Error estimates of finite element methods for nonlinear fractional stochastic differential equations

Abstract

In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes.

1 Introduction

In the last few years, fractional calculus has attracted lots of attention. The increasing interest in fractional equations is motivated by their applications in various fields of science such as fluid mechanics, heat conduction in materials with memory, physics, chemistry, and engineering [1–5]. As we know, fractional differential equations are highly effective mathematical tools to describe complex behaviors and phenomena of memory processes because of the convolution integral with the power-law memory kernel introduced in the fractional derivatives [6–8]. On the other hand, stochastic perturbations cannot be avoided in physical systems and sometimes even cannot be ignored, so that the corresponding stochastic terms need to be added to the deterministic governing equations. Hence stochastic differential equations with fractional time derivatives have been proposed, which are a more realistic mathematical model of the real-world situations [9], just like the equations (1.1) we are going to discuss in this paper naturally arise from the consideration of the heat equation in a material with thermal memory [10].

In this paper, we consider the following initial value problem for the nonlinear fractional stochastic partial differential equation (SPDE) with multiplicative noise:

$$\begin{aligned} \textstyle\begin{cases} D^{\alpha}_{t}u(t)+Au(t)=f(u(t))+g(u(t))\frac{dW(t)}{dt},\quad \alpha\in(0,1), t\in[0, T],\\ u(0)=u_{0}. \end{cases}\displaystyle \end{aligned}$$

(1.1)

The random process $\{u(t)\}_{t\in[0,T]}$, defined on a filtered probability space $(\Omega,\mathcal{F},\mathbb{P},\{\mathcal{F}_{t}\} _{t\geq0})$ with normal filtration $\{\mathcal{F}_{t}\}_{t\geq0}$, takes values in a separable Hilbert space H with inner product $(\cdot ,\cdot)$ and norm $\|\cdot\|$. The initial value $u_{0}$ is an H-valued and $\mathcal{F}_{0}$-measurable random variable. The operator $A:\mathcal{D}(A)\subset H\rightarrow H$ is not necessarily a bounded, linear, densely defined, and selfadjoint operator with compact inverse. The nonlinear operators f and g are Lipschitz continuous in an appropriate sense. The process W with values in some separable Hilbert space U is a nuclear Q-Wiener process with respect to the filtration. The covariance operator Q is assumed to be selfadjoint and positive semidefinite with finite trace. Here, we denote the Caputo fractional derivative of order α $(0<\alpha<1)$ with respect to t by $D^{\alpha}_{t}$ and define it as [11, 12]

$$\begin{aligned} D^{\alpha}_{t}u(t)=\frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-s)^{-\alpha}\frac {d}{ds}u(s) \,ds. \end{aligned}$$

It is known that the fractional derivative $D^{\alpha}_{t}$ recovers the canonical first-order derivative $\frac{d}{dt}u(t)$ for the fractional order $\alpha=1$, and thus model (1.1) evolves into the standard stochastic partial differential equation (SPDE), whose numerical approximation has been extensively discussed in the literature; see, for example, [13–16].

Stochastic partial differential equations have been applied in many fields such as viscoelasticity, turbulence, electromagnetic theory, heterogeneous flows, and materials [17–22], so the study of stochastic partial differential equations has recently attracted a lot of attention. In particular, as in [10, 23–26], equations of type (1.1) can be used to model random effects on transport of particles in medium with thermal memory. In [10], a class of SPDEs with time-fractional derivatives was introduced, and the existence and uniqueness of solutions to these equations was proved. The existence of mild solutions for a class of nonlinear fractional stochastic partial differential equations has been discussed in [24]. Foondun and Nane [23] studied asymptotic properties of space–time fractional SPDEs. In [25], the existence and uniqueness of mild solutions for a class of nonlinear fractional Sobolev-type stochastic differential equations under non-Lipschitz conditions was discussed by employing Picard-type approximate sequences. The approximate controllability problem for fractional stochastic differential inclusions with nonlocal conditions and infinite delay has been researched in [26]. Since the random effects on transport of particles in medium with thermal memory can be exactly modeled by fractional stochastic differential systems, it is important and necessary to discuss numerical schemes and error estimation for stochastic fractional equations. However, numerical methods for these kinds of fractional SPDEs are rarely studied, and we only note [27–30]. To the authors’ knowledge, no result has been reported on the error estimation of nonlinear fractional stochastic partial differential equations with multiplicative noise based on the form of mild solutions proposed in [24], so the motivation of this paper is to fill this gap.

The main difficulty in the analysis is estimation of nonlinear terms; see Lemmas 3.6 and 3.7. Estimation of a discrete solution operator with limited smoothing properties is also a challenge; see Lemma 4.3. Our main results are as follows. First, in Theorem 3.1, denoting by $u_{h}(t)$ and $u(t)$ the mild solutions to (3.2) and (1.1), we derive a strong convergence error bound for the semidiscrete scheme:

$$\begin{aligned} \bigl\Vert u(t)-u_{h}(t) \bigr\Vert _{L_{2(\Omega, H)}} \leq Ch^{2}. \end{aligned}$$

Second, for $\alpha\in(0,1)$, we obtain am $L_{2(\Omega, H)}$-norm error estimate for the fully discrete scheme in Theorem 4.1:

$$\begin{aligned} \bigl\Vert u(t_{n})-u^{n}_{h} \bigr\Vert _{L_{2(\Omega;H)}}\leq C\bigl[k^{\alpha}+h^{2}\bigr], \end{aligned}$$

where $u^{n}_{h}$ denotes an approximation of the mild solution $u(t)$ at time $t_{n}$. The parameters h and k, which will be detailed in Sects. 3 and 4, represent the maximal meshsize and time step, respectively.

The rest of the paper is organized as follows: In Sect. 2, we introduce some basic notation, present the Laplace transform, and give a representation of the mild solution of equation (1.1) by using basic properties of the Mittag–Leffler function. In Sect. 3, we first give a short review of Galerkin finite element methods and then study the space semidiscrete scheme and derive error estimates for the standard Galerkin finite element method with smooth initial data. Finally, in Sect. 4, using the GMMP scheme, we prove strong error estimates for the fully discrete scheme.

2 Preliminaries

In this section, we recall some useful properties on the Mittag–Leffler function, introduce the Laplace transform and present a representation of the mild solution of problem (1.1). Besides, we use the letter C to denote a constant that may vary from one occurrence to another and denote by $L(U,H)$ the space of bounded linear operators from U to H, where U and H are real separable Hilbert spaces with inner product $(\cdot, \cdot)$ and norms $\|\cdot\|_{U}$ and $\|\cdot\|_{H}$.

2.1 Mittag–Leffler function

The Mittag–Leffler function is defined by

$$\begin{aligned} E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty} \frac{z^{k}}{\Gamma(k\alpha+\beta )},\quad z\in\mathbb{C}, \end{aligned}$$

where $\Gamma(\cdot)$ is the standard gamma function

$$\begin{aligned} \Gamma(z)= \int_{0}^{\infty}t^{z-1}e^{-t}\,dt,\quad \mathfrak{R}(z)>0. \end{aligned}$$

We give important properties of the Mittag–Leffler function $E_{\alpha ,\beta}(z)$ essential in our analysis.

Lemma 2.1

([31])

Let $0<\alpha<2$ and $\beta\in\mathbb{R}$ be arbitrary, and let $\frac {\pi\alpha}{2}<\mu<\min(\pi,\alpha\pi)$. Then there exists a constant $C=C(\alpha,\beta,\mu)>0$ such that, for $\mu\leq|\arg(z)|\leq\pi$,

$$\begin{aligned} \bigl\vert E_{\alpha,\beta}(z) \bigr\vert \leq \textstyle\begin{cases}\frac{C}{1+ \vert z \vert ^{2}}, &\beta-\alpha\in\mathbb{Z}^{-}\cup \{0\},\\ \frac{C}{1+ \vert z \vert } &\textit{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Moreover, for $\lambda>0$, $\alpha>0$, and $t>0$, we have

$$\begin{aligned} D_{t}^{\alpha}E_{\alpha,1}\bigl(-\lambda t^{\alpha}\bigr)=-\lambda E_{\alpha ,1}\bigl(-\lambda t^{\alpha}\bigr)\quad \textit{and}\quad \frac{d}{dt}E_{\alpha,1}\bigl(-\lambda t^{\alpha}\bigr)=-\lambda t^{\alpha-1} E_{\alpha,\alpha}\bigl(-\lambda t^{\alpha}\bigr). \end{aligned}$$

In our analysis, we also use the Laplace transform. Let $\pi:\mathbb {R}_{+}\rightarrow H$ be subexponential, that is, for any $\varepsilon >0$, the function $t\rightarrow\pi(t)e^{-\varepsilon t}$ belongs to $L^{1}(\mathbb{R}_{+}, H)$. The Laplace transform of $\hat{\pi}:\mathbb {C}_{+}\rightarrow H$ is denoted by

$$\hat{\pi}(z)= \int_{0}^{+\infty}\pi(t)e^{-zt}\,{d}t,\quad \mathfrak{R}(z)>0, $$

where the same notation H represents the complexification of H. Further, we denote by ∗ the Laplace convolution product on $[0,t]$ of two locally integrable subexponential functions $\pi,\sigma\in L_{\mathrm{loc}}^{1}(\mathbb{R}_{+}, H)$, that is,

$$(\pi\ast\sigma) (t)= \int_{0}^{t}\pi(t-s)\sigma(s)\,{d}s. $$

It is well known that $\pi\ast\sigma\in L_{\mathrm{loc}}^{1}(\mathbb{R}_{+}, H)$ is subexponential and

$$\widehat{\pi\ast\sigma}=\hat{\pi}(z)\hat{\sigma}(z). $$

2.2 Solution representation

In order to study the representation of the solution of (1.1), we introduce some notation.

Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space. By $L_{2}(\Omega, H)$ we denote the space of H-valued square-integrable random variables with norm

$$\Vert v \Vert _{L_{2(\Omega,H)}}=\bigl(E \Vert v \Vert _{H}^{2} \bigr)^{\frac{1}{2}}= \biggl( \int _{\Omega} \bigl\Vert v(w) \bigr\Vert _{H}^{2} \mathbf{P}(w) \biggr)^{\frac{1}{2}}, $$

where E stands for expected value. Let $Q\in\mathcal{L}(U)$ be a selfadjoint positive semidefinite operator with $\operatorname{Tr}(Q)<\infty$, where $\operatorname{Tr}(Q)$ is the trace of Q. Let $\{(\gamma_{j},e_{j})\} _{j=1}^{\infty }$ be the eigenpairs of Q with orthonormal eigenvectors. The U-valued Q-Wiener process $W(t)$, defined on the probability space $(\Omega, \mathcal{F}, \mathbf{P})$, has the orthogonal expansion

$$W(t)=\sum_{j=1}^{\infty}\gamma_{j}^{1/2} \beta_{j}(t)e_{j}, $$

where $\{\beta_{j}(t)\}_{j=1}^{\infty}$ are real-valued mutually independent standard Brownian motions. Further, the set $L_{2}^{0}=HS(Q^{1/2}(H), H)$ expresses the space of all Hilbert–Schmidt operators from $Q^{1/2}(H)$ to H with norm $\|\psi\|_{L_{2}^{0}}=(\sum_{j=1}^{\infty}\|\psi Q^{1/2}e_{j}\|^{2})^{1/2}$, and the subset $L^{0}_{2,r}\subset L^{0}_{2}, r\geq0$ is the subspace of all Hilbert–Schmidt operators from $Q^{1/2}(H)$ to $\dot{H}^{r}$ with norm $\|\psi\|_{L_{2,r}^{0}}=\|A^{\frac{r}{2}}\psi\|_{L^{0}_{2}}$. It is then possible to define the stochastic integral $\int_{0}^{t}\psi(s) \,{d}W(s)$ together with Itô’s isometry

$$\begin{aligned} E \biggl\Vert \int^{t}_{0}\psi(s)\,{d}W(s) \biggr\Vert _{H}^{2}= \int^{t}_{0}E \bigl\Vert \psi(s) \bigr\Vert ^{2}_{L_{2}^{0}}\,{d}s. \end{aligned}$$

(2.1)

In a standard way, we present the fractional powers $A^{s}$, $s\in \mathbb{R}$, of A as

$$A^{s}v=\sum_{j=1}^{\infty} \lambda_{j}^{s}(v,\varphi_{j}) \varphi_{j},\qquad D\bigl(A^{\frac{s}{2}}\bigr)=\Biggl\{ v\in H: \bigl\Vert A^{\frac{s}{2}}v \bigr\Vert ^{2}=\sum _{j=1}^{\infty }\lambda_{j}^{s}(v, \varphi_{j})^{2}< \infty\Biggr\} , $$

where $\{\lambda_{j}\}^{\infty}_{j=1}$ and $\{\varphi_{j}\}_{j=1}^{\infty}$ are respectively the eigenvalues and the orthonormal eigenfunctions of A, that is,

$$A\varphi_{j}=\lambda_{j}\varphi_{j}\quad\text{and}\quad(\varphi_{i}, \varphi _{j})= \delta_{i,j} \quad\text{for }i,j\geq1. $$

In addition, the sequence $\{\lambda_{j}\}^{\infty}_{j=1}$ is an increasing sequence of real numbers, that is, $0\leq\lambda_{1}\leq\lambda_{2}\leq \cdots$ . Let $\dot{H}^{s}=D(A^{\frac{s}{2}})$ with norm

$$\begin{aligned} \Vert v \Vert _{s}= \bigl\Vert A^{\frac{s}{2}}v \bigr\Vert = \Biggl(\sum_{j=1}^{\infty}\lambda _{j}^{s}(v,\varphi_{j})^{2} \Biggr)^{1/2},\quad v \in\dot{H}^{s}. \end{aligned}$$

We define the operators $E(t)$ and $\bar{E}(t)$ by

$$\begin{aligned} &E(t)v=\sum_{j=1}^{\infty}E_{\alpha,1} \bigl(-\lambda_{j} t^{\alpha}\bigr) (v,\varphi _{j}) \varphi_{j},\quad v \in\dot{H}^{s}, \\ &\bar{E}(t)v=\sum_{j=1}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}\bigl(-\lambda _{j} t^{\alpha}\bigr) (v,\varphi_{j})\varphi_{j},\quad v \in \dot{H}^{s}, \end{aligned}$$

where $\alpha\in(0,1)$ indicates the order of Caputo fractional derivative. Then, we present the mild solution $u(t)$ of (1.1) [24]:

$$\begin{aligned} u(t)=E(t)u_{0}+ \int_{0}^{t}\bar{E}(t-s)f\bigl(u(s)\bigr)\,ds+ \int_{0}^{t}\bar{E}(t-s)g\bigl(u(s)\bigr)\,dW(s). \end{aligned}$$

(2.2)

Next, we impose the following conditions on f, g, and $u(t)$, which are the conditions of existence and uniqueness of the mild solution u [24].

Assumption 2.1

For the nonlinear operator $f: H \rightarrow H$, there exists a constant C such that

$$\begin{aligned} \bigl\Vert f(x)-f(y) \bigr\Vert \leq C \Vert x-y \Vert ,\quad \bigl\Vert f(x) \bigr\Vert \leq C \Vert x \Vert . \end{aligned}$$

(2.3)

Assumption 2.2

For the nonlinear operator $g: H \rightarrow L^{0}_{2}$, there exists a constant C such that

$$\begin{aligned} \bigl\Vert g(x)-g(y) \bigr\Vert _{L^{0}_{2}}\leq C \Vert x-y \Vert ,\quad \bigl\Vert g(x) \bigr\Vert _{L^{0}_{2}}\leq C \Vert x \Vert . \end{aligned}$$

(2.4)

Assumption 2.3

The mild solution $u: [0,T]\times\Omega\rightarrow H$ satisfies

$$\begin{aligned} \sup_{0\leq t\leq T}E\bigl( \bigl\Vert u(t) \bigr\Vert ^{2}_{s}\bigr)< +\infty, \end{aligned}$$

(2.5)

where $s\in[0,2]$.

Some properties of the operators $E(t)$ and $\bar{E}(t)$, which are crucial for the semidiscrete error estimates, will be introduced later.

Lemma 2.2

([32])

For $\alpha\in(0,1)$, we have the following estimates:

$$\begin{aligned} \bigl\Vert \bigl(D_{t}^{\alpha}\bigr)^{\ell}E(t)v \bigr\Vert _{p}\leq Ct^{-\alpha(\ell+\frac{p-q}{2})} \Vert v \Vert _{q},\quad t>0, \end{aligned}$$

where $0\leq q\leq p\leq2$ for $\ell=0$, and $0\leq p\leq q\leq2$ and $q\leq p+2$ for $\ell=1$.

Lemma 2.3

([31])

For any $t>0$ and $0\leq p-q\leq4$, we have

$$\begin{aligned} \bigl\Vert \bar{E}(t)v \bigr\Vert _{p}\leq Ct^{-1+\alpha(1+\frac{q-p}{2})} \Vert v \Vert _{q}. \end{aligned}$$

3 Error estimates for spatially semidiscrete approximation

In this section, we first review the Galerkin finite element methods and some basic estimates for the finite element projection operators. Then we introduce a representation of the semidiscrete scheme of the mild solution $u(t)$ and some smoothing properties of the operators $E_{h}(t)$ and $\bar{E}_{h}(t)$. We close this section with the proof of the semidiscrete error estimates.

3.1 Space discretization

Let $\{\mathcal{T}_{h}\}_{h\in(0,1]}$ denote a regular family of triangulations of $\mathcal{D}$, where h is the maximal meshsize, and let $V_{h}$ denote the space of piecewise linear continuous functions with respect to $\mathcal{T}_{h}$ vanishing on $\partial\mathcal{D}$. Thereby, $V_{h}\subset H_{0}^{1}(\mathcal{D})=\dot {H}^{1}=\{v\in L_{2}(\mathcal{D}), \nabla v\in L_{2}(\mathcal{D}), v|_{\partial\mathcal{D}}=0\}$. Denote by $R_{h}:\dot{H}^{1}\rightarrow V_{h}$ the Ritz projector onto $V_{h}$ with respect to the inner product

$$\begin{aligned} a(v,w)=\bigl(A^{\frac{1}{2}}v,A^{\frac{1}{2}}w\bigr),\quad v,w \in \dot{H}^{1}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} a(R_{h}v,\chi)=a(v,\chi),\quad v \in\dot{H}^{1}, \chi\in V_{h}. \end{aligned}$$

Meanwhile, the following error estimate is established:

$$\begin{aligned} \Vert R_{h}v-v \Vert \leq Ch^{s} \Vert v \Vert _{s},\quad v \in\dot{H}^{s}, 1\leq s\leq2. \end{aligned}$$

(3.1)

The semidiscrete problem corresponding to (1.1) is to find a process $u_{h}(t)\in V_{h}$ such that

$$\begin{aligned} D^{\alpha}_{t}u_{h}(t)+A_{h}u_{h}(t)=P_{h}f \bigl(u_{h}(t)\bigr)+P_{h}g\bigl(u_{h}(t)\bigr) \frac {dW}{dt},\quad u_{h}(0)=P_{h}u_{0}, \end{aligned}$$

(3.2)

where the mapping $A_{h}:V_{h}\rightarrow V_{h}$ is a discrete version of the operator A defined by

$$\begin{aligned} a(\varphi,\chi)=(A_{h}\varphi,\chi),\quad\forall\varphi, \chi\in V_{h}, \end{aligned}$$

and $P_{h}$ is the orthogonal projector

$$\begin{aligned} P_{h}: H\rightarrow V_{h},\quad (P_{h}v,\chi)=(v, \chi),\quad \forall v\in H, \forall \chi\in V_{h}. \end{aligned}$$

Depending on the eigenvalues and eigenfunctions $\{\lambda^{h}_{j}\} _{j=1}^{N}$ and $\{\varphi^{h}_{j}\}_{j=1}^{N}$ of the discrete operator $A_{h}$, we can introduce a representation of the solution of (3.2). Firstly, we present the discrete analogues of operators $E(t)$ and $\bar {E}(t)$ as follows:

$$\begin{aligned} &E_{h}(t)v_{h}=\sum _{j=1}^{N} E_{\alpha,1}\bigl(- \lambda^{h}_{j} t^{\alpha}\bigr) \bigl(v_{h}, \varphi ^{h}_{j}\bigr)\varphi^{h}_{j}, \end{aligned}$$

(3.3)

$$\begin{aligned} &\bar{E}_{h}(t)v_{h}=\sum _{j=1}^{N} t^{\alpha-1}E_{\alpha,\alpha}\bigl(- \lambda ^{h}_{j} t^{\alpha}\bigr) \bigl(v_{h}, \varphi^{h}_{j}\bigr)\varphi^{h}_{j}. \end{aligned}$$

(3.4)

Analogously, the unique solution of the finite element problem (3.2) can be given by

$$\begin{aligned} u_{h}(t)=E_{h}(t)P_{h}u_{0}+ \int_{0}^{t}\bar{E}_{h}(t-s)P_{h}f \bigl(u_{h}(s)\bigr)\,ds+ \int_{0}^{t}\bar {E}_{h}(t-s)P_{h}g \bigl(u_{h}(s)\bigr)\,dW(s). \end{aligned}$$

(3.5)

Then, similarly to Lemmas 2.2 and 2.3, we show some vital properties of $E_{h}(t)$ and $\bar{E}_{h}(t)$ in the following:

Lemma 3.1

([32])

Let $E_{h}(t)$ be defined by (3.3), and let $\chi\in V_{h}$. Then, for $\alpha\in(0,1)$ and $p, q \in[-1,1]$, we have

$$\begin{aligned} \bigl\Vert \bigl(D^{\alpha}_{t}\bigr)^{\ell}E_{h}(t)\chi \bigr\Vert _{p}\leq Ct^{-\alpha(\ell+\frac {p-q}{2})} \Vert \chi \Vert _{q}, \end{aligned}$$

where $q\leq p$ and $0\leq p-q\leq2$ for $\ell=0$, and $p\leq q\leq p+2$ for $\ell=1$.

Lemma 3.2

([32])

Let $\bar{E}_{h}$ be defined by (3.4), and let $\chi\in V_{h}$. Then, for all $t>0$,

$$\begin{aligned} \bigl\Vert \bar{E}_{h}(t)\chi \bigr\Vert _{p}\leq \textstyle\begin{cases}Ct^{-1+\alpha(1+\frac{q-p}{2})} \Vert \chi \Vert _{q},&p-2\leq q\leq p,\\ Ct^{-1+\alpha} \Vert \chi \Vert _{q},&p< q, \end{cases}\displaystyle \end{aligned}$$

where $p,q \in[-1,1]$.

Based on this lemma, we have the following conclusion.

Lemma 3.3

Let $\bar{E}_{h}$ be defined by (3.4), and let $v\in H, P_{h}v=v_{h}$. For all $t>0$, we have

$$\begin{aligned} \bigl\Vert \bar{E}_{h}(t)P_{h}v \bigr\Vert \leq Ct^{\alpha-1} \Vert v \Vert . \end{aligned}$$

Proof

By Lemma 3.2 with $p=q=0$ we get

$$\begin{aligned} \bigl\Vert \bar{E}_{h}(t)v_{h} \bigr\Vert \leq Ct^{\alpha-1} \Vert v_{h} \Vert . \end{aligned}$$

Since $v_{h}=P_{h}v$, we get

$$\begin{aligned} \bigl\Vert \bar{E}_{h}(t)v_{h} \bigr\Vert \leq Ct^{\alpha-1} \Vert P_{h}v \Vert \leq Ct^{\alpha-1} \Vert v \Vert , \end{aligned}$$

which completes the proof. □

Moreover, we need the following estimate of $u_{h}(t)$.

Lemma 3.4

For any $t \in[0, T]$ and $\alpha\in(\frac{1}{2},1)$, let $u_{h}(t)$ be the mild solution of (3.2). Then there exists a constant $C > 0$ such that

$$\begin{aligned} \sup_{0\leq t\leq T} \bigl\Vert u_{h}(t) \bigr\Vert ^{2}_{L_{2(\Omega;H)}}\leq C \Vert u_{0} \Vert ^{2}_{L_{2(\Omega;H)}}. \end{aligned}$$

Proof

For any $t \in[0, T]$, from (3.5) by Lemma 3.1 with $\ell=p=q=0$, Lemma 3.3, Assumptions 2.1 and 2.2, and Itô’s isometry we obtain

$$\begin{aligned} E \bigl\Vert u_{h}(t) \bigr\Vert ^{2}\leq{}&4E \bigl\Vert E_{h}(t)P_{h}u_{0} \bigr\Vert ^{2}+4E \biggl\Vert \int^{t}_{0}\bar {E}_{h}(t-s)P_{h}f \bigl(u_{h}(s)\bigr)\,ds \biggr\Vert ^{2} \\ &{}+4E \biggl\Vert \int^{t}_{0}\bar{E}_{h}(t-s)P_{h}g \bigl(u_{h}(s)\bigr)\,dW(s) \biggr\Vert ^{2} \\ \leq{}&4E \bigl\Vert E_{h}(t)P_{h}u_{0} \bigr\Vert ^{2}+4 \int^{t}_{0}E \bigl\Vert \bar{E}_{h}(t-s)P_{h}f \bigl(u_{h}(s)\bigr) \bigr\Vert ^{2}\,ds \\ &{}+4 \int^{t}_{0}E \bigl\Vert \bar{E}_{h}(t-s)P_{h}g \bigl(u_{h}(s)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ \leq{}& CE \Vert u_{0} \Vert ^{2}+C \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert u_{h}(s) \bigr\Vert ^{2}\,ds \\ &{}+ C \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert u_{h}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

Thus, applying the integral version of Gronwall’s lemma, we deduce that

$$\begin{aligned} \sup_{0\leq t\leq T}E \bigl\Vert u_{h}(t) \bigr\Vert ^{2}\leq CE \Vert u_{0} \Vert ^{2}\exp{\biggl(C \int ^{t}_{0}(t-s)^{2\alpha-2}\,ds\biggr)}\leq CE \Vert u_{0} \Vert ^{2}. \end{aligned}$$

 □

3.2 Semidiscrete finite element approximation

In this subsection, we first present and prove some lemmas, which are crucial for the derivation of the semidiscrete error estimate for the nonlinear fractional stochastic differential equation. Then we give a detailed proof of the semidiscrete error estimate.

Lemma 3.5

([28])

Let $0\leq\nu\leq\mu\leq2$ and $F_{h}(t)=E(t)-E_{h}(t)P_{h}$. Then, for $\alpha\in(0,1)$, there exists a constant C such that

$$\begin{aligned} \bigl\Vert F_{h}(t)x \bigr\Vert \leq Ch^{\mu}t^{-\alpha\frac{\mu-\nu}{2}} \Vert x \Vert _{\nu},\quad x\in \dot{H}^{\nu}. \end{aligned}$$

Lemma 3.6

Let $1< q\leq2$ and $\bar{F}_{h}(t)=\bar{E}(t)-\bar{E}_{h}(t)P_{h}$. Then, for $t\in[0,T]$, there exists a constant C such that

$$\begin{aligned} \biggl\Vert \int_{0}^{T}\bar{F}_{h}(t)h(t)\,dt \biggr\Vert ^{2}\leq Ch^{2q} \int_{0}^{T} \bigl\Vert h(t) \bigr\Vert ^{2}_{q-2}\,dt,\quad h(t)\in\dot{H}^{q-2}. \end{aligned}$$

Proof

By the definition of $\bar{F}_{h}(t)$ we split $\int_{0}^{t}\bar {F}_{h}(t-s)h(t)\,ds$ into two additional terms:

$$\begin{aligned} \int_{0}^{t}\bar{F}_{h}(t-s)h(t)\,ds&= \int_{0}^{t}\bar{E}(t-s)h(t)\,ds- \int_{0}^{t}\bar {E}_{h}(t-s)P_{h}h(t) \,ds\\ &=v(t)-v_{h}(t) \\ &=\bigl(v(t)-P_{h}v(t)\bigr)+\bigl(P_{h}v(t)-v_{h}(t) \bigr) \\ &=\eta(t)+\xi(t), \end{aligned}$$

where $v(t)$ and $v_{h}(t)$ are the solutions of the following equations:

$$\begin{aligned} &\textstyle\begin{cases}D^{\alpha}_{t}v(t)+Av(t)=h(t),\\ v(0)=0, \end{cases}\displaystyle \end{aligned}$$

(3.6)

$$\begin{aligned} & \textstyle\begin{cases}D^{\alpha}_{t}v_{h}(t)+A_{h}v_{h}(t)=P_{h}h(t),\\ v_{h}(0)=P_{h}v(0)=0. \end{cases}\displaystyle \end{aligned}$$

(3.7)

To bound ξ, we note that by our definitions

$$\begin{aligned} D^{\alpha}_{t}\xi(t)+A_{h}\xi(t)=A_{h} \bigl(R_{h}v(t)-P_{h}v(t)\bigr),\quad \xi(0)=0. \end{aligned}$$

By the Laplace transforms of both sides of this equation, we recover

$$\begin{aligned} z^{a}\hat{\xi}(z)+A_{h}\hat{\xi}(z)=A_{h}(R_{h}-P_{h}) \hat{v}(z). \end{aligned}$$

Therefore

$$\begin{aligned} \hat{\xi}(z)=\bigl(z^{\alpha}I+A_{h}\bigr)^{-1}A_{h}(R_{h}-P_{h}) \hat{v}(z). \end{aligned}$$

Since the operator $A_{h}$ generates an analytic contraction semigroup, there exists a constant C, depending only on ϕ and α, such that

$$\begin{aligned} \bigl\Vert \bigl(z^{\alpha}I+A_{h}\bigr)^{-1} \bigr\Vert \leq Cz^{-\alpha},\quad \forall z\in\Sigma_{\phi}, \end{aligned}$$

where $\Sigma_{\phi}=\{z\in\mathbb{C}:|\operatorname{arg} z|\leq\phi\}$. By the identity

$$\begin{aligned} \bigl(z^{\alpha}I+A_{h}\bigr)^{-1}A_{h}=I-z^{\alpha}\bigl(z^{\alpha}I+A_{h}\bigr)^{-1} \end{aligned}$$

we get

$$\begin{aligned} \bigl\Vert \bigl(z^{\alpha}I+A_{h}\bigr)^{-1}A_{h} \bigr\Vert \leq1+ \bigl\Vert z^{\alpha}\bigl(z^{\alpha}I+A_{h}\bigr)^{-1} \bigr\Vert \leq1+C\leq C. \end{aligned}$$

Using the inverse Laplace transform and inequality (3.1), we obtain

$$\begin{aligned} \bigl\Vert \xi(t) \bigr\Vert &\leq C \bigl\Vert (R_{h}-P_{h})v(t) \bigr\Vert \\ &\leq C \bigl\Vert (R_{h}-I)v(t) \bigr\Vert +C \bigl\Vert (I-P_{h})v(t) \bigr\Vert \leq Ch^{q} \bigl\Vert v(t) \bigr\Vert _{q}. \end{aligned}$$

Then by Theorem 2.1 of [31] we get

$$\begin{aligned} \int^{T}_{0} \bigl\Vert \xi(t) \bigr\Vert ^{2}\,dt\leq Ch^{2q} \int^{T}_{0} \bigl\Vert v(t) \bigr\Vert ^{2}_{q}\,dt\leq Ch^{2q} \int ^{T}_{0} \bigl\Vert h(t) \bigr\Vert ^{2}_{q-2}\,dt. \end{aligned}$$

According to inequality (3.1) and Theorem 2.1 of [31], the estimate of η yields

$$\begin{aligned} \int^{T}_{0} \bigl\Vert \eta(t) \bigr\Vert ^{2}\,dt&\leq C \int^{T}_{0} \bigl\Vert (R_{h}-I)v(t) \bigr\Vert ^{2}\,dt\leq Ch^{2q} \int ^{T}_{0} \bigl\Vert v(t) \bigr\Vert ^{2}_{q}\,dt\\ &\leq Ch^{2q} \int^{T}_{0} \bigl\Vert h(t) \bigr\Vert ^{2}_{q-2}\,dt. \end{aligned}$$

Since

$$\begin{aligned} \int^{T}_{0} \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)h\,ds \biggr\Vert ^{2}\,dt&= C(T) \biggl\Vert \int^{t}_{0}\bar {F}_{h}(t)h\,ds \biggr\Vert ^{2} \\ &= \int^{T}_{0} \bigl\Vert \eta(t)+\xi(t) \bigr\Vert ^{2}\,dt \\ &\leq 2 \int^{T}_{0} \bigl\Vert \eta(t) \bigr\Vert ^{2}\,dt+2 \int^{T}_{0} \bigl\Vert \xi(t) \bigr\Vert ^{2}\,dt \\ &\leq Ch^{2q} \int^{T}_{0} \bigl\Vert h(t) \bigr\Vert ^{2}_{q-2}\,dt, \end{aligned}$$

we get the conclusion

$$\begin{aligned} \biggl\Vert \int_{0}^{T}\bar{F}_{h}(t)h\,dt \biggr\Vert ^{2}\leq Ch^{2q} \int_{0}^{T} \bigl\Vert h(t) \bigr\Vert ^{2}_{q-2}\,dt. \end{aligned}$$

 □

Lemma 3.7

Let $1< q\leq2$ and $\bar{F}_{h}(t)=\bar{E}(t)-\bar{E}_{h}(t)P_{h}$. Then, for $t\in[0,T]$ and $\tilde{h}(s)\in\dot{H}^{q}$, there exists a constant C such that

$$\begin{aligned} E \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)\tilde{h}(s) \,dW(s) \biggr\Vert ^{2}\leq Ch^{2q} \int ^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert \tilde{h}(s) \bigr\Vert ^{2}_{L^{0}_{2,q}}\,ds. \end{aligned}$$

Proof

Just like in the proof of Lemma 3.6, we split $\int_{0}^{t}\bar {F}_{h}(t-s)h(t)\,ds$ into two additional terms:

$$\begin{aligned} \int_{0}^{t}\bar{F}_{h}(t-s)\tilde{h}(s) \,dW(s)&= \int_{0}^{t}\bar{E}(t-s)\tilde {h}(s)\,dW(s)- \int_{0}^{t}\bar{E}_{h}(t-s)P_{h} \tilde{h}(s)\,dW(s) \\ &=\tilde{v}(t)-\tilde{v}_{h}(t)=\bigl(\tilde{v}(t)-P_{h} \tilde{v}(t)\bigr)+\bigl(P_{h}\tilde {v}(t)-\tilde{v}_{h}(t) \bigr) \\ &=\varrho(t)+\vartheta(t), \end{aligned}$$

where $\tilde{v}(t)$ and $\tilde{v}_{h}(t)$ are the solutions of the following equations:

$$\begin{aligned} &\textstyle\begin{cases} D^{\alpha}_{t}\tilde{v}(t)+A\tilde{v}(t)=\tilde{h}(t)\frac {dW(t)}{dt},\\ \tilde{v}(0)=0, \end{cases}\displaystyle \end{aligned}$$

(3.8)

$$\begin{aligned} &\textstyle\begin{cases} D^{\alpha}_{t}\tilde{v}_{h}(t)+A_{h}\tilde{v}_{h}(t)=P_{h}\tilde {h}(t)\frac{dW(t)}{dt},\\ \tilde{v}_{h}(0)=P_{h}\tilde{v}(0)=0. \end{cases}\displaystyle \end{aligned}$$

(3.9)

To bound ϑ, we note that by our definitions

$$\begin{aligned} D^{\alpha}_{t}\vartheta(t)+A_{h}\vartheta(t)=A_{h} \bigl(R_{h}\tilde{v}(t)-P_{h}\tilde {v}(t)\bigr),\quad \vartheta(0)=0. \end{aligned}$$

As in the proof of Lemma 3.6, taking the Laplace transform and inverse Laplace transform on both sides of this equation, we eventually get

$$\begin{aligned} \bigl\Vert \vartheta(t) \bigr\Vert &\leq C \bigl\Vert (R_{h}-P_{h})\tilde{v}(t) \bigr\Vert \\ &\leq C \bigl\Vert (R_{h}-I)\tilde{v}(t) \bigr\Vert +C \bigl\Vert (I-P_{h})\tilde{v}(t) \bigr\Vert \leq Ch^{q} \bigl\Vert \tilde {v}(t) \bigr\Vert _{q}. \end{aligned}$$

Thus by Lemma 2.3 with $p=q\in(1,2]$ and Itô’s isometry we derive

$$\begin{aligned} E \bigl\Vert \vartheta(t) \bigr\Vert ^{2}&\leq Ch^{2q}E \bigl\Vert \tilde{v}(t) \bigr\Vert ^{2}_{q}=Ch^{2q}E \biggl\Vert \int ^{t}_{0}\bar{E}(t-s)\tilde{h}(s)\,dW(s) \biggr\Vert ^{2}_{q} \\ &= Ch^{2q}E \biggl\Vert \int^{t}_{0}A^{\frac{q}{2}}\bar{E}(t-s)\tilde{h}(s) \,dW(s) \biggr\Vert ^{2} \\ &= Ch^{2q} \int^{t}_{0}E \bigl\Vert A^{\frac{q}{2}} \bar{E}(t-s)\tilde{h}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ &\leq Ch^{2q} \int^{t}_{0}(t-s)^{2\alpha-2}E \Biggl\Vert \sum _{j=1}^{\infty}\tilde {h}(s)Q^{\frac{1}{2}}e_{j} \Biggr\Vert ^{2}_{q} \,ds \\ &= Ch^{2q} \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert A^{\frac{q}{2}}\tilde{h}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds. \end{aligned}$$

According to inequality (3.1) and Lemma 2.3, the estimate of ϱ yields

$$\begin{aligned} E \bigl\Vert \varrho(t) \bigr\Vert ^{2}&\leq C \bigl\Vert (R_{h}-I)\tilde{v}(t) \bigr\Vert ^{2}\leq Ch^{2q}E \bigl\Vert \tilde {v}(t) \bigr\Vert ^{2}_{q} \\ &\leq Ch^{2q} \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert A^{\frac{q}{2}}\tilde{h}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds. \end{aligned}$$

Thereby,

$$\begin{aligned} E \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)\tilde{h}(s) \,dW(s) \biggr\Vert ^{2}\leq Ch^{2q} \int ^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert \tilde{h}(s) \bigr\Vert ^{2}_{L^{0}_{2,q}}\,ds. \end{aligned}$$

 □

Now, we will give the semidiscrete error estimate in space for the stochastic fractional differential equation (1.1).

Theorem 3.1

Let $u(t)$ and $u_{h}(t)$ be the solutions of (1.1) and (3.2), respectively. Then, for $t\geq0$, $\alpha\in(\frac {1}{2},1)$, and $u_{0}\in L_{2}(\Omega, \dot{H}^{s}), s\in[0,2]$, we have

$$\begin{aligned} \bigl\Vert u(t)-u_{h}(t) \bigr\Vert _{L_{2(\Omega, H)}} \leq Ch^{2}. \end{aligned}$$

Proof

For $t\in[0,T]$, by (1.1) and (3.2) we have

$$\begin{aligned} & \bigl\Vert u(t)-u_{h}(t) \bigr\Vert _{L_{2(\Omega, H)}} \\ &\quad\leq \bigl\Vert \bigl(E(t)-E_{h}(t)P_{h} \bigr)u_{0} \bigr\Vert _{L_{2(\Omega, H)}} \\ &\qquad{}+ \biggl\Vert \int_{0}^{t}\bigl(\bar{E}(t-s)f\bigl(u(s)\bigr)- \bar{E}_{h}(t-s)P_{h} f\bigl(u_{h}(s)\bigr)\bigr) \,ds \biggr\Vert _{L_{2(\Omega, H)}} \\ &\qquad{}+ \biggl\Vert \int_{0}^{t}\bigl(\bar{E}(t-s)g\bigl(u(s)\bigr)- \bar{E}_{h}(t-s)P_{h} g\bigl(u_{h}(s)\bigr)\bigr) \,dW(s) \biggr\Vert _{L_{2(\Omega, H)}} \\ &\quad =I+II+III. \end{aligned}$$

For I, by Lemma 3.5 with $\nu=\mu=1+r\ (r\in(0,1])$ we have

$$\begin{aligned} I\leq Ch^{1+r} \Vert u_{0} \Vert _{L_{2(\Omega;\dot{H}^{1+r})}}. \end{aligned}$$

We dominate II by two additional terms:

$$\begin{aligned} II={}& \biggl\Vert \int^{t}_{0}\bar{E}(t-s)f\bigl(u(s)\bigr)- \bar{E}_{h}(t-s)P_{h}f\bigl(u_{h}(s)\bigr)\,ds \biggr\Vert _{L_{2(\Omega, H)}} \\ \leq{}& \biggl\Vert \int^{t}_{0}\bar{E}_{h}(t-s)P_{h} \bigl(f\bigl(u(s)\bigr)-f\bigl(u_{h}(s)\bigr)\bigr)\,ds \biggr\Vert _{L_{2(\Omega, H)}} \\ &{}+ \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)f\bigl(u(s) \bigr)\,ds \biggr\Vert _{L_{2(\Omega, H)}} \\ ={}&I_{1}+I_{2}. \end{aligned}$$

We estimate each term separately. First, note that by Lemma 3.3 and Assumption 2.1 we have

$$\begin{aligned} I_{1}&\leq \int^{t}_{0} \bigl\Vert \bar{E}_{h}(t-s)P_{h} \bigl(f\bigl(u(s)\bigr)-f\bigl(u_{h}(s)\bigr)\bigr) \bigr\Vert _{L_{2(\Omega, H)}}\,ds \\ &\leq C \int^{t}_{0}(t-s)^{\alpha-1} \bigl\Vert f \bigl(u(s)\bigr)-f\bigl(u_{h}(s)\bigr) \bigr\Vert _{L_{2(\Omega, H)}}\,ds \\ &\leq C \int^{t}_{0}(t-s)^{\alpha-1} \bigl\Vert u(s)-u_{h}(s) \bigr\Vert _{L_{2(\Omega, H)}}\,ds. \end{aligned}$$

The term $I_{2}$ is reckoned by applying Lemma 3.6, Assumptions 2.1 and 2.3. Then we get

$$\begin{aligned} I^{2}_{2}&= E \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)f\bigl(u(s) \bigr)\,ds \biggr\Vert ^{2} \\ &\leq Ch^{4} \int^{t}_{0}E \bigl\Vert f\bigl(u(s)\bigr) \bigr\Vert ^{2}\,ds \\ &\leq Ch^{4} \int^{t}_{0}\sup_{0\leq s\leq T}E \bigl\Vert u(s) \bigr\Vert ^{2}\,ds \\ &\leq Ch^{4}. \end{aligned}$$

A combination of the estimates $I_{1}$ and $I_{2}$ gives

$$\begin{aligned} II^{2}\leq Ch^{4}+C \int^{t}_{0}(t-s)^{2(\alpha-1)} \bigl\Vert u(s)-u_{h}(s) \bigr\Vert ^{2}_{L_{2(\Omega , H)}}\,ds. \end{aligned}$$

In a similar way as for II, we dominate $III$ by two additional terms:

$$\begin{aligned} III={}& \biggl\Vert \int^{t}_{0}\bar{E}(t-s)g\bigl(u(s)\bigr)- \bar{E}_{h}(t-s)P_{h}g\bigl(u_{h}(s)\bigr)\,dW(s) \biggr\Vert _{L_{2(\Omega, H)}} \\ \leq{}& \biggl\Vert \int^{t}_{0}\bar{E}_{h}(t-s)P_{h} \bigl(g\bigl(u(s)\bigr)-g\bigl(u_{h}(s)\bigr)\bigr)\,dW(s) \biggr\Vert _{L_{2(\Omega , H)}} \\ &{}+\biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)g\bigl(u(s) \bigr)\,dW(s) \biggr\Vert _{L_{2(\Omega, H)}} \\ ={}&I_{3}+I_{4}. \end{aligned}$$

As in an estimate for $I_{1}$, we can get an estimate for $I_{3}$ by using Lemma 3.3 together with Assumption 2.2 and Itô’s isometry:

$$\begin{aligned} I^{2}_{3}&= E \biggl\Vert \int^{t}_{0}\bar{E}_{h}(t-s)P_{h} \bigl(g\bigl(u(s)\bigr)-g\bigl(u_{h}(s)\bigr)\bigr)\,dW(s) \biggr\Vert ^{2} \\ &= \int^{t}_{0}E \bigl\Vert \bar{E}_{h}(t-s)P_{h} \bigl(g\bigl(u(s)\bigr)-g\bigl(u_{h}(s)\bigr)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}}\,ds \\ &= \int^{t}_{0}E\Biggl(\sum _{j=1}^{\infty} \bigl\Vert \bar {E}_{h}(t-s)P_{h} \bigl(g\bigl(u(s)\bigr)-g\bigl(u_{h}(s)\bigr)\bigr)Q^{\frac{1}{2}}e_{j} \bigr\Vert ^{2}\Biggr)\,ds \\ &\leq C \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert g \bigl(u(s)\bigr)-g\bigl(u_{h}(s)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ &\leq C \int^{t}_{0}(t-s)^{2\alpha-2} \bigl\Vert u(s)-u_{h}(s) \bigr\Vert ^{2}_{L_{2(\Omega, H)}}\,ds. \end{aligned}$$

For the estimate of term $I_{4}$, we apply Lemma 3.7, Assumptions 2.2 and 2.3, and Itô’s isometry:

$$\begin{aligned} I^{2}_{4}&= E \biggl\Vert \int^{t}_{0}\bar{F}_{h}(t-s)g\bigl(u(s) \bigr)\,dW(s) \biggr\Vert ^{2} \\ &\leq Ch^{4} \int^{t}_{0}(t-s)^{2\alpha-2}E \bigl\Vert u(s) \bigr\Vert ^{2}_{2}\,ds \\ &\leq Ch^{4} \int^{t}_{0}(t-s)^{2\alpha-2}\,ds \int^{t}_{0}\sup_{0\leq s\leq T}E \bigl\Vert u(s) \bigr\Vert ^{2}_{2}\,ds \\ &\leq Ch^{4}. \end{aligned}$$

In total, we have by $I_{3}$ and $I_{4}$ that

$$\begin{aligned} III^{2}\leq Ch^{4}+C \int^{t}_{0}(t-s)^{2\alpha-2} \bigl\Vert u(s)-u(t) \bigr\Vert ^{2}_{L_{2(\Omega, H)}}\,ds. \end{aligned}$$

Let $\varphi(t)=\|u(s)-u_{h}(s)\|^{2}_{L_{2(\Omega, H)}}$. Since

$$\begin{aligned} &I^{2}\leq Ch^{2+2r} \Vert u_{0} \Vert _{L_{2(\Omega;\dot{H}^{1+r})}}=Ch^{4} \Vert u_{0} \Vert _{L_{2(\Omega;\dot{H}^{2})}}, \\ &II^{2}\leq Ch^{4}+C \int^{t}_{0}(t-s)^{2(\alpha-1)} \bigl\Vert u(s)-u_{h}(s) \bigr\Vert ^{2}_{L_{2(\Omega, H)}}\,ds, \\ &III^{2}\leq Ch^{4}+C \int^{t}_{0}(t-s)^{2\alpha-2} \bigl\Vert u(s)-u(t) \bigr\Vert ^{2}_{L_{2(\Omega , H)}}\,ds, \end{aligned}$$

according to the integral version of Gronwall’s lemma, we get

$$\begin{aligned} \varphi(t)\leq Ch^{4}. \end{aligned}$$

Then we have

$$\begin{aligned} \bigl\Vert u(s)-u_{h}(s) \bigr\Vert _{L_{2(\Omega, H)}}\leq Ch^{2}. \end{aligned}$$

 □

4 Error estimates for fully discrete approximation

In this section, we first introduce the GMMP scheme. Then we give a fully discrete scheme and the corresponding fully discrete error estimate, together with some lemmas, which are significant in the proof of the fully discrete error estimate.

4.1 The GMMP scheme

We denote the time mesh points by $t_{n} = nk, n = 0, 1, \ldots,N$, with a fixed time step $k > 0$, such that $0\leq t_{n} \leq T$ and $k =\frac{T}{N}$. Now let us present the GMMP scheme derived by Gorenflo, Mainardi, Moretti, and Paradisi [33]. The Caputo fractional derivative (when $0<\alpha<1$) can be approximated by

$$\begin{aligned} D^{\alpha}_{t}u(t_{n})& \approx \frac{1}{k^{\alpha}}\sum^{n}_{m=0}w^{\alpha}_{m}\bigl[u(t_{n-m})-u(0)\bigr] \\ &= \frac{1}{k^{\alpha}}\Biggl[\sum^{n}_{m=0}w^{\alpha}_{m}u(t_{n-m})- \phi_{n}u(0)\Biggr], \end{aligned}$$

(4.1)

where

$$\begin{aligned} &w^{\alpha}_{m}=\frac{\Gamma(m-\alpha)}{\Gamma(-\alpha)\Gamma(m+1)}, \end{aligned}$$

(4.2)

$$\begin{aligned} &\phi_{n}=\sum^{n}_{m=0}w^{\alpha}_{m}= \frac{\Gamma(n+1-\alpha)}{\Gamma(1-\alpha )\Gamma(n+1)},\quad n\geq0. \end{aligned}$$

(4.3)

Moreover, $w^{\alpha}_{m}$ and $\phi_{n}$ have the following properties.

Lemma 4.1

([34, 35])

For $\alpha>0,n=1,2,\ldots$ , we have:

• (1) $w^{\alpha}_{0}=1, w^{\alpha}_{n}<0, |w^{\alpha}_{n+1}|\leq|w^{\alpha}_{n}|$, and $0<-\sum^{n}_{m=1}w^{\alpha}_{m}<-\sum^{\infty}_{m=1}w^{\alpha}_{m}=w^{\alpha}_{0}$;

• (2) $\phi_{n}-\phi_{n-1}=w^{\alpha}_{n}<0$, that is, $\phi_{n}<\phi_{n-1}<\phi _{n-2}<\cdots<\phi_{0}=1$.

4.2 Error estimates

By using the GMMP scheme (4.1) we indicate the approximation of $u(t_{n})$ by $u^{n} \approx u(t_{n})$. Then the fully discrete scheme for equation (1.1) can be defined by

$$\begin{aligned} \frac{1}{k^{\alpha}}\Biggl[\sum^{n}_{m=0}w^{\alpha}_{m}u^{n-m}_{h}- \phi _{n}u^{0}_{h}\Biggr]+A_{h}u^{n}_{h}=P_{h}f \bigl(u^{n}_{h}\bigr)+\frac{1}{k} \int ^{t_{n}}_{t_{n-1}}P_{h}g\bigl(u^{n-1}_{h} \bigr)\,dW(s). \end{aligned}$$

(4.4)

Furthermore, we define $R(\lambda,X)=(\lambda I-X)^{-1},\lambda>0$, and $\tilde{E}_{kh}=R(k^{-\alpha},-A_{h})=(k^{-\alpha}I+A_{h})^{-1}$. Then scheme (4.4) can be rewritten as

$$\begin{aligned} u^{n}_{h}={}&k^{-\alpha}\phi_{n} \tilde{E}_{kh}u^{0}_{h}-k^{-\alpha} \tilde{E}_{kh}\sum^{n}_{m=1}w^{\alpha}_{m}u^{n-m}_{h}+ \tilde{E}_{kh}P_{h}f\bigl(u^{n}_{h} \bigr) \\ &{}+\frac{1}{k} \int ^{t_{n}}_{t_{n-1}}\tilde{E}_{kh}P_{h}g \bigl(u^{n-1}_{h}\bigr)\,dW(s). \end{aligned}$$

(4.5)

Besides, the semidiscretized version of mild solution (3.5) at time $t_{n}$ should be shown:

$$\begin{aligned} u_{h}(t_{n})={}&E_{h}(t_{n})P_{h}u_{0}+ \int^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h}f \bigl(u_{h}(s)\bigr)\,ds \\ &{}+ \int ^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h}g \bigl(u_{h}(s)\bigr)\,dW(s). \end{aligned}$$

(4.6)

Now let us introduce and prove some lemmas, which will play an important role later on.

Lemma 4.2

([30])

For any $k>0$ and $h\in(0,1)$, there exists a constant $C>0$ such that

$$\begin{aligned} \Vert \tilde{E}_{kh}v \Vert \leq Ck^{\alpha} \Vert v \Vert , \qquad \Vert \tilde{E}_{kh}P_{h}v \Vert \leq Ck^{\alpha} \Vert v \Vert , \quad \forall v\in H. \end{aligned}$$

Lemma 4.3

For any $t>0$ and $p,q\in[-1,1]$ such that $0\leq p-q<2$, we have

$$\begin{aligned} \bigl\Vert E_{h}(t)v_{h}-v_{h} \bigr\Vert _{p}\leq Ct^{\frac{(2+q-p)\alpha}{2}} \Vert v_{h} \Vert _{q+2}, \quad \forall v_{h}\in V_{h}. \end{aligned}$$

Proof

The definition of $E_{h}(t)v_{h}$ in (3.3) and Lemma 2.1 yield

$$\begin{aligned} &\bigl\Vert E_{h}(t)v_{h}-v_{h} \bigr\Vert ^{2}_{p}\\ &\quad=\sum_{j=1}^{N} \bigl(\lambda^{h}_{j}\bigr)^{p} \bigl(1-E_{\alpha ,1}\bigl(-\lambda^{h}_{j}t^{\alpha}\bigr)\bigr)^{2}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad= t^{(q-p)\alpha}\sum_{j=1}^{N}\bigl( \lambda^{h}_{j}t^{\alpha}\bigr)^{p-q} \bigl(1-E_{\alpha ,1}\bigl(-\lambda^{h}_{j}t^{\alpha}\bigr)\bigr)^{2}\bigl(\lambda^{h}_{j} \bigr)^{q}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad= t^{(q-p)\alpha}\sum_{j=1}^{N}\bigl( \lambda^{h}_{j}t^{\alpha}\bigr)^{p-q}\biggl( \int ^{t}_{0}\lambda^{h}_{j}s^{\alpha-1}E_{\alpha,\alpha} \bigl(-\lambda^{h}_{j}s^{\alpha}\bigr)\,ds \biggr)^{2}\bigl(\lambda^{h}_{j} \bigr)^{q}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad\leq Ct^{(q-p)\alpha}\sum_{j=1}^{N} \bigl(\lambda^{h}_{j}t^{\alpha}\bigr)^{p-q} \biggl( \int ^{t}_{0}\lambda^{h}_{j}s^{\alpha-1} \frac{1}{1+(\lambda^{h}_{j}s^{\alpha })^{2}}\,ds\biggr)^{2}\bigl(\lambda^{h}_{j} \bigr)^{q}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad= Ct^{(q-p)\alpha}\sum_{j=1}^{N}\bigl( \lambda^{h}_{j}t^{\alpha}\bigr)^{p-q}\biggl( \int ^{t}_{0}\frac{\lambda^{h}_{j}s^{\alpha-1}}{(\lambda^{h}_{j}s^{\alpha})^{(\frac {p-q}{2})}}\frac{(\lambda^{h}_{j}s^{\alpha})^{(\frac{p-q}{2})}}{1+(\lambda ^{h}_{j}s^{\alpha})^{2}}\,ds \biggr)^{2}\bigl(\lambda^{h}_{j} \bigr)^{q}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad\leq Ct^{(q-p)\alpha}\sum_{j=1}^{N} \bigl(\lambda^{h}_{j}t^{\alpha}\bigr)^{p-q} \biggl( \int ^{t}_{0}\frac{\lambda^{h}_{j}s^{\alpha-1}}{(\lambda^{h}_{j}s^{\alpha})^{(\frac {p-q}{2})}}\,ds \biggr)^{2}\bigl(\lambda^{h}_{j} \bigr)^{q}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad= Ct^{(q-p)\alpha}\sum_{j=1}^{N} \bigl(t^{\alpha}\bigr)^{p-q}\biggl( \int^{t}_{0}\frac {s^{\alpha-1}}{(s^{\alpha})^{\frac{p-q}{2}}}\,ds \biggr)^{2}\bigl(\lambda ^{h}_{j} \bigr)^{q+2}\bigl(v_{h},\varphi^{h}_{j} \bigr)^{2} \\ &\quad\leq Ct^{(q-p)\alpha}\sum_{j=1}^{N} \bigl(t^{\alpha}\bigr)^{p-q}\cdot t^{2\alpha -(p-q)\alpha}\bigl( \lambda^{h}_{j}\bigr)^{q+2}\bigl(v_{h}, \varphi^{h}_{j}\bigr)^{2} \\ &\quad=Ct^{(2+q-p)\alpha} \Vert v_{h} \Vert _{q+2}. \end{aligned}$$

 □

Lemma 4.4

([30])

For any $\lambda> 0$ and $\mu\in R$, there exists a constant C such that

$$\begin{aligned} \bigl\Vert \bigl[\mu R(\lambda,A_{h})-I\bigr]P_{h}v \bigr\Vert \leq C\lambda^{-1} \Vert v \Vert . \end{aligned}$$

Based on the previous discussion, we are ready to prove the error estimates for the fully discrete approximation.

Theorem 4.1

Let $u^{n}_{h}$ and $u(t_{n})$ be solutions of (4.4) and (1.1), respectively, for $t\geq0$, $\alpha\in(\frac{1}{2},1)$, and $u_{0}\in L_{2}(\Omega, \dot{H}^{s}), s\in[0,2]$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \bigl\Vert u(t_{n})-u^{n}_{h} \bigr\Vert ^{2}_{L_{2(\Omega;H)}}\leq C\bigl[k^{2\alpha}+h^{4}\bigr]. \end{aligned}$$

Proof

By the triangle inequality we have

$$\begin{aligned} \bigl\Vert u(t_{n})-u^{n}_{h} \bigr\Vert _{L_{2(\Omega;H)}}&\leq\bigl\Vert u(t_{n})-u_{h}(t_{n}) \bigr\Vert _{L_{2(\Omega ;H)}}+ \bigl\Vert u_{h}(t_{n})-u^{n}_{h} \bigr\Vert _{L_{2(\Omega;H)}} \\ &= \bigl\Vert \rho^{n} \bigr\Vert _{L_{2(\Omega;H)}}+ \bigl\Vert \theta^{n} \bigr\Vert _{L_{2(\Omega;H)}}. \end{aligned}$$

Since we have estimated the error of $\|\rho^{n}\|_{L_{2(\Omega;H)}}$ in Theorem 3.1, we only need to estimate $\|\theta^{n}\| _{L_{2(\Omega;H)}}$. Using equations (4.6) and (4.5), we obtain

$$\begin{aligned} \bigl\Vert \theta^{n} \bigr\Vert _{L_{2(\Omega;H)}}\leq{}& \bigl\Vert E_{h}(t_{n})P_{h}u_{0}-k^{-\alpha} \phi _{n}\tilde{E}_{kh}P_{h}u_{0} \bigr\Vert _{L_{2(\Omega;H)}} \\ &{}+ \Biggl\Vert -k^{-\alpha}\tilde{E}_{k h}\sum _{m=1}^{n}w^{\alpha}_{m}u^{n-m}_{h} \Biggr\Vert _{L_{2(\Omega;H)}} \\ &{}+ \biggl\Vert \int^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h}f \bigl(u_{h}(s)\bigr)\,ds \biggr\Vert _{L_{2(\Omega;H)}} \\ &{}+ \bigl\Vert -\tilde{E}_{k h}P_{h}f\bigl(u^{n}_{h} \bigr) \bigr\Vert _{L_{2(\Omega;H)}} \\ &{}+ \biggl\Vert \int^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h}g \bigl(u_{h}(s)\bigr)\,dW(s) \biggr\Vert _{L_{2(\Omega;H)}} \\ &{}+ \biggl\Vert -\frac{1}{k} \int^{t_{n}}_{t_{n-1}}\tilde{E}_{k h}P_{h}g \bigl(u^{n-1}_{h}\bigr)\,dW(s) \biggr\Vert _{L_{2(\Omega;H)}} \\ ={}&I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}. \end{aligned}$$

For $I_{1}$, by the triangle inequality, we separate $I^{2}_{1}$ into two additional terms:

$$\begin{aligned} I^{2}_{1}&= E \bigl\Vert E_{h}(t_{n})P_{h}u_{0}-k^{-\alpha} \phi_{n}\tilde{E}_{k h}P_{h}u_{0} \bigr\Vert ^{2} \\ &= E \bigl\Vert \bigl[E_{h}(t_{n})P_{h}u_{0}-P_{h}u_{0} \bigr]+\bigl[P_{h}u_{0}-k^{-\alpha}\phi_{n} \tilde{E}_{k h}P_{h}u_{0}\bigr] \bigr\Vert ^{2} \\ &\leq 2E \bigl\Vert \bigl[E_{h}(t_{n})P_{h}u_{0}-P_{h}u_{0} \bigr] \bigr\Vert ^{2}+2E \bigl\Vert \bigl[P_{h}u_{0}-k^{-\alpha} \phi _{n}\tilde{E}_{k h}P_{h}u_{0}\bigr] \bigr\Vert ^{2} \\ &= I_{11}+I_{12}. \end{aligned}$$

For $I_{11}$, by Lemma 4.3 with $p=q=0$ we get

$$\begin{aligned} I_{11}&= 2E \bigl\Vert \bigl[E_{h}(t_{n})P_{h}u_{0}-P_{h}u_{0} \bigr] \bigr\Vert ^{2}\leq Ct^{2\alpha}_{n}E \Vert P_{h}Au_{0} \Vert ^{2} \\ &\leq Ck^{2\alpha}E \Vert u_{0} \Vert ^{2}_{2}. \end{aligned}$$

For $I_{12}$, setting $\mu=k^{-\alpha}\phi_{n}$ and using Lemma 4.4, we have

$$\begin{aligned} I_{12}&= 2E \Vert \mu\tilde{E}_{k h}P_{h}u_{0}-P_{h}u_{0} \Vert ^{2} \\ &= 2E \bigl\Vert \bigl[\mu R\bigl(k^{-\alpha},A_{h}\bigr)-I \bigr]P_{h}u_{0} \bigr\Vert ^{2} \\ &\leq Ck^{2\alpha}E \Vert u_{0} \Vert ^{2}. \end{aligned}$$

By Lemma 4.1 we have $\sum_{m=1}^{n}|w^{\alpha}_{m}|< w^{\alpha}_{0}=1$. Together with Lemmas 4.2 and 3.4, we obtain

$$\begin{aligned} I^{2}_{2}&= E \Biggl\Vert k^{-\alpha} \tilde{E}_{k h}\sum_{m=1}^{n}w^{\alpha}_{m}u^{n-m}_{h} \Biggr\Vert ^{2} \\ &= E \Biggl\Vert k^{-\alpha}\tilde{E}_{k h}\sum _{m=1}^{n}w^{\alpha}_{m}\bigl[ \bigl(u^{n-m}_{h}-u_{h}(t_{n-m}) \bigr)+u_{h}(t_{n-m})\bigr] \Biggr\Vert ^{2} \\ &\leq C\sum_{m=1}^{n}E \bigl\Vert \theta^{n-m} \bigr\Vert ^{2}+C\sum _{m=1}^{n} E \bigl\Vert \tilde {E}_{kh}u_{h}(t_{n-m}) \bigr\Vert ^{2}) \\ &\leq C\sum_{m=1}^{n}E \bigl\Vert \theta^{n-m} \bigr\Vert ^{2}+Ck^{2\alpha}\bigl(E \Vert u_{0} \Vert ^{2}\bigr). \end{aligned}$$

The term $I_{3}$ is estimated by applying Lemma 3.2, Assumption 2.1, and Lemma 3.4: for $0< t_{n}\leq T=Nk$, we get

$$\begin{aligned} I^{2}_{3}&= E \biggl\Vert \int^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h} f\bigl(u_{h}(s)\bigr)\,ds \biggr\Vert ^{2} \\ &\leq \int^{t_{n}}_{0}E \bigl\Vert \bar{E}_{h}(t_{n}-s)P_{h} f\bigl(u_{h}(s)\bigr) \bigr\Vert ^{2}\,ds \\ &\leq C \int^{t_{n}}_{0}(t_{n}-s)^{2\alpha-2}E \bigl\Vert u_{h}(s) \bigr\Vert ^{2}\,ds \\ &\leq Ck^{2\alpha}E \Vert u_{0} \Vert ^{2}. \end{aligned}$$

By Lemma 4.2, Lemma 3.4, and Assumption 2.1 we get the following estimate for $I_{4}$:

$$\begin{aligned} I^{2}_{4}&\leq 2E \bigl\Vert \tilde{E}_{k h}P_{h} \bigl(f\bigl(u^{n}_{h}\bigr)-f\bigl(u_{h}(t_{n}) \bigr)\bigr) \bigr\Vert ^{2}+2E \bigl\Vert \tilde {E}_{k h}P_{h}f\bigl(u_{h}(t_{n})\bigr) \bigr\Vert ^{2} \\ &\leq Ck^{2\alpha}E \bigl\Vert f\bigl(u^{n}_{h} \bigr)-f\bigl(u_{h}(t_{n})\bigr) \bigr\Vert ^{2}+Ck^{2\alpha}E \bigl\Vert f\bigl(u_{h}(t_{n}) \bigr) \bigr\Vert ^{2} \\ &\leq Ck^{2\alpha}E \bigl\Vert u^{n}_{h}-u_{h}(t_{n}) \bigr\Vert ^{2}+Ck^{2\alpha}E \bigl\Vert u_{h}(t_{n}) \bigr\Vert ^{2} \\ &\leq Ck^{2\alpha}E \bigl\Vert \theta^{n} \bigr\Vert ^{2}+Ck^{2\alpha}E \Vert u_{0} \Vert ^{2}. \end{aligned}$$

For $I_{5}$, by Lemma 3.2, Assumption 2.2, Lemma 3.4, and Itô’s isometry, we obtain

$$\begin{aligned} I^{2}_{5}&= E \biggl\Vert \int^{t_{n}}_{0}\bar{E}_{h}(t_{n}-s)P_{h} g\bigl(u_{h}(s)\bigr)\,dW(s) \biggr\Vert ^{2} \\ &= \int^{t_{n}}_{0}E \bigl\Vert \bar{E}_{h}(t_{n}-s)P_{h} g\bigl(u_{h}(s)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}}\,ds \\ &\leq C \int^{t_{n}}_{0}(t_{n}-s)^{2\alpha-2}E \bigl\Vert u_{h}(s) \bigr\Vert ^{2}\,ds \\ &\leq Ck^{2\alpha}\bigl(E \Vert u_{0} \Vert ^{2} \bigr). \end{aligned}$$

For $I_{6}$, by Lemma 4.2, Lemma 3.4, Assumption 2.2, and Itô’s isometry we have

$$\begin{aligned} I_{6}^{2}={}&E \biggl\Vert -\frac{1}{k} \int^{t_{n}}_{t_{n-1}}\tilde{E}_{k h}P_{h}g \bigl(u^{n-1}_{h}\bigr)\,dW(s) \biggr\Vert ^{2} \\ \leq{}&2E \biggl\Vert \frac{1}{k} \int^{t_{n}}_{t_{n-1}}\tilde{E}_{k h}P_{h} \bigl(g\bigl(u^{n-1}_{h}\bigr)-g\bigl(u_{h}(t_{n-1}) \bigr)\bigr)\,dW(s) \biggr\Vert ^{2} \\ &{}+2E \biggl\Vert \frac{1}{k} \int^{t_{n}}_{t_{n-1}}\tilde{E}_{k h}P_{h}g \bigl(u_{h}(t_{n-1})\bigr)\,dW(s) \biggr\Vert ^{2} \\ ={}&\frac{2}{k} \int^{t_{n}}_{t_{n-1}}E \bigl\Vert \tilde{E}_{k h}P_{h} \bigl(g\bigl(u^{n-1}_{h}\bigr)-g\bigl(u_{h}(t_{n-1}) \bigr)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}}\,ds \\ &{}+\frac{2}{k} \int^{t_{n}}_{t_{n-1}}E \bigl\Vert \tilde{E}_{k h}P_{h}g \bigl(u_{h}(t_{n-1})\bigr) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds \\ \leq{}&Ck^{2\alpha}E \bigl\Vert \theta^{n-1} \bigr\Vert ^{2}+Ck^{2\alpha}\bigl(E \Vert u_{0} \Vert ^{2}\bigr). \end{aligned}$$

Therefore, coming back to $\|\theta^{n}\|_{L_{2(\Omega;H)}}$, combining $I_{1}, I_{2}, I_{3}, I_{4}, I_{5}$, and $I_{6}$ and applying a discrete version of Gronwall’s lemma, we have

$$\begin{aligned} \bigl\Vert \theta^{n} \bigr\Vert ^{2}_{L_{2(\Omega;H)}}\leq Ck^{2\alpha}. \end{aligned}$$

By the triangle inequality we obtain

$$\begin{aligned} \bigl\Vert u(t_{n})-u^{n}_{h} \bigr\Vert ^{2}_{L_{2(\Omega;H)}}\leq \bigl\Vert \theta^{n} \bigr\Vert ^{2}_{L_{2(\Omega ;H)}}+ \bigl\Vert \rho^{n} \bigr\Vert ^{2}_{L_{2(\Omega;H)}}\leq C\bigl(k^{2\alpha}+h^{4}\bigr), \end{aligned}$$

which completes the proof. □

5 Conclusions and discussions

In this paper, we have studied semidiscrete and fully discrete schemes for nonlinear time-fractional SPDEs. The semidiscrete scheme employs a standard Galerkin finite element method, and the time direction of the fully discrete scheme is based on the GMMP scheme. The strong convergence error estimates for the semidiscrete and fully discrete schemes in the $L_{2}$-norm are demonstrated. However, there are several possible extensions of the work. First, we only consider the initial value condition in our given problem; the complex boundary condition in our future study will be discussed. Second, numerical investigations on time-space fractional SPDEs are an interesting direction for our future research.