}-\mathcal{O}_{t_{m}}^{N}\big{\|}_{L^{p}(\Omega,\dot{H}^{-\beta})}&\leq C~{}\Big{(}\int_{0}^{t_{m}}\|\Psi_{\tau}^{M,N}(t_{m}-s)A^{-\frac{\beta}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}^{2}\mathrm{d}s\Big{)}^{\frac{1}{2}}\\ &\leq C\,\tau^{\frac{3+\beta}{4}}\|A^{\frac{1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}\leq C\,\tau^{\frac{3+\beta}{4}}.\end{split}

(93)

This finishes the proof. ∎

The following theorem shows the weak convergence rate of the temporal semi-discretization.

Theorem 4.1 (Weak convergence rate of the temporal approximation).

Suppose Assumptions 1.1-1.4 are satisfied. Let $X^{N}(T)$ and $X_{T}^{M,N}$ be given by (40) and (79), respectively. Then, we have for $\Phi\in C_{b}^{2}$ ,

\big{|}\mathbb{E}[\Phi(X^{N}(T))]-\mathbb{E}[\Phi(X_{T}^{M,N})]\big{|}\leq C\,\tau.

(94)

Proof.

At first, we define $\bar{X}_{T}^{M,N}={X}_{T}^{M,N}-\mathcal{O}_{T}^{M,N}$ and separate the above error into

\displaystyle\begin{split}\mathbb{E}\Big{[}\Phi(X^{N}(T))\Big{]}-\mathbb{E}\Big{[}\Phi(X_{T}^{M,N})\Big{]}&=\Big{(}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{M,N})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}_{T}^{M,N}+\mathcal{O}_{T}^{M,N})\big{]}\Big{)}\\ &\quad+\Big{(}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{N})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{M,N})\big{]}\Big{)}\\ &=:K_{1}+K_{2}.\end{split}

(95)

To estimate $K_{1}$ , it suffices to bound $\big{\|}\bar{X}^{N}(T)-\bar{X}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}$ . To this end, we introduce an auxiliary process $Y_{t_{m}}^{M,N}$ by

Y_{t_{m}}^{M,N}=E_{\tau,N}^{m}X_{0}-\tau\sum_{j=1}^{m}E_{\tau,N}^{m-j+1}AF(X^{N}(t_{j}))+\sum_{j=1}^{m}E_{\tau,N}^{m-j+1}\Delta W_{j}

(96)

and define $\bar{Y}_{t_{m}}^{M,N}=Y_{t_{m}}^{M,N}-\mathcal{O}_{t_{m}}^{M,N}$ . Note that the application of an appropriate auxiliary process was used in [16, 28] to deduce the strong convergence rates for the numerical approximations of similar SPDEs. Owning to (3), (4), (46), (80), (81) and discrete Burkholder-Davis-Gundy-type inequality, one can easily derive that for any $m\in\{0,1,,2,\cdots,M\}$ ,

\|Y_{t_{m}}^{M,N}\|_{L^{p}(\Omega,\dot{H}^{3})}<\infty.

(97)

Subsequently, by the triangle inequality, we have

\big{\|}\bar{X}^{N}(T)-\bar{X}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}\leq\big{\|}\bar{X}^{N}(T)-\bar{Y}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}+\big{\|}\bar{Y}_{T}^{M,N}-\bar{X}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}.

(98)

The error term $\big{\|}\bar{X}^{N}(T)-\bar{Y}_{T}^{M,N}\big{\|}_{L^{p}(\Omega,\dot{H})}$ can be further divided into three terms

\begin{split}\big{\|}&\bar{X}^{N}(T)-\bar{Y}_{T}^{M,N}\big{\|}_{L^{p}(\Omega,\dot{H})}=\Big{\|}(E(T)P_{N}-E_{\tau,N}^{M})X_{0}\\ &\quad-\Big{(}\int_{0}^{T}E(T-s)P_{N}AF(X^{N}(s))\mathrm{d}s-\tau\sum_{j=1}^{M}E_{\tau,N}^{M-j+1}AF(X^{N}(t_{j}))\Big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\leq\big{\|}(E(T)P_{N}-E_{\tau,N}^{M})X_{0}\big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\Big{\|}\int_{0}^{T}\big{(}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{)}AF(X^{N}(s))\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\big{(}F(X^{N}(s))-F(X^{N}(\lceil s\rceil))\big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &=:K_{11}+K_{12}+K_{13}.\end{split}

(99)

By (ii) of Lemma 4.1 with $\beta=4$ and Assumption 1.4, we deduce

K_{11}\leq C\,\tau|X_{0}|_{4}\leq C\,\tau.

(100)

Concerning the term $K_{12}$ , by use of (vi) and (iv) in Lemma 4.1, (15), (44), (45), (46), we obtain

\begin{split}K_{12}&\leq\Big{\|}\int_{0}^{T}\big{(}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{)}APF(X^{N}(T))\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\int_{0}^{T}\Big{\|}\big{(}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{)}AP(F(X^{N}(s))-F(X^{N}(T)))\Big{\|}_{L^{p}(\Omega,\dot{H})}\mathrm{d}s\\ &\leq C\,\tau\,\|F(X^{N}(T))\|_{L^{p}(\Omega,\dot{H}^{2})}+C\,\tau\,\int_{0}^{T}(T-s)^{-1}\|P(F(X^{N}(s))-F(X^{N}(T)))\|_{L^{p}(\Omega,\dot{H}^{2})}\mathrm{d}s\\ &\leq C\,\tau+C\,\tau\,\Big{(}1+\sup_{r\in[0,t]}\|X^{N}(r)\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}\Big{)}\cdot\int_{0}^{T}(T-s)^{-1}\|X^{N}(s)-X^{N}(T)\|_{L^{2p}(\Omega,\dot{H}^{2})}\mathrm{d}s\\ &\leq C\,\tau+C\,\tau\,\int_{0}^{T}(T-s)^{-1}(T-s)^{\frac{1}{4}}\mathrm{d}s\\ &\leq C\,\tau.\end{split}

(101)

To handle $K_{13}$ , we decompose it into four terms with the aid of the Taylor expansion and the mild form of $X^{N}(t)$ ,

\begin{split}K_{13}&\leq\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\Big{(}F^{\prime}(X^{N}(s))(E(\lceil s\rceil-s)-I)X^{N}(s)\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\Big{(}F^{\prime}(X^{N}(s))\int_{s}^{\lceil s\rceil}\!\!\!E(\lceil s\rceil-r)P_{N}AF(X^{N}(r))\mathrm{d}r\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\Big{(}F^{\prime}(X^{N}(s))\int_{s}^{\lceil s\rceil}E(\lceil s\rceil-r)P_{N}\mathrm{d}W(r)\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\Big{(}\int_{0}^{1}F^{\prime\prime}\big{(}X^{N}(s)+\lambda(X^{N}(\lceil s\rceil)-X^{N}(s))\big{)}\\ &\qquad\qquad\qquad\big{(}X^{N}(\lceil s\rceil)-X^{N}(s),X^{N}(\lceil s\rceil)-X^{N}(s)\big{)}(1-\lambda)\mathrm{d}\lambda\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &=:K_{131}+K_{132}+K_{133}+K_{134}.\end{split}

(102)

The smoothness of $E_{\tau,N}^{m}$ in (80), (18), (58) and the regularity of $X^{N}(t)$ lead to

\begin{split}K_{131}&=\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A^{\frac{3}{2}}A^{-\frac{1}{2}}\Big{(}F^{\prime}(X^{N}(s))(E(\lceil s\rceil-s)-I)X^{N}(s)\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\leq C\!\int_{0}^{T}\!\!\!(T-\lfloor s\rfloor)^{-\frac{3}{4}}\left\|\big{(}1+|X^{N}(s)|_{2}^{2}\big{)}\big{|}(E(\lceil s\rceil-s)-I)X^{N}(s)\big{|}_{-1}\right\|_{L^{p}(\Omega,\mathbb{R})}\!\!\!\mathrm{d}s\\ &\leq C\,\tau.\end{split}

(103)

Following similar approach as above and utilizing (46) yield

\begin{split}K_{132}&\leq C\int_{0}^{T}(T-\lfloor s\rfloor)^{-\frac{3}{4}}\Big{\|}\big{(}1+|X^{N}(s)|_{2}^{2}\big{)}\int_{s}^{\lceil s\rceil}\big{|}E(\lceil s\rceil-r)P_{N}AF(X^{N}(r))\big{|}_{-1}\mathrm{d}r\Big{\|}_{L^{p}(\Omega,\mathbb{R})}\mathrm{d}s\\ &\leq C\,\tau\int_{0}^{T}(T-\lfloor s\rfloor)^{-\frac{3}{4}}\mathrm{d}s\sup_{r\in[0,T]}\|F(X^{N}(r))\|_{L^{2p}(\Omega,\dot{H}^{2})}\\ &\leq C\,\tau.\end{split}

(104)

From stochastic Fubini theorem, the Burkholder-Davis-Gundy-type inequality and Hölder’s inequality, it follows that

\begin{split}K_{133}&=\Big{\|}\sum_{j=1}^{M}\int_{t_{j-1}}^{t_{j}}\!\int_{t_{j-1}}^{t_{j}}\!\!\chi_{[s,t_{j})}(r)E_{\tau,N}^{M-[s]}AF^{\prime}(X^{N}(s))E(\lceil s\rceil-r)P_{N}\mathrm{d}W(r)\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &=\Big{\|}\sum_{j=1}^{M}\int_{t_{j-1}}^{t_{j}}\!\int_{t_{j-1}}^{t_{j}}\!\!\chi_{[s,t_{j})}(r)E_{\tau,N}^{M-[s]}AF^{\prime}(X^{N}(s))E(\lceil s\rceil-r)P_{N}\mathrm{d}s\mathrm{d}W(r)\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\leq\left(\sum_{j=1}^{M}\int_{t_{j-1}}^{t_{j}}\Big{\|}\int_{t_{j-1}}^{t_{j}}\chi_{[s,t_{j})}(r)E_{\tau,N}^{M-[s]}AF^{\prime}(X^{N}(s))E(\lceil s\rceil-r)\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\mathcal{L}_{2}^{0})}^{2}\,\mathrm{d}r\right)^{\frac{1}{2}}\\ &\leq C\,\tau^{\frac{1}{2}}\left(\sum_{j=1}^{M}\int_{t_{j-1}}^{t_{j}}\!\int_{t_{j-1}}^{t_{j}}\!\!\Big{\|}E_{\tau,N}^{M-[s]}A^{\frac{1}{2}}A^{\frac{1}{2}}F^{\prime}(X^{N}(s))E(\lceil s\rceil-r)\Big{\|}_{L^{p}(\Omega,\mathcal{L}_{2}^{0})}^{2}\mathrm{d}s\,\mathrm{d}r\right)^{\frac{1}{2}}\\ &\leq C\,\tau^{\frac{1}{2}}\left(\sum_{j=1}^{M}\int_{t_{j-1}}^{t_{j}}\!\int_{t_{j-1}}^{t_{j}}\!\!(T-\lfloor s\rfloor)^{-\frac{1}{2}}\big{(}1+\|X^{N}(s)\|_{L^{2p}(\Omega,\dot{H}^{2})}^{4}\big{)}\big{\|}A^{\frac{1}{2}}Q^{\frac{1}{2}}\big{\|}_{\mathcal{L}_{2}}^{2}\mathrm{d}s\,\mathrm{d}r\right)^{\frac{1}{2}}\\ &\leq C\,\tau,\end{split}

(105)

where $\chi_{[0,t]}$ denotes the indicate function on $[0,t]$ . Additionally, (57), (80) and the stability of $E(\lceil s\rceil-r)$ were used in the third inequality and in the last inequality we used (3) and (44). Owing to Hölder’s inequality, the Sobolev embedding inequality $\dot{H}^{\delta}\subset V$ for $\frac{3}{2}<\delta<2$ and Proposition 3.1, we obtain

\begin{split}K_{134}&=\Big{\|}\int_{0}^{T}E_{\tau,N}^{M-[s]}A\Big{(}\int_{0}^{1}F^{\prime\prime}\big{(}X^{N}(s)+\lambda(X^{N}(\lceil s\rceil)-X^{N}(s))\big{)}\\ &\qquad\qquad\qquad\big{(}X^{N}(\lceil s\rceil)-X^{N}(s),X^{N}(\lceil s\rceil)-X^{N}(s)\big{)}(1-\lambda)\mathrm{d}\lambda\Big{)}\mathrm{d}s\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\leq C\int_{0}^{T}(T-\lfloor s\rfloor)^{-\frac{2+\delta}{4}}\|X^{N}(\lceil s\rceil)-X^{N}(s)\|_{L^{4p}(\Omega,\dot{H})}^{2}\big{(}1+\sup_{s\in[0,T]}\|X^{N}(s)\|_{L^{2p}(\Omega,V)}\big{)}\mathrm{d}s\\ &\leq C\,\tau.\end{split}

(106)

Gathering the above estimates and (99) together gives

\big{\|}\bar{X}^{N}(T)-\bar{Y}_{T}^{M,N}\big{\|}_{L^{p}(\Omega,\dot{H})}\leq C\,\tau.

(107)

Finally, we turn to remaining error term $\big{\|}\bar{Y}_{T}^{M,N}-\bar{X}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}$ in (98). Denoting $e_{t}^{M,N}=\bar{Y}_{t}^{M,N}-\bar{X}_{t}^{M,N}$ , we have

e_{t_{m}}^{M,N}-e_{t_{m-1}}^{M,N}+\tau A^{2}e_{t_{m}}^{M,N}=\tau P_{N}AF(X_{t_{m}}^{M,N})-\tau P_{N}AF(X^{N}(t_{m})).

(108)

Multiplying both sides by $A^{-1}e_{t_{m}}^{M,N}$ shows

\begin{split}\langle e_{t_{m}}^{M,N}-&e_{t_{m-1}}^{M,N},A^{-1}e_{t_{m}}^{M,N}\rangle+\tau\langle A^{2}e_{t_{m}}^{M,N},A^{-1}e_{t_{m}}^{M,N}\rangle\\ &=\tau\langle F(\bar{X}_{t_{m}}^{M,N}+\mathcal{O}_{t_{m}}^{M,N})-F(\bar{Y}_{t_{m}}^{M,N}+\mathcal{O}_{t_{m}}^{M,N}),e_{t_{m}}^{M,N}\rangle\\ &\quad+\tau\langle F(\bar{Y}_{t_{m}}^{M,N}+\mathcal{O}_{t_{m}}^{M,N})-F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{M,N}),e_{t_{m}}^{M,N}\rangle\\ &\quad+\tau\langle F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{M,N})-F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{N}),e_{t_{m}}^{M,N}\rangle.\end{split}

(109)

Following similar approach in (66) and using the inequality $\langle e_{t_{m}}^{M,N}-e_{t_{m-1}}^{M,N},A^{-1}e_{t_{m}}^{M,N}\rangle\geq\tfrac{1}{2}\big{(}|e_{t_{m}}^{M,N}|_{-1}^{2}-|e_{t_{m-1}}^{M,N}|_{-1}^{2}\big{)}$ and the monotonicity of $F$ in (21), we further obtain

\begin{split}\tfrac{1}{2}\big{(}|e_{t_{m}}^{M,N}|_{-1}^{2}-|e_{t_{m-1}}^{M,N}|_{-1}^{2}\big{)}+\tau|e_{t_{m}}^{M,N}|_{1}^{2}&\leq\tfrac{3}{4}\tau|e_{t_{m}}^{M,N}|_{1}^{2}+\tfrac{9}{8}\tau|e_{t_{m}}^{M,N}|_{-1}^{2}\\ &\quad+C\,\tau\big{\|}F(\bar{Y}_{t_{m}}^{M,N}+\mathcal{O}_{t_{m}}^{M,N})-F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{M,N})\big{\|}^{2}\\ &\quad+C\,\tau\big{|}F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{M,N})-F(\bar{X}^{N}(t_{m})+\mathcal{O}_{t_{m}}^{N})\big{|}_{-1}^{2}.\end{split}

(110)

By iteration in $m$ and Gronwall’s inequality, we obtain

\begin{split}|e_{T}^{M,N}|_{-1}^{2}+\tau\sum_{j=1}^{M}|e_{t_{j}}^{M,N}|_{1}^{2}&\leq C\,\tau\sum_{j=1}^{M}\Big{(}\big{\|}F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})-F(\bar{X}^{N}(t_{j})+\mathcal{O}_{t_{j}}^{M,N})\big{\|}^{2}\Big{)}\\ &\quad+C\,\tau\sum_{j=1}^{M}\Big{(}\big{|}F(\bar{X}^{N}(t_{j})+\mathcal{O}_{t_{j}}^{M,N})-F(\bar{X}^{N}(t_{j})+\mathcal{O}_{t_{j}}^{N})\big{|}_{-1}^{2}\Big{)}.\end{split}

(111)

It is worth mentioning that (107) also holds for arbitrary $t_{j},j\in\{1,\cdots,M\}$ by repeating the same argument from (99) to (107). Then, employing (22), (44), (58), (97) and Lemma 4.2 results in

\begin{split}\Big{\|}\tau\sum_{j=1}^{M}|e_{t_{j}}^{M,N}|_{1}^{2}\Big{\|}_{L^{p}(\Omega,\mathbb{R})}&\leq C~{}\tau\sum_{j=1}^{M}\!\Big{\|}\big{\|}\bar{Y}_{t_{j}}^{M,N}\!-\!\bar{X}^{N}(t_{j})\big{\|}^{2}\big{(}1\!+\!\|\bar{Y}_{t_{j}}^{M,N}\|_{V}^{4}+\|\bar{X}^{N}(t_{j})\|_{V}^{4}+\|\mathcal{O}_{t_{j}}^{M,N}\|_{V}^{4}\big{)}\Big{\|}_{L^{p}(\Omega,\mathbb{R})}\\ &\quad+C~{}\tau\sum_{j=1}^{M}\!\Big{\|}\big{|}\mathcal{O}_{t_{j}}^{M,N}-\mathcal{O}_{t_{j}}^{N}\big{|}_{-1}^{2}\big{(}1\!+\!|\bar{X}^{N}(t_{j})|_{2}^{4}+|\mathcal{O}_{t_{j}}^{M,N}|_{2}^{4}+|\mathcal{O}_{t_{j}}^{N}|_{2}^{4}\big{)}\Big{\|}_{L^{p}(\Omega,\mathbb{R})}\\ &\leq C\,\tau^{2}.\end{split}

(112)

Furthermore, since

e_{T}^{M,N}=\bar{Y}_{T}^{M,N}-\bar{X}_{T}^{M,N}=\tau\sum_{j=1}^{M}E_{\tau,N}^{M-j+1}A\big{(}F(X_{t_{j}}^{M,N})-F(X^{N}(t_{j}))\big{)},

(113)

we split $\|e_{T}^{M,N}\|_{L^{p}(\Omega,\dot{H})}$ into three parts

\begin{split}\|e_{T}^{M,N}\|_{L^{p}(\Omega,\dot{H})}&=\tau\Big{\|}\sum_{j=1}^{M}E_{\tau,N}^{M-j+1}A\big{(}F(X^{N}(t_{j}))-F(X_{t_{j}}^{M,N})\big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\leq\tau\sum_{j=1}^{M}\Big{\|}E_{\tau,N}^{M-j+1}A\big{(}F(\bar{X}^{N}(t_{j})+\mathcal{O}_{t_{j}}^{N})-F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{N})\big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\tau\sum_{j=1}^{M}\Big{\|}E_{\tau,N}^{M-j+1}A\big{(}F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{N})-F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})\big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &\quad+\tau\Big{\|}\sum_{j=1}^{M}E_{\tau,N}^{M-j+1}A\big{(}F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})-F(\bar{X}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})\big{)}\Big{\|}_{L^{p}(\Omega,\dot{H})}\\ &=:Err_{1}+Err_{2}+Err_{3}.\end{split}

(114)

Taking (80), (22), (107), Hölder’s inequality and moment bounds of $Y_{t_{m}}^{M,N}$ and $X^{N}(t)$ into account, we arrive at

\begin{split}Err_{1}&\leq C\,\tau\sum_{j=1}^{M}t_{M-j+1}^{-\frac{1}{2}}\|\bar{X}^{N}(t_{j})-\bar{Y}_{t_{j}}^{M,N}\|_{L^{2p}(\Omega,\dot{H})}\\ &\qquad\qquad\big{(}1+\|\bar{X}^{N}(t_{j})\|_{L^{4p}(\Omega,V)}^{2}+\|\bar{Y}_{t_{j}}^{M,N}\|_{L^{4p}(\Omega,V)}^{2}+\|\mathcal{O}_{t_{j}}^{N}\|_{L^{4p}(\Omega,V)}^{2}\big{)}\\ &\leq C\,\tau\,\sum_{j=1}^{M}t_{M-j+1}^{-\frac{1}{2}}\,\tau\leq C\,\tau.\end{split}

(115)

Analogously to the above estimate but with (58) instead, we derive

\begin{split}Err_{2}&\leq C\tau\sum_{j=1}^{M}t_{M-j+1}^{-\frac{3}{4}}\|\mathcal{O}_{t_{j}}^{N}-\mathcal{O}_{t_{j}}^{M,N}\|_{L^{2p}(\Omega,\dot{H}^{-1})}\\ &\quad\big{(}1+\|\bar{Y}_{t_{j}}^{M,N}\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}+\|\mathcal{O}_{t_{j}}^{N}\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}+\|\mathcal{O}_{t_{j}}^{M,N}\|_{L^{4p}(\Omega,\dot{H}^{2})}^{2}\big{)}\\ &\leq C\,\tau.\end{split}

(116)

At last, combining (57), Hölder’s inequality, (112) and regularity of $Y_{t_{m}}^{M,N}$ and $X_{t_{m}}^{M,N}$ leads to

\begin{split}Err_{3}&\leq C\left\|\tau\sum_{j=1}^{M}t_{M-j+1}^{-\frac{1}{4}}\big{|}F(\bar{Y}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})-F(\bar{X}_{t_{j}}^{M,N}+\mathcal{O}_{t_{j}}^{M,N})\big{|}_{1}\right\|_{L^{p}(\Omega,\mathbb{R})}\\ &\leq C\left\|\tau\sum_{j=1}^{M}t_{M-j+1}^{-\frac{1}{4}}|e_{t_{j}}^{M,N}|_{1}\big{(}1+|\bar{Y}_{t_{j}}^{M,N}|_{2}^{2}+|\bar{X}_{t_{j}}^{M,N}|_{2}^{2}+|\mathcal{O}_{t_{j}}^{M,N}|_{2}^{2}\big{)}\right\|_{L^{p}(\Omega,\mathbb{R})}\\ &\leq C\left\|\tau\sum_{j=1}^{M}|e_{t_{j}}^{M,N}|_{1}^{2}\right\|_{L^{p}(\Omega,\mathbb{R})}^{\frac{1}{2}}\times\left\|\tau\sum_{j=1}^{M}t_{M-j+1}^{-\frac{1}{2}}\big{(}1+|\bar{Y}_{t_{j}}^{M,N}|_{2}^{4}+|\bar{X}_{t_{j}}^{M,N}|_{2}^{4}+|\mathcal{O}_{t_{j}}^{M,N}|_{2}^{4}\big{)}\right\|_{L^{p}(\Omega,\mathbb{R})}^{\frac{1}{2}}\\ &\leq C\,\tau.\end{split}

(117)

Combining the above estimates together yields

\big{\|}\bar{X}^{N}(T)-\bar{X}_{T}^{M,N}\big{\|}_{L^{2}(\Omega,\dot{H})}\leq C\,\tau,

(118)

and thus $|K_{1}|\leq C\,\tau$ . The estimate of $K_{2}$ relies on a second-order Taylor expansion and the triangle inequality:

\begin{split}|K_{2}|&=\Big{|}\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{N})\big{]}-\mathbb{E}\big{[}\Phi(\bar{X}^{N}(T)+\mathcal{O}_{T}^{M,N})\big{]}\Big{|}\\ &\leq\Big{|}\mathbb{E}\Big{[}\Phi^{\prime}(X^{N}(T))\big{(}\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\big{)}\Big{]}\Big{|}\\ &\quad+\Big{|}\mathbb{E}\Big{[}\int_{0}^{1}\Phi^{\prime\prime}\big{(}X^{N}(T)+\lambda(\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N})\big{)}\big{(}\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N},\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\big{)}(1-\lambda)\mathrm{d}\lambda\Big{]}\Big{|}\\ &\leq\Big{|}\mathbb{E}\big{[}\Phi^{\prime}(X^{N}(T))\big{(}\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\big{)}\big{]}\Big{|}+C\,\mathbb{E}\big{[}\|\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\|^{2}\big{]}.\end{split}

(119)

Thanks to Lemma 4.2 with $\beta=0$ , we have

\mathbb{E}\big{[}\|\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\|^{2}\big{]}\leq(C\tau^{\frac{3}{4}})^{2}\leq C\,\tau^{\frac{3}{2}}.

(120)

Then, we turn our attention to the first term,

\begin{split}\Big{|}\mathbb{E}\big{[}\Phi^{\prime}(X^{N}(T))\big{(}\mathcal{O}_{T}^{M,N}-\mathcal{O}_{T}^{N}\big{)}\big{]}\Big{|}&=\Big{|}\mathbb{E}\Big{[}\Big{\langle}\int_{0}^{T}\big{(}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{)}\mathrm{d}W(s),\Phi^{{}^{\prime}}(X^{N}(T))\Big{\rangle}\Big{]}\Big{|}\\ &=\Big{|}\mathbb{E}\int_{0}^{T}\big{\langle}E(T-s)P_{N}-E_{\tau,N}^{M-[s]},\mathcal{D}_{s}\Phi^{\prime}(X^{N}(T))\big{\rangle}_{\mathcal{L}_{2}^{0}}\mathrm{d}s\Big{|}\\ &\leq\mathbb{E}\int_{0}^{T}\big{\|}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{\|}_{\mathcal{L}_{2}^{0}}\|\Phi^{\prime\prime}(X^{N}(T))\mathcal{D}_{s}X^{N}(T)\|_{\mathcal{L}_{2}^{0}}\mathrm{d}s\\ &\leq C\int_{0}^{T}\big{\|}\big{(}E(T-s)P_{N}-E_{\tau,N}^{M-[s]}\big{)}A^{-\frac{1}{2}}\big{\|}_{\mathcal{L}}\|A^{\frac{1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_{2}}\mathrm{d}s\\ &\leq C\tau\int_{0}^{T}(T-s)^{-\frac{3}{4}}\mathrm{d}s\leq C\,\tau,\end{split}

(121)

where (3), the Malliavin integration by parts formula (36), (47) in Proposition 3.2 and (iv) in Lemma 4.1 with $\mu=1$ were used. Therefore, we obtain $|K_{1}|\leq C\,\tau$ and $|K_{2}|\leq C\,\tau$ . The proof is thus complete. ∎

Corollary 4.1.

As a by-product of the weak error analysis, one can easily obtain the rates of the strong error, for $N\in\mathbb{N}$ and $m\in\{1,2,\cdots,M\}$ ,

\begin{split}\|X(t_{m})-X_{t_{m}}^{M,N}\|_{L^{2}(\Omega,\dot{H})}&\leq\|\bar{X}(t_{m})-\bar{X}_{t_{m}}^{M,N}\|_{L^{2}(\Omega,\dot{H})}+\|\mathcal{O}_{t_{m}}-\mathcal{O}_{t_{m}}^{M,N}\|_{L^{2}(\Omega,\dot{H})}\\ &\leq\|\bar{X}(t_{m})-\bar{X}^{N}(t_{m})\|_{L^{2}(\Omega,\dot{H})}+\|\bar{X}^{N}(t_{m})-\bar{X}_{t_{m}}^{M,N}\|_{L^{2}(\Omega,\dot{H})}\\ &\quad+\|\mathcal{O}_{t_{m}}-\mathcal{O}_{t_{m}}^{N}\|_{L^{2}(\Omega,\dot{H})}+\|\mathcal{O}_{t_{m}}^{N}-\mathcal{O}_{t_{m}}^{M,N}\|_{L^{2}(\Omega,\dot{H})}\\ &\leq C(\lambda_{N}^{-2}+\tau+\lambda_{N}^{-\frac{3}{2}}+\tau^{\frac{3}{4}})\\ &\leq C(\lambda_{N}^{-\frac{3}{2}}+\tau^{\frac{3}{4}}),\end{split}

(122)

where the third inequality follows from (74), (118), (76) and (120) with $t_{m}$ instead of $T$ , successively. The strong error estimates here, the same as that in [16, 27, 28], coincide with the spatial regularity of $X(t)$ , and thus are optimal.

Remark 4.1.

It is worthwhile to mention that the obtained weak convergence rate in time (i.e., $\mathcal{O}(\tau)$ ) is optimal for the Euler–type method applying to stochastic differential equation.

Acknowledgements

M. Cai and S. Gan are supported by NSF of China (No. 11971488). M. Cai is supported by the China Scholarship Council. Y. Hu is supported by an NSERC discovery grant. We are very grateful to the referees for the interesting and constructive comments and suggestions.

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