Milstein Scheme for a Stochastic Semilinear Subdiffusion Equation Driven by Fractionally Integrated Multiplicative Noise

Abstract

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties of the mild solution are derived using the semigroup approach. For spatial discretization, the standard Galerkin finite element method is employed, while the Grünwald–Letnikov method is used for time discretization. The Milstein scheme is utilized to approximate the multiplicative noise. For sufficiently smooth noise, the proposed scheme achieves the temporal strong convergence order of $O({\tau }^{\alpha })$, $\alpha \in (0,1)$. Numerical experiments are presented to verify that the computational results are consistent with the theoretical predictions.

1. Introduction

This paper considers the Milstein scheme for solving the following stochastic semilinear subdiffusion equation driven by the fractionally integrated multiplicative noise, with $0<\alpha <1$:

$${}_0{}^{C}D_t^{\alpha }u(t)=-Au(t)+F(u(t))+{}_0{}^{R}D_t^{\alpha -1}G(u(t)){\displaystyle \frac{dW(t)}{dt}},\mathrm{for}0<t\le T,\mathrm{with}u(0)=u_0,$$

(1)

where $A=-\Delta$ with $\mathcal{D}(A)=H_0^1(D)\cap H^2(D)$ and $\Delta$ is the Laplacian. Here, $D\subset {\mathbb{R}}^d,$ where $d=1,2,3,$ is a regular domain with a $C^1$-boundary. Here, ${}_0{}^{C}D_t^{\alpha }v(t)$ denotes the left-sided Caputo fractional derivative with order $\alpha \in (0,1)$,

$${}_0{}^{C}D_t^{\alpha }v(t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{v}^{\prime }(s)ds,$$

and ${}_0{}^{R}D_t^{\alpha -1}v(t)$ is the Riemann–Liouville fractional integral of order $1-\alpha \in (0,1)$,

$${}_0{}^{R}D_t^{\alpha -1}v(t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }vs.(s)ds.$$

The term ${\displaystyle \frac{dW(t)}{dt}}$ denotes the noise on a complete filtered probability space $(\Omega ,\mathcal{F},\mathcal{P},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0})$; see Section 2 for details. We assume that the multiplicative noise term $G(u(t)){\displaystyle \frac{dW(t)}{dt}}$ in (1) is commutative ([1,2]). In this case, the Lévy area, which typically arises in Milstein-type time discretization schemes, vanishes. The nonlinear term $F:\mathbb{R}\to \mathbb{R}$ is a real-valued function, and the initial value $u_0$ is a given function.

Stochastic subdiffusion equations are extensively employed to model anomalous diffusion phenomena, including those observed in highly heterogeneous aquifers [3], random walks [4], underground environmental issues [5], and thermal diffusion in media with fractional geometries [6], among others. The fractionally integrated noise term, ${}_0{}^{R}D_t^{\alpha -1}G(u(t)){\displaystyle \frac{dW(t)}{dt}}$, captures the random effects on particle motion in media with memory, accounting for behaviors such as particle sticking and trapping [7] or the dependence of internal energy on past random fluctuations.

To clarify the physical motivation behind model (1), we briefly introduce a representative application; see [7] for further details. Let $v(t,x)$, $P(t,x)$, and $\overrightarrow{S}(t,x)$ denote the body temperature, energy, and flux density, respectively. In a homogeneous medium, with constants $a_1>0$ and $a_2>0$, the following relations are satisfied:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial P(t,x)}{\partial t}}=-div\overrightarrow{S}(t,x),\hfill \\ P(t,x)=a_{1}v(t,x),\hfill \\ \overrightarrow{S}(t,x)=-a_2\nabla v(t,x).\hfill \end{array}\right.$$

From these equations, we obtain the heat equation:

$$a_1{\displaystyle \frac{\partial v(t,x)}{\partial t}}=a_2\Delta v(t,x).$$

In practical inhomogeneous media, for example, in the case of heat conduction in materials with thermal memory subjected to multiplicative noise, the energy term is expressed as follows, with $\dot{W}(r)={\displaystyle \frac{dW(r)}{dr}}$ and $\alpha \in (0,1)$:

$$P(t,x)={\int }_0^t{h}_1(t-r)v(r,x)dr+{\int }_0^t{h}_2(t-r)(a_2(v(r,x))+g(v(r,x))\dot{W}(r,x))dr,$$

(2)

where $h_1(t)=\Gamma {(1-\alpha )}^{-1}{t}^{-\alpha }$, $h_2(t)=-\Gamma (1-\alpha )\Gamma {(2-\alpha )}^{-1}t{h}_1(t)=-\Gamma {(2-\alpha )}^{-1}{t}^{1-\alpha }$. Differentiating (2), we obtain the model (1):

$$\begin{array}{cc}\hfill -div\overrightarrow{S}& ={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{(t-r)}^{-\alpha }v(r,x)dr\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma (2-\alpha )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{(t-r)}^{1-\alpha }(a_2(v(r,x))+g(v(r,x))\dot{W}(r,x))dr\hfill \\ \hfill & ={}_0{}^{C}D_t^{\alpha }v(t,x)-{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{\int }_0^{\tau }{(\tau -r)}^{1-\alpha }(a_2(v(r,x))+g(v(r,x))\dot{W}(r,x))drd\tau \hfill \\ \hfill & ={}_0{}^{C}D_t^{\alpha }v(t,x)-{}_0{}^{R}D_t^{\alpha -1}(a_2(v(t,x))+g(v(t,x))\dot{W}(t,x)).\hfill \end{array}$$

(3)

This equation characterizes heat conduction in a non-homogeneous medium influenced by the fractional-order dynamics driven by the integrated multiplicative noise. We note that other types of fractional derivatives, such as the Caputo–Fabrizio and Atangana–Baleanu derivatives, also appear in the literature. The study of (1) with these alternative fractional derivatives will be the subject of our future work.

Let us briefly recall some theoretical results related to (1). Anh et al. [8] discussed sufficient conditions for the existence of a solution (in the mean-square sense) and established the temporal and spatial Hölder continuities of the solution to (1) for $F=0$ with additive noise. Mijena and Nane [9] proved the existence and uniqueness of mild solutions to (1) defined on the whole domain $\mathbb{R}$ with $F=0$ and provided conditions ensuring the continuity of the solution. Furthermore, Mijena and Nane [10] showed that the absolute moments of the solution to (1) with $F=0$ (defined on $\mathbb{R}$) grow exponentially and that the distances to the origin of the farthest high peaks of those moments increase linearly with time. Liu et al. [11] studied the existence and uniqueness of solutions to (1) involving fairly general quasi-linear elliptic operators; see also [12] for additional analytic results. Chen [13] analyzed the moments, Hölder continuity, and intermittency of the solution for the one-dimensional nonlinear stochastic subdiffusion equation. Chen et al. [14] studied the existence and uniqueness of solutions to (1) with $F=0$ defined on the entire domain $\mathbb{R}$. Kang et al. [15] investigated the existence and uniqueness of mild solutions to (1) with additive noise, assuming that F satisfies a global Lipschitz condition.

Numerical methods for solving stochastic linear and semilinear subdiffusion equations driven by fractionally integrated noise have been extensively studied in the literature. For instance, Jin et al. [16] proposed a fully discrete numerical scheme for the linear stochastic subdiffusion equation with fractionally integrated additive Gaussian noise, employing the Galerkin finite element method for spatial discretization and the Grünwald–Letnikov method for temporal discretization; see also [17,18]. Wu et al. [19] applied the $L^1$ scheme in combination with the finite element method to solve the linear stochastic subdiffusion equation driven by fractionally integrated additive noise, and established optimal convergence rates for the fully discrete scheme; see also [20]. Hu et al. [21] investigated the weak convergence of stochastic subdiffusion problems using the $L^1$ scheme for time discretization. Noupelah et al. [22] studied strong convergence rates for a fully discrete spatio-temporal scheme for stochastic semilinear subdiffusion, using a fractional exponential integrator for temporal discretization and the finite element method for spatial discretization. Karaa et al. [23] analyzed strong approximations for a stochastic time-fractional Allen–Cahn equation with additive noise, employing the standard finite element method for spatial discretization, the Grünwald–Letnikov method for temporal discretization, and the Euler scheme for approximating the noise term.

Although a few numerical methods have been developed for the stochastic subdiffusion equation, to the best of our knowledge, there are no existing higher-order numerical methods for solving the stochastic subdiffusion equation driven by fractionally integrated multiplicative noise. Recently, Qiao et al. [24] studied a numerical scheme for the classical Allen–Cahn equation with multiplicative noise. They employed the Galerkin finite element method for spatial discretization and the Milstein scheme for time integration. It was shown that, for sufficiently smooth noise, the Milstein scheme achieves a higher convergence order—twice that of the standard Euler scheme. Inspired by the approach in [24], we propose an efficient Milstein scheme for solving a semilinear stochastic subdiffusion equation driven by fractionally integrated multiplicative noise. The standard Galerkin finite element method is used for spatial discretization, while the time-fractional derivative is approximated using the Grünwald–Letnikov scheme. The multiplicative noise is approximated via the Milstein scheme. The optimal error estimates for the proposed fully discrete scheme are established using a discrete convolution argument.

Let us introduce the main results in this paper below. Let $N>0$ be a positive integer, $0=t_0<t_1<\dots <t_N=T$ the time partition of $[0,T]$, and $\tau$ the time step size. Let $u_h^j,j=0,1,\dots ,N$ be the approximate solutions of $u(t_j)$ in the finite element space $S_h$. Let $P_h:H=L^2(D)\to S_h$ and $A_h:S_h\to S_h$ denote the $L^2$-projection operator and the discrete Laplacian operator, respectively. The Milstein scheme is defined as follows (see Section 4 for detail):

$${\tau }^{-\alpha }\sum _{j=0}^n{\omega }_{n-j}^{(\alpha )}(u_h^j-u_h^0)+A_h{u}_h^n=P_{h}F(u_h^n)+{\tau }^{1-\alpha }\sum _{j=0}^n{\omega }_{n-j}^{(\alpha -1)}{P}_h{g}^j,n\ge 1,$$

where ${\omega }_j^{(\alpha )}$ are some suitable weights, $g^j={\rho }_1^j+{\rho }_2^j$, and $j=1,2,\dots ,n$. Here,

$${\rho }_1^j=G(u_h^{j-1}){\displaystyle \frac{W(t_j)-W(t_{j-1})}{\tau }},{\rho }_2^j=G^{\prime }(u_h^{j-1})G(u_h^{j-1}){\int }_{t_{j-1}}^{t_j}{\displaystyle \frac{W(s)-W(t_{j-1})}{\tau }}dW(s).$$

With the application of the discrete Laplace transform, the solution $u_h^n$ takes the following form:

$$\begin{array}{cc}\hfill u_h^n& ={\overline{u}}_h^n+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{R}_{n-j}{P}_{h}F(u_h^{j+1})ds+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_{h}G(u_h^j)dW(s)\hfill \\ \hfill & +\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_h{G}^{\prime }(u_h^j)G(u_h^j)(W(s)-W(t_j))dW(s),\hfill \end{array}$$

(4)

where ${\overline{u}}_h^n$ denotes the approximate solution of the corresponding homogeneous problem. Here, $R_j$ and ${\overline{R}}_j$ (where $j=1,2,\dots ,n$) are some suitable discrete operators specified in Section 4.

Let u and $u_h^n$ be the solutions of (1) and (4), respectively. We obtain the following error estimates: with $1/2<\alpha <1$, $1\le \sigma \le 2$, $0\le \kappa \le 2$, and $0\le \sigma -\kappa \le 1$,

$$\begin{array}{cc}\hfill \parallel u(t_n)-u_h^n{\parallel }_{L^2(\Omega ,H)}& \le C\tau t_n^{-1+{\displaystyle \frac{\sigma \alpha }{2}}}\parallel u_0{\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}\hfill \\ \hfill & +C{h}^{\{2-\epsilon ,\mu \}}+C{\tau }^{min\{{\displaystyle \frac{\alpha \sigma }{2}},\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},\alpha (\sigma -1),\alpha (\sigma -\kappa )+1-\alpha \}},\hfill \end{array}$$

where $0<\mu \le 2$ and $\alpha (\mu -(\sigma -\kappa ))\le 1.$

Under the assumption that the noise is sufficiently smooth, e.g., $\sigma =2$, $\sigma -\kappa =1$, and $u_0\in L^2(\Omega ,{\dot{H}}^2)$, we have the following error estimate, with $\epsilon >0$:

$$\parallel u(t_n)-u_h^n{\parallel }_{L^2(\Omega ,H)}\le C{h}^{2-\epsilon }+C{\tau }^{min\{\alpha ,{\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{1}{2}}\}}=C({\tau }^{\alpha }+h^{2-\epsilon }).$$

The main contributions of this paper are summarized as follows:

• The existence and uniqueness of solutions to stochastic semilinear subdiffusion equations driven by integrated multiplicative noise are established via the Banach fixed point theorem.
• The temporal and spatial regularity of mild solutions to the stochastic semilinear subdiffusion equations driven by integrated multiplicative noise are rigorously analyzed.
• A higher-order fully discrete numerical scheme is developed for solving the stochastic semilinear subdiffusion equations with integrated multiplicative noise. Specifically, the standard Galerkin finite element method is employed for spatial discretization, the Grünwald–Letnikov method is used for time discretization, and the Milstein scheme is adopted to approximate the multiplicative noise.
• Error estimates for the proposed fully discrete scheme are derived using the semigroup approach.

The paper is organized as follows. In Section 2, we present the formulation of the mild solution. The existence, uniqueness, and regularity of the solution in both space and time are established in Section 3. Section 4 introduces the fully discrete scheme, which combines the finite element method for spatial discretization, the Grünwald–Letnikov scheme for temporal approximation, and the Milstein scheme for handling the multiplicative noise. Detailed error estimates for the fully discrete scheme are provided in Section 5. In Section 6, the numerical simulations are presented to validate the theoretical results.

Throughout the paper, C denotes a generic positive constant, which may differ at each occurrence and may depend on T and $\alpha$ but remains independent of the step sizes $\tau$ and h.

2. Preliminaries

Let $(·,·)$, $\parallel ·\parallel$ denote the inner product and the norm in $H=L^2(D)$. Let ${\{{\lambda }_j,e_j\}}_{j=1}^{\infty }$ be the eigenpairs of the operator A. Denote as ${\dot{H}}^r$, $\forall r\in \mathbb{R}$, the space

$${\dot{H}}^r:=D(A^{{\displaystyle \frac{r}{2}}})={\{v\in H:|v|}_r^2=\parallel A^{{\displaystyle \frac{r}{2}}}v{\parallel }^2=\sum _{j=1}^{\infty }{\lambda }_j^r{(v,e_j)}^2<\infty \}.$$

Let $\mathbb{E}$ stand for the expectation in the probability space $(\Omega ,\mathcal{F},\mathcal{P},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0})$. We assume the noise ${\displaystyle \frac{dW(t)}{dt}}$ can be expressed as follows:

$${\displaystyle \frac{dW(t)}{dt}}:=\sum _{l=1}^{\infty }{\gamma }_l^{1/2}{\dot{\beta }}_l(t)e_l,$$

(5)

where ${\beta }_l(t),l=1,2,\dots$ are the real-valued mutually independent Brownian motions and ${\{{\gamma }_j,e_j\}}_{j=1}^{\infty }$ are the eigenpairs of the covariance operator Q in H.

Define as ${\mathcal{L}}_2^0:=HS(Q^{{\displaystyle \frac{1}{2}}}(H),H)$ ([25,26]) the space of Hilbert–Schmidt operators from $Q^{{\displaystyle \frac{1}{2}}}(H)$ to H equipped with the inner product $\langle ·,·\rangle$ and norm ${\parallel ·\parallel }_{{\mathcal{L}}_2^0}$:

$$\langle {\varphi }_1,{\varphi }_2\rangle :=\sum _{j=1}^{\infty }({\varphi }_1{Q}^{{\displaystyle \frac{1}{2}}}{e}_j,{\varphi }_2{Q}^{{\displaystyle \frac{1}{2}}}{e}_j){,\parallel \varphi \parallel }_{{\mathcal{L}}_2^0}^2:=\sum _{j=1}^{\infty }{\parallel \varphi Q^{{\displaystyle \frac{1}{2}}}{e}_j\parallel }^2<\infty ,\mathrm{for}{\varphi }_1,{\varphi }_2,\varphi \in {\mathcal{L}}_2^0.$$

Let $HS(Q^{{\displaystyle \frac{1}{2}}}(H),{\mathcal{L}}_2^0)$ denote the Hilbert–Schmidt operator space with norm ${\parallel ·\parallel }_{HS(Q^{{\displaystyle \frac{1}{2}}}(H),{\mathcal{L}}_2^0)}$ defined by

$${\parallel \varphi \parallel }_{HS(Q^{{\displaystyle \frac{1}{2}}}(H),{\mathcal{L}}_2^0)}^2:=\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\gamma }_i{\gamma }_j\parallel \varphi e_i{e}_j\parallel <\infty ,\varphi \in HS(Q^{{\displaystyle \frac{1}{2}}}(H),{\mathcal{L}}_2^0).$$

We also use the operator space ${\mathcal{L}}^{\otimes 2}(H,{\mathcal{L}}_2^0)$, in which the norm is defined by

$${\parallel \varphi \parallel }_{{\mathcal{L}}^{\otimes 2}(H,{\mathcal{L}}_2^0)}:=\underset{v_1\in H,v_2\in H}{sup}{\displaystyle \frac{\parallel \varphi v_1{v}_2{\parallel }_{{\mathcal{L}}_2^0}}{\parallel v_1\parallel \parallel v_2\parallel }}<\infty ,\varphi \in {\mathcal{L}}^{\otimes 2}(H,{\mathcal{L}}_2^0).$$

Assumption 1. 

The elliptic operator A satisfies the following resolvent estimate, with $\theta \in (\pi /2,\pi )$:

$${\parallel (zI+A)}^{-1}{\parallel \le C\left|z\right|}^{-1}forz\in {\Sigma }_{\theta }=\{z\ne 0:|argz|<\theta \},$$

(6)

which implies that, with $0<\alpha <1$,

$$\parallel {(z^{\alpha }I+A)}^{-1}{\parallel \le C\left|z\right|}^{-\alpha },\forall z\in {\Sigma }_{\theta }=\{z\ne 0:|argz|<\theta \}.$$

(7)

Assumption 2 

([27]). For the nonlinear term $F(u)$, we assume that there exist $1\le \nu <2$ and $1\le \theta <2$ such that

$$\begin{array}{cc}\hfill \parallel F(u)-F(v)\parallel \le C\parallel u-v\parallel ,& u,v\in H,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \parallel F(u)\parallel \le C(1+\parallel u\parallel ),& u\in H,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \parallel F^{\prime }(u)v\parallel \le C\parallel v\parallel ,& u,v\in H,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill |F^{\prime }{(u)v|}_{-\nu }\le {C(1+|u|}_{{\sigma }_1}{)|v|}_{-{\sigma }_1},& u\in {\dot{H}}^{{\sigma }_1},v\in {\dot{H}}^{-{\sigma }_1},0\le {\sigma }_1<\nu ,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill |F^{\prime \prime }(u)(v_1,v_2){|}_{-\theta }\le C\parallel v_1\parallel \parallel v_2\parallel ,& v_1,v_2\in H.\hfill \end{array}$$

(12)

Assumption 3. 

For $1\le \sigma \le 2,0\le \kappa \le 2,0\le \sigma -\kappa \le 1$, we assume

$$\parallel A^{{\displaystyle \frac{\sigma -\kappa }{2}}}{G(u)\parallel }_{{\mathcal{L}}_2^0}\le {C(1+|u|}_{\sigma -\kappa }),$$

(13)

where κ is defined by [16],

$$\kappa :=\left\{\begin{array}{cc}\hfill & 2,0<\alpha <{\displaystyle \frac{1}{2}},\hfill \\ \hfill & {\displaystyle \frac{1-\epsilon }{\alpha }},{\displaystyle \frac{1}{2}}\le \alpha <1,0<\epsilon <1,\hfill \end{array}\right.$$

(14)

which ensures ${\int }_0^t\parallel A^{{\displaystyle \frac{\kappa }{2}}}E(s){\parallel }^{2}ds<\infty$. Here, $E(t)$ represents the Mittag-Leffler function as defined in (20). Moreover, the functions $G(u)$, $G^{\prime }(u)$, and $G^{\prime \prime }(u)$ satisfy certain properties that play a crucial role in establishing the strong convergence rates of the Milstein-type scheme (see Assumption 5.2 in [24]):

$$\begin{array}{cc}\hfill & \parallel A^{{\displaystyle \frac{s}{2}}}{(G(u)-G(v))\parallel }_{{\mathcal{L}}_2^0}\le C{|u-v|}_s,0\le s\le 1,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \parallel G^{\prime }{(v)\parallel }_{\mathcal{L}(H,{\mathcal{L}}_2^0)}\le C,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \parallel G^{\prime \prime }{(v)\parallel }_{{\mathcal{L}}^{\otimes 2}(H,{\mathcal{L}}_2^0)}\le C,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \parallel G^{\prime }(u)G(u)-G^{\prime }{(v)G(v)\parallel }_{HS(Q^{{\displaystyle \frac{1}{2}}}(H),{\mathcal{L}}_2^0)}\le C\parallel u-v\parallel .\hfill \end{array}$$

(18)

Denote $f(t)=F(u(t)),h(t)=G(u(t)){\displaystyle \frac{dW(t)}{dt}}$, then (1) can be written as

$${}_0{}^{C}D_t^{\alpha }u(t)+Au(t)=f(t)+{}_0{}^{R}D_t^{\alpha -1}h(t),for0<t\le T,withu(0)=u_0.$$

(19)

Taking the Laplace transform of (19), we have

$$z^{\alpha }\widehat{u}(z)-z^{\alpha -1}{u}_0+A\widehat{u}(z)=\widehat{f}(z)+z^{\alpha -1}\widehat{h}(z),$$

which implies that

$$\widehat{u}(z)=z^{\alpha -1}{(z^{\alpha }+A)}^{-1}{u}_0+{(z^{\alpha }+A)}^{-1}\widehat{f}(z)+z^{\alpha -1}{(z^{\alpha }+A)}^{-1}\widehat{h}(z).$$

By the inverse Laplace transform, we have

$$u(t)=E(t)u_0+{\int }_0^t\overline{E}(t-s)F(u(s))ds+{\int }_0^{t}E(t-s)G(u(s))dW(s),$$

(20)

where $E(t)=E_{\alpha ,1}(-t^{\alpha }A) \mathrm{and} \overline{E}(t)=t^{\alpha -1}{E}_{\alpha ,\alpha }(-t^{\alpha }A)$, respectively, are Mittag-Leffler functions that also have the following integration forms:

$$E(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{zt}{(z^{\alpha }+A)}^{-1}{z}^{\alpha -1}dz,\overline{E}(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{zt}{(z^{\alpha }+A)}^{-1}dz,$$

where $\Im z$ increases along $\Gamma$ and

$$\Gamma =\{z:|argz|=\theta ,{\displaystyle \frac{\pi }{2}}<\theta <{\theta }_0<\pi \}.$$

Under Assumption 1, and following an approach similar to the one in the proof of Lemma 4.1 in [16], the Mittag-Leffler functions $E(t)$ and $\overline{E}(t)$ in (20) satisfy the following estimates: $p\ge 0$, $q\ge 0$, $0\le p-q\le 2,l=0,1$,

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{d^{l}E(t)}{d{t}^l}}v{|}_p\le C{t}^{-\alpha {\displaystyle \frac{p-q}{2}}-l}{\left|v\right|}_q,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{d^l\overline{E}(t)}{d{t}^l}}v{|}_p\le C{t}^{-\alpha {\displaystyle \frac{p-q}{2}}+\alpha -1-l}{\left|v\right|}_q.\hfill \end{array}$$

(22)

We present a useful lemma that will be applied in handling stochastic integrals.

Lemma 1 

([16,28]). Let $p\ge 2$ and ${(\psi (t))}_{t\in [0,T]}$ be a predictable and ${\mathcal{L}}_2^0$-valued stochastic process such that ${\parallel \psi \parallel }_{L^p(\Omega ;L^2(0,T;{\mathcal{L}}_2^0))}<\infty$. Then, it holds that

$$\parallel {\int }_0^T\psi (t)dW(t){\parallel }_{L^p(\Omega ,H)}\le C{\parallel \psi \parallel }_{L^p(\Omega ;L^2(0,T,{\mathcal{L}}_2^0))}<\infty .$$

(23)

We also utilize the following fractional Grönwall inequality to analyze the spatial regularity of the mild solution (20).

Lemma 2 

([23]). Let $\alpha \in (0,1)$. Suppose that y is nonnegative and y satisfies the inequality

$${}_0{}^{C}D_t^{\alpha }y(t)\le \beta y(t)+\sigma (t),$$

where the function $\sigma \in L^{\infty }(0,T)$ and the constant $\beta >0$. Then,

$$y(t)\le C(y(0)+\parallel \sigma {\parallel }_{L^{\infty }(0,T)}).$$

3. The Well-Posedness of the Mild Solution

In this section, we investigate the well-posedness of problem (1), including the existence, uniqueness, and spatial and temporal regularity of the mild solution (20). The proof of existence and uniqueness for the mild solution (20) is inspired by the corresponding theorem presented in [15]. In Theorem 1, we apply the Banach fixed point theorem to establish the existence and uniqueness of solutions to the stochastic semilinear subdiffusion equations driven by integrated multiplicative noise. In Theorem 2, we employ the semigroup approach to establish the spatial regularity of the mild solution. In Theorem 3, we again use the semigroup approach to demonstrate the temporal regularity of the mild solution.

Theorem 1. 

Let $0\le \sigma \le 2$. Let Assumptions A2 and A3 hold. Assume that the initial dataset $u_0\in L^{2p}(\Omega ,{\dot{H}}^{\sigma })$ is an ${\mathcal{F}}_0$-measurable random variable. Then, there exists a unique mild solution (20) such that $u(t)\in C([0,T];L^{2p}(\Omega ,{\dot{H}}^{\sigma }))$.

Proof. 

See the proof in Appendix A.1 in Appendix A. □

Next, we shall consider the spatial and temporal regularities of solution $u(t)$. First, we show the spatial regularity.

Theorem 2. 

Let $u(t)$ be the solution of (1). Suppose that Assumptions A2 and A3 hold. Furthermore, assume $u_0\in L^{2p}(\Omega ,{\dot{H}}^{\sigma })$ with $\sigma \in [1,2]$. Then, there exists a positive constant C such that the following estimates hold:

$$\underset{0\le t\le T}{sup}{\parallel u(t)\parallel }_{L^{2p}(\Omega ,{\dot{H}}^{\sigma })}\le C(1+\parallel u_0{\parallel }_{L^{2p}(\Omega ,{\dot{H}}^{\sigma })}).$$

Proof. 

See the proof in Appendix A.2 in Appendix A. □

Next, we provide the temporal regularity of the mild solution $u(t)$.

Theorem 3. 

Let $0<\alpha <1.$ Let Assumptions 1–3 hold. Assume $u_0\in L^{2p}(\Omega ,{\dot{H}}^{\sigma })(1\le \sigma \le 2)$. Then, for $0\le t_1<t_2\le T$, we have, for $0\le \sigma -\kappa \le 1$,

$$\parallel u(t_2)-u(t_1){\parallel }_{L^{2p}(\Omega ,H)}\le C{(t_2-t_1)}^{min\{\alpha {\displaystyle \frac{\sigma }{2}},\alpha ,{\displaystyle \frac{1}{2}}\}}=:C{(t_2-t_1)}^{\eta },$$

(24)

$$\parallel u(t_2)-u(t_1){\parallel }_{L^{2p}(\Omega ,{\dot{H}}^1)}\le C{(t_2-t_1)}^{min\{\alpha {\displaystyle \frac{\sigma -1}{2}},{\displaystyle \frac{\alpha }{2}},{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{2}}\}}=:C{(t_2-t_1)}^{\beta }.$$

(25)

Furthermore, we have the important estimate that will be used in the Milstein scheme error analysis:

$$\begin{array}{cc}\hfill & \parallel u(t_2)-u(t_1)-{\int }_{t_1}^{t_2}G(u(s))dW(s){\parallel }_{L^{2p}(\Omega ,H)}\hfill \\ \hfill & \le C{(t_2-t_1)}^{min\{\alpha {\displaystyle \frac{\sigma }{2}},\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}}\}}=C{(t_2-t_1)}^{\beta +{\displaystyle \frac{\alpha }{2}}}.\hfill \end{array}$$

Proof. 

See the proof in Appendix A.3 in Appendix A. □

4. Full Discretization Scheme

In this section, we propose an efficient fully discrete spatio-temporal scheme for solving problem (1). The numerical method is based on the standard piecewise linear finite element method for spatial discretization, the Grünwald–Letnikov method for temporal discretization, and the Milstein scheme for noise approximation.

Let $S_h\subset H_0^1(D)$ denote the space of piecewise linear finite elements associated with a triangulation ${\mathcal{T}}_h$ of the domain $D\subset {\mathbb{R}}^d,d\le 3$. Let $P_h:H\to S_h$ denote the $L^2$ projection defined by

$$(P_{h}v,\chi )=(v,\chi ),\forall \chi \in S_h.$$

(26)

Let $A_h:S_h\to S_h$ be the discrete approximation of the elliptic operator $A:\mathcal{D}(A)\subset H\to H$, defined by

$$(A_h\psi ,\chi )=A(\psi ,\chi ),\forall \chi \in S_h,$$

(27)

where $A(·,·)$ is the bilinear form associated with the elliptic operator A. Here, $A_h$ satisfies, with $-{\displaystyle \frac{1}{2}}\le r\le {\displaystyle \frac{1}{2}}$;

$$\parallel A_h^r{P}_{h}v\parallel \le c\parallel A^{r}v\parallel ,\forall v\in H,$$

see [16].

The spatial semidiscrete scheme of the problem (1) is to find the solution $u_h(t)\in S_h$ such that, with $u_h(0)=P_h{u}_0$,

$${}_0{}^{C}D_t^{\alpha }{u}_h(t)+A_h{u}_h(t)=P_{h}F(u_h(t))+{}_0{}^{R}D_t^{\alpha -1}{P}_{h}G(u_h(t)){\displaystyle \frac{dW(t)}{dt}},0<t\le T.$$

(28)

Similar to the continuous case, one can likewise derive the mild solution of the spatial semidiscrete problem (28), given by

$$u_h(t)=E_h(t)P_h{u}_0+{\int }_0^t{\overline{E}}_h(t-s)P_{h}F(u_h(s))ds+{\int }_0^t{E}_h(t-s)P_{h}G(u_h(s))dW(s).$$

(29)

These operators,

$$E_h(t)=E_{\alpha ,1}(-t^{\alpha }{A}_h),{\overline{E}}_h(t)=t^{\alpha -1}{E}_{\alpha ,\alpha }(-t^{\alpha }{A}_h),$$

have similar smoothing properties to $E(t)$ and $\overline{E}(t)$. Thus, $u_h(t)$ and $u(t)$ have the same temporal and spatial regularity estimate results.

We now introduce the time discretization scheme of the spatially semidiscrete solution (29). Let $u_h^n\approx u_h(t_n)$ be the approximation of $u_h(t_n)$. Let $g^j={\rho }_1^j+{\rho }_2^j$, where, for $j=1,2,\dots ,N$,

$${\rho }_1^j=G(u_h^{j-1}){\displaystyle \frac{W(t_j)-W(t_{j-1})}{\tau }},{\rho }_2^j=G^{\prime }(u_h^{j-1})G(u_h^{j-1}){\int }_{t_{j-1}}^{t_j}{\displaystyle \frac{W(s)-W(t_{j-1})}{\tau }}dW(s).$$

We consider the time-stepping scheme: given the initial value $u_h^0=P_h{u}_0$, find $u_h^n$ such that

$${\partial }_{\tau }^{\alpha }(u_h^n-u_h^0)+A_h{u}_h^n=P_{h}F(u_h^n)+{\partial }_{\tau }^{\alpha -1}{P}_h{g}^n,$$

(30)

where the Riemann–Liouville integral/derivative at time $t_n$ is approximated using the Grüwald–Letnikov scheme:

$${\partial }_{\tau }^{\alpha }v(t_n)\approx {\tau }^{-\alpha }\sum _{j=0}^n{\omega }_{n-j}^{(\alpha )}{v}^j=:{\partial }_{\tau }^{\alpha }{v}^n,{\partial }_{\tau }^{\alpha -1}v(t_n)\approx {\tau }^{(1-\alpha )}\sum _{j=0}^n{\omega }_{n-j}^{(\alpha -1)}{v}^j=:{\partial }_{\tau }^{-(1-\alpha )}{v}^n.$$

Here, the weights ${\omega }_{n-j}^{(\alpha )}$ and ${\omega }_{n-j}^{(\alpha -1)}$ are generated by the power series expansion ($\delta (\xi )=1-\xi$):

$$\delta {(\xi )}^{\alpha }=\sum _{j=0}^{\infty }{\omega }_j^{(\alpha )}{\xi }^j,\delta {(\xi )}^{\alpha -1}=\sum _{j=0}^{\infty }{\omega }_j^{(\alpha -1)}{\xi }^j,$$

respectively. Then, the numerical scheme (30) reads, with $g^0=0$,

$${\tau }^{-\alpha }\sum _{j=0}^n{\omega }_{n-j}^{(\alpha )}(u_h^j-u_h^0)+A_h{u}_h^n=P_{h}F(u_h^n)+{\tau }^{1-\alpha }\sum _{j=0}^n{\omega }_{n-j}^{(\alpha -1)}{P}_h{g}^j,n\ge 1.$$

(31)

Next, we introduce the new operators $R_j$ and ${\overline{R}}_j$, which satisfy

$$\tilde{R}(\xi ):=\sum _{j=0}^{\infty }{R}_j{\xi }^j,\tilde{\overline{R}}(\xi ):=\sum _{j=0}^{\infty }{\overline{R}}_j{\xi }^j,$$

where $\tilde{R}(\xi )=1+\xi {({\tau }^{-\alpha }\delta {(\xi )}^{\alpha }+A_h)}^{-1}{\tau }^{-1}$, while $\tilde{\overline{R}}(\xi )=1+\xi {({\tau }^{-\alpha }\delta {(\xi )}^{\alpha }+A_h)}^{-1}{\tau }^{-\alpha }\delta {(\xi )}^{\alpha -1}$.

Following the analysis in [16] as applied to (31), we obtain

$$u_h^n={\overline{u}}^n+\tau \sum _{j=0}^{n-1}{R}_{n-j}{P}_{h}F(u_h^{j+1})+\tau \sum _{j=0}^{n-1}{\overline{R}}_{n-j}{P}_h({\rho }_1^{j+1}+{\rho }_2^{j+1}),$$

(32)

where ${\overline{u}}^n$ represents the discrete solution of the homogeneous problem in (31). Equivalently, we have

$$\begin{array}{cc}\hfill u_h^n& ={\overline{u}}^n+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{R}_{n-j}{P}_{h}F(u_h^{j+1})ds+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_{h}G(u_h^j)dW(s)\hfill \\ \hfill & +\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_h{G}^{\prime }(u_h^j)G(u_h^j)(W(s)-W(t_j))dW(s).\hfill \end{array}$$

(33)

5. The Error Estimates

In this section, we provide detailed proofs of the error estimates for the proposed fully discrete scheme (33). Using the semigroup approach, we establish the error estimates presented in Theorems 4 and 5 below.

To complete the proofs of Theorems 4 and 5 below, we need the following lemmas.

Lemma 3 

([16]). Denote $F_h(t)=E(t)-E_h(t)P_h$ and ${\overline{F}}_h(t)=\overline{E}(t)-{\overline{E}}_h(t)P_h$. Let $0\le \nu \le \mu \le 2$ and $v\in {\dot{H}}^{\nu }$. For $t>0$, it holds true that

$$\begin{array}{cc}\hfill & \parallel F_h(t)v\parallel \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -\nu }{2}}}{\left|v\right|}_{\nu },\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \parallel {\overline{F}}_h(t)v\parallel \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -\nu }{2}}+\alpha -1}{\left|v\right|}_{\nu }.\hfill \end{array}$$

(35)

Lemma 4 

([23]). Let $\alpha \in (0,1)$. Let $p_1$ and N be positive integers and $\tau >0$. Let $t_n=n\tau$ for $0\le n\le N$. Let ${(y_n)}_{n=1}^N$ be a nonnegative sequence. Assume that there exist ${\eta }_1,\dots ,{\eta }_{p_1}\in [0,1)$ and $s_1,\dots ,s_{p_1},b>0$ such that

$$y_n\le \sum _{j=1}^{p_1}{s}_j{t}_n^{-{\eta }_j}+b\tau \sum _{j=1}^{n-1}{t}_{n-j}^{\alpha -1}{y}_j,1\le n\le N.$$

Then, there exists a constant $C=C({\eta }_1,\dots ,{\eta }_{p_1},\alpha ,b,t_N)$ such that

$$y_n\le C\sum _{j=1}^{p_1}{s}_j{t}_n^{-{\eta }_j}for1\le n\le N.$$

For the fully discrete error estimation in $L^2(\Omega ,H)$, we introduce an intermediate solution ${\tilde{u}}_h^n$ defined by

$$\begin{array}{cc}\hfill {\tilde{u}}_h^n& ={\overline{u}}^n+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{R}_{n-j}{P}_{h}F(u(t_{j+1}))ds+\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_{h}G(u(t_j))dW(s)\hfill \\ \hfill & +\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{\overline{R}}_{n-j}{P}_h{G}^{\prime }(u(t_j))G(u(t_j))(W(s)-W(t_j))dW(s);\hfill \end{array}$$

(36)

then, we have

$$\begin{array}{cc}\hfill u(t_n)-{\tilde{u}}_h^n& =(E(t_n)u_0-{\overline{u}}^n)+\left[{\int }_0^{t_n}\overline{E}(t_n-s)F(u(s))ds-\sum _{j=0}^{n-1}{\int }_{t_j}^{t_{j+1}}{R}_{n-j}{P}_{h}F(u(t_{j+1}))ds\right]\hfill \\ \hfill & +\left[{\int }_0^{t_n}E(t_n-s)G(u(s))dW(s)-\tau \sum _{j=0}^{n-1}{\overline{R}}_{n-j}{P}_h({\rho }_1(t_{j+1})+{\rho }_2(t_{j+1}))\right]\hfill \\ \hfill & =I_1+I_2+I_3.\hfill \end{array}$$

Lemma 5 

([16]). (Error estimate of $I_1$ ) Assume $u_0\in L^{2p}(\Omega ,{\dot{H}}^{\sigma })$, where $1\le \sigma \le 2$ and $p\ge 1$, is an ${\mathcal{F}}_0$-measurable random variable. Then, there exists a constant C such that

$$\parallel I_1{\parallel }_{L^2(\Omega ,H)}\le C\tau t_n^{-1+{\displaystyle \frac{\sigma \alpha }{2}}}\parallel u_0{\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}.$$

Lemma 6. 

(Error estimate of $I_2$) Let $\epsilon >0$ and β be defined as in Theorem 3. Then, we have, with ${\displaystyle \frac{1}{2}}<\alpha <1$,

$$\begin{array}{cc}\hfill \parallel I_2{\parallel }_{L^2(\Omega ,H)}& \le C{\tau }^{{\displaystyle \frac{\alpha \sigma }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^{2-\epsilon }+C{\tau }^{min\{\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},2\beta \}}.\hfill \end{array}$$

Proof. 

See the proof in Appendix A.5 in Appendix A. □

Lemma 7. 

(Error estimate of $I_3$) Let β and η be defined as in Theorem 3. For $I_3$, we have the following estimate with $0\le \sigma -\kappa \le 1:$

$$\parallel I_3{\parallel }_{L^2(\Omega ,H)}\le C{h}^{\mu }+C{\tau }^{min\{1,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},\beta +{\displaystyle \frac{\alpha }{2}},\eta +{\displaystyle \frac{1}{2}},2\beta \}},$$

where $0<\mu \le 2$ and $\alpha (\mu -(\sigma -\kappa ))\le 1$.

Proof. 

See the proof in Appendix A.6 in Appendix A. □

Combing Lemmas 5 and 6 with Lemma 7, we obtain the following theorem, designated Theorem 4.

Theorem 4. 

Let $p\ge 1$. Let $u_0\in L^{2p}(\Omega ,{\dot{H}}^{\sigma })$ with $\sigma \in [1,2]$. Let β and η be defined as in Theorem 3. We assume that Assumptions A1–A3 hold. Let u be the mild solution defined in (20), and let ${\tilde{u}}_h^n$ be the numerical solution (36). Then, there exists a positive constant C such that the following estimates hold for $0\le \sigma -\kappa \le 1$, ${\displaystyle \frac{1}{2}}<\alpha <1$, and $\epsilon >0$:

$$\begin{array}{cc}\hfill \parallel u(t_n)-{\tilde{u}}_h^n{\parallel }_{L^2(\Omega ,H)}& \le C\tau t_n^{-1+{\displaystyle \frac{\sigma \alpha }{2}}}\parallel u_0{\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}\hfill \\ \hfill & +C{h}^{\{2-\epsilon ,\mu \}}+C{\tau }^{min\{{\displaystyle \frac{\alpha \sigma }{2}},\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},\beta +{\displaystyle \frac{\alpha }{2}},\eta +{\displaystyle \frac{1}{2}},2\beta \}}\hfill \\ \hfill & =C\tau t_n^{-1+{\displaystyle \frac{\sigma \alpha }{2}}}\parallel u_0{\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}\hfill \\ \hfill & +C{h}^{\{2-\epsilon ,\mu \}}+C{\tau }^{min\{{\displaystyle \frac{\alpha \sigma }{2}},\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},\alpha (\sigma -1),\alpha (\sigma -\kappa )+1-\alpha \}}.\hfill \end{array}$$

where $0<\mu \le 2$ and $\alpha (\mu -(\sigma -\kappa ))\le 1.$

Now, we turn to the main theorem in this paper.

Theorem 5. 

Let Assumptions A1–A3 hold. Let u be the mild solution in (20), and let $u_h^n$ be the numerical solution (33). Then, for ${\displaystyle \frac{1}{2}}<\alpha <1$, there exists a constant C independent of τ and h such that, for $0\le \sigma -\kappa \le 1$ and $\epsilon >0$,

$$\begin{array}{cc}\hfill \parallel u(t_n)-u_h^n{\parallel }_{L^2(\Omega ,H)}& \le C\tau t_n^{-1+{\displaystyle \frac{\sigma \alpha }{2}}}\parallel u_0{\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}\hfill \\ \hfill & +C{h}^{\{2-\epsilon ,\mu \}}+C{\tau }^{min\{{\displaystyle \frac{\alpha \sigma }{2}},\alpha ,{\displaystyle \frac{\alpha (\sigma -\kappa )}{2}}+{\displaystyle \frac{1}{2}},\alpha (\sigma -1),\alpha (\sigma -\kappa )+1-\alpha \}},\hfill \end{array}$$

where $0<\mu \le 2$ and $\alpha (\mu -(\sigma -\kappa ))\le 1.$

Proof. 

See the proof in Appendix A.4 in Appendix A. □

Remark 1. 

Let ${\displaystyle \frac{1}{2}}<\alpha <1$. Let $u_0\in L^2(\Omega ,{\dot{H}}^{\sigma })$ with $1\le \sigma \le 2$. We have, for $\sigma =\kappa$,

$$\parallel u(t)-u_h{(t)\parallel }_{L^2(\Omega ,H)}\le C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^{\{2-\epsilon ,\mu \}},t\in [0,T],$$

where $0<\mu \le 2$ and $\alpha \mu \le 1$.

Remark 2. 

Let ${\displaystyle \frac{1}{2}}<\alpha <1$. Let $u_0\in L^2(\Omega ,{\dot{H}}^{\sigma })$ with $1\le \sigma \le 2$. We have, for $\sigma -\kappa =1$,

$$\parallel u(t)-u_h{(t)\parallel }_{L^2(\Omega ,H)}\le C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\sigma \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{h}^{2-\epsilon },t\in [0,T].$$

Remark 3. 

Let ${\displaystyle \frac{1}{2}}<\alpha <1$. Let $u_0\in L^2(\Omega ,{\dot{H}}^{\sigma })$ with $1\le \sigma \le 2$. We have, for $\sigma -\kappa =1$,

$$\parallel u_h(t_n)-u_h^n{\parallel }_{L^2(\Omega ,H)}\le C\tau t_n^{-1+{\displaystyle \frac{\nu \alpha }{2}}}{\parallel u_0\parallel }_{L^2(\Omega ,{\dot{H}}^{\sigma })}+C{\tau }^{\alpha }.$$

6. Numerical Simulations

In this section, we consider the numerical simulations for the following semilinear stochastic subdiffusion problem, with $\alpha \in (0,1)$,

$$\begin{array}{cc}\hfill & {}_0{}^{C}D_t^{\alpha }u(t,x)-{\displaystyle \frac{{\partial }^{2}u(t,x)}{\partial x^2}}=f(u(t,x))+{}_0{}^{R}D_t^{\alpha -1}G(u(t,x)){\displaystyle \frac{dW(t)}{dt}},0\le t\le T,0<x<1,\hfill \end{array}$$

(37)

with the initial condition $u(0,x)=u_0(x)=x(1-x)$ and the boundary conditions: $u(t,0)=u(t,1)=0.$ Here, $f(u)=sinu$ and $G(u)=sinu$ and

$${\displaystyle \frac{dW(t)}{dt}}=\sum _{m=1}^{\infty }{\gamma }_m^{1/2}{e}_m{\displaystyle \frac{d{\beta }_m(t)}{dt}},$$

(38)

where ${\beta }_m(t),m=1,2,\dots$ are the Brownian motions and $e_m(x)=\sqrt{2}sinm\pi x$ denote the eigenfunctions of the operator $A=-{\displaystyle \frac{{\partial }^2}{\partial x^2}}$ with $D(A)=H_0^1(0,1)\cap H^2(0,1)$. Here, ${\gamma }_m,m=1,2,\dots$ are the eigenvalues of the covariance operator Q of the stochastic process $W(t)$, that is, $Q{e}_m={\gamma }_m{e}_m.$

With $v(t,x)=u(t,x)-u_0(x)$, we rewrite (37) as the following equivalent form: with $v(0,x)=0$ and $v(t,0)=v(t,1)=0,$

$$\begin{array}{cc}\hfill & {}_0{}^{C}D_t^{\alpha }vs.(t,x)-{\displaystyle \frac{{\partial }^{2}vs.(t,x)}{\partial x^2}}={\displaystyle \frac{{\partial }^2{u}_0(x)}{\partial x^2}}+f(u(t,x))+g(t,x),0\le t\le T,0<x<1,\hfill \end{array}$$

(39)

where $g(t,x)={}_0{}^{R}D_t^{\alpha -1}G(u(t,x)){\displaystyle \frac{dW(t)}{dt}}$.

Let $0=t_0<t_1<\dots <t_N=T$ be a partition of the time interval $[0,T]$ with uniform time step size $\tau$, and let $0=x_0<x_1<\dots <x_M=1$ be a partition of the spatial interval $[0,1]$ with uniform space step size h. Let $S_h$ denote the piecewise linear finite element space. We approximate $v(t_n,x)$ by $V^n\in S_h$. The Milstein scheme for (39) is then defined as follows: find $V^n\in S_h$ for $n=0,1,\dots ,N$, such that, with $V^0=0$,

$$\begin{array}{cc}\hfill & \left({\tau }^{-\alpha }\sum _{j=1}^n{w}_{n-j}{V}^j,\chi \right)+(\nabla V^n,\nabla \chi )=-(\nabla P_h{u}_0,\nabla \chi )+(f(U^{n-1}),\chi )+(g_{n-1},\chi ),\forall \chi \in S_h,\hfill \end{array}$$

(40)

where $U^{n-1}=V^{n-1}+u_0$ is the approximation of $u(t_{n-1},x)$ and $P_h:H\to S_h$ denotes the $L^2$ projection operator. For simplicity, we adopt an explicit scheme to approximate the nonlinear term $f(u)$. The weights are generated by the Grüwald–Letnikov scheme for $\alpha \in (0,1)$,

$${(1-z)}^{\alpha }=\sum _{j=0}^{\infty }{w}_j{z}^j.$$

Here

$$\begin{array}{cc}\hfill g_{n-1}& ={}_0{}^{R}D_t^{\alpha -1}G(U^{n-1})\sum _{m=1}^{\infty }{\gamma }_m^{1/2}{e}_m(x){\displaystyle \frac{{\beta }_m(t_n)-{\beta }_m(t_{n-1})}{\tau }}\hfill \\ \hfill & +{}_0{}^{R}D_t^{\alpha -1}{G}^{\prime }(U^{n-1})G(U^{n-1})\sum _{m=1}^{\infty }{\gamma }_m^{1/2}{e}_m(x){\int }_{t_{n-1}}^{t_n}{\displaystyle \frac{{\beta }_m(s)-{\beta }_m(t_{n-1})}{\tau }}d{\beta }_m(s)\hfill \\ \hfill & =:g_{n-1}^{(1)}+g_{n-1}^{(2)}.\hfill \end{array}$$

Let $M>1$ be a positive integer and Let ${\phi }_k,k=1,2,\dots ,M-1$ denote the finite element basis functions. The kth component $\left(f(U^{n-1}),{\phi }_k\right),k=1,2,\dots ,M-1$ can be approximated by using the following formula:

$$\begin{array}{cc}\hfill & \left(f(U^{n-1}),{\phi }_k\right)={\int }_{x_{k-1}}^{x_k}f(U^{n-1}){\phi }_{k}dx+{\int }_{x_k}^{x_{k+1}}f(U^{n-1}){\phi }_{k}dx\hfill \\ \hfill & \approx {\displaystyle \frac{h}{2}}\left[f(U^{n-1}(x_{k-1}))+f(U^{n-1}(x_{k+1}))\right]=:{\displaystyle \frac{h}{2}}\left[F_{-1}+F_1\right],\hfill \end{array}$$

(41)

where, with $k=1,2,\dots ,M-1$,

$$\begin{array}{cc}\hfill & F_{-1}=f(U^{n-1}(x_{k-1}))=f(V^{n-1}(x_{k-1})+u_0(x_{k-1})),\hfill \\ \hfill & F_1=f(U^{n-1}(x_{k+1}))=f(V^{n-1}(x_{k+1})+u_0(x_{k+1})).\hfill \end{array}$$

(42)

The kth component is

$$\left(g_{n-1},{\phi }_k\right)=\left(g_{n-1}^{(1)},{\phi }_k\right)+\left(g_{n-1}^{(2)},{\phi }_k\right),$$

where, with $k=1,2,\dots ,M-1$,

$$\left(g_{n-1}^{(1)},{\phi }_k\right)\approx {\tau }^{1-\alpha }\sum _{j=1}^n{w}_{n-j}^{(\alpha -1)}\left[\sum _{m=1}^{M-1}{\gamma }_m^{1/2}(G(U^{j-1})e_m,{\phi }_k){\displaystyle \frac{{\beta }_m(t_j)-{\beta }_m(t_{j-1})}{\tau }}\right],$$

(43)

and

$$\begin{array}{cc}\hfill & \left(g_{n-1}^{(2)},{\phi }_k\right)\hfill \\ \hfill & \approx {\tau }^{1-\alpha }\sum _{j=1}^n{w}_{n-j}^{(\alpha -1)}\left[\sum _{m=1}^{M-1}{\gamma }_m^{1/2}(G^{\prime }(U^{j-1})G(U^{j-1})e_m,{\phi }_k){\displaystyle \frac{{({\beta }_m(t_j)-{\beta }_m(t_{j-1}))}^2}{\tau }}\right].\hfill \end{array}$$

(44)

Here, $w_j^{(\alpha -1)},j=0,1,2,\dots ,n$ are generated by

$${(1-\zeta )}^{\alpha -1}=\sum _{j=0}^{\infty }{w}_j^{(1-\alpha )}{\zeta }^j,$$

and

$$\begin{array}{cc}\hfill & \left(G(U^{j-1}),{\phi }_k\right)={\int }_{x_{k-1}}^{x_k}G(U^{j-1}){\phi }_{k}dx+{\int }_{x_k}^{x_{k+1}}G(U^{j-1}){\phi }_{k}dx\hfill \\ \hfill & \approx {\displaystyle \frac{h}{2}}\left[G(U^{j-1}(x_{k-1}))e_m(x_{k-1})+G(U^{j-1}(x_k))e_m(x_k)\right],\hfill \end{array}$$

(45)

and

$$\begin{array}{cc}\hfill & \left(G^{\prime }(U^{j-1})G(U^{j-1}),{\phi }_k\right)\hfill \\ \hfill & \approx {\int }_{x_{k-1}}^{x_k}{G}^{\prime }(U^{j-1})G(U^{j-1}){\phi }_{k}dx+{\int }_{x_k}^{x_{k+1}}{G}^{\prime }(U^{j-1})G(U^{j-1}){\phi }_{k}dx\hfill \\ \hfill & \approx {\displaystyle \frac{h}{2}}\left[G^{\prime }(U^{j-1}(x_{k-1}))G(U^{j-1}(x_{k-1}))e_m(x_{k-1})+G^{\prime }(U^{j-1}(x_k))G(U^{j-1}(x_k))e_m(x_k)\right].\hfill \end{array}$$

The computational complexity of (40) is relatively low, as it is an explicit method. We remark that in (43) and (44), the noise series is truncated after the first $M-1$ terms, which does not affect the convergence order with respect to h. Additionally, the integrals in (41) are approximated using the trapezoidal rule, which likewise does not impact the convergence order of h.

Let $N_{\mathrm{ref}}=2^7$ and $T=1$. Let $d{t}_{\mathrm{ref}}=T/N_{\mathrm{ref}}$ denote the reference time step size. To obtain the reference solution $v_{\mathrm{ref}}$, we set the space step size to $h=2^{-6}$ and the time step size to $d{t}_{\mathrm{ref}}=2^{-7}$. To observe the time convergence orders, we consider the different time step sizes $\tau =kappa·d{t}_{\mathrm{ref}}$, where $kappa=[2^5,2^4,2^3,2^2]$, and obtain the approximate solution $V^N$. We then calculate the error in $L^2$ norm at $T=1$ for the different time step sizes using $M_1=20$ simulations:

$$\parallel v_{ref}(t_N)-V^N{\parallel }_{L^2(\Omega ,H)}=\sqrt{\mathbb{E}\parallel v_{ref}(t_N)-V^N{\parallel }^2}.$$

We shall consider the following two cases for the numerical simulations using the proposed Milstein scheme.

Case 1: The smooth noise case with ${\gamma }_l=l^{-{\beta }_1}$, ${\beta }_1>1$ which implies that

$$tr(Q)=\sum _{l=1}^{\infty }{\gamma }_l=\sum _{l=1}^{\infty }{l}^{-{\beta }_1}<\infty ,$$

where $tr(Q)$ denotes the trace of the operator Q.

Case 2: The nonsmooth noise case with ${\beta }_1=0$.

By Theorem 5, the theoretical convergence orders with $\sigma -\kappa =1$ in Case 1 shows

$$\parallel v_{\mathrm{ref}}(t_N)-V^N{\parallel }_{L^2(\Omega ,H)}=O({\tau }^{min\{\alpha ,{\displaystyle \frac{\alpha +1}{2}}\}})=C{\tau }^{\alpha }.$$

(46)

In

Table 1. Time convergence orders at $T=1$ for smooth noise in Case 1 with various $\alpha >{\displaystyle \frac{1}{2}}$, where ${\gamma }_l=l^{-3}$ for $l=1,2,\dots$.

$\mathit{\alpha }$	$1-\mathit{\alpha }$	$\mathit{\tau }=1/4$	$1/8$	$1/16$	$1/32$	Order
$0.5$	0.5	3.89 × 10−3	2.19 × 10−3	1.73 × 10−3	1.21 × 10−3
			0.83	0.34	0.51	0.56 (0.50)
$0.6$	0.4	4.90 × 10−3	2.69 × 10−3	2.01 × 10−3	1.26 × 10−3
			0.86	0.42	0.67	0.65 (0.60)
$0.7$	0.3	6.11 × 10−3	3.25 × 10−3	2.30 × 10−3	1.28 × 10−3
			0.91	0.50	0.84	0.75 (0.70)
$0.8$	0.2	7.41 × 10−3	3.81 × 10−3	2.58 × 10−3	1.31 × 10−3
			0.96	0.56	0.98	0.83 (0.80)
$0.9$	0.1	8.46 × 10−3	4.17 × 10−3	2.73 × 10−3	1.31 × 10−3
			1.02	0.61	1.06	0.90 (0.90)

, we consider Case 1 with ${\gamma }_l=l^{-3}$ and $\alpha =0.5,0.6,0.7,0.8,0.9$. The observed time convergence rates are consistent with the theoretical predictions, as indicated in brackets.

In

Table 2. Time convergence orders at $T=1$ for nonsmooth noise in Case 2 with various $\alpha >{\displaystyle \frac{1}{2}}$, where ${\gamma }_l=1$ for $l=1,2,\dots$.

$\mathit{\alpha }$	$1-\mathit{\alpha }$	$\mathit{\tau }=1/4$	$1/8$	$1/16$	$1/32$	Order
$0.5$	0.5	3.52 × 10−3	2.37 × 10−3	1.47 × 10−3	1.1 × 10−3
			0.57	0.45	0.65	0.56
$0.6$	0.4	4.42 × 10−3	2.87 × 10−3	2.16 × 10−3	1.38 × 10−3
			0.62	0.41	0.65	0.56
$0.7$	0.3	5.65 × 10−3	3.61 × 10−3	2.75 × 10−3	1.83 × 10−3
			0.65	0.39	0.59	0.54
$0.8$	0.2	7.20 × 10−3	4.68 × 10−3	3.55 × 10−3	2.53 × 10−3
			0.62	0.4	0.49	0.50
$0.9$	0.1	8.56 × 10−3	5.69 × 10−3	4.25 × 10−3	3.26 × 10−3
			0.60	0.42	0.39	0.47

, we consider Case 2 with ${\gamma }_l=1$ and $\alpha =0.5,0.6,0.7,0.8,0.9$. As expected, the experimentally observed time convergence rates of the Milstein scheme are higher than those achieved by the standard Euler scheme.

We next consider (37) with $f(u)={\displaystyle \frac{u}{1+\left|u\right|}}$ and $g(u)=u$, which corresponds to a one-dimensional model discussed in Section 6 of [22]. In [22], the authors proposed a fully discrete scheme for solving (37) and established a temporal convergence order of $O\left({\tau }^{{\displaystyle \frac{min(\alpha (2\overline{H}+\overline{\beta }-1),2-(2-\overline{\beta })\alpha )}{2}}}\right)$, where $\overline{H}\in (0,1)$ denotes the Hurst parameter and $\overline{\beta }\in [0,1]$ describes the smoothness of the noise (see Theorem 3 in [22]). When $\overline{H}=1/2$ (corresponding to the standard Wiener process, as in our paper) and $\overline{\beta }=1$ (corresponding to the trace-class noise), the time convergence order obtained in [22] reduces to $O({\tau }^{\alpha /2})$, which is lower than our achieved order of $O({\tau }^{\alpha })$. In

Table 3. Time convergence orders at $T=1$ for smooth noise in Case 1 with various $\alpha >{\displaystyle \frac{1}{2}}$, where ${\gamma }_l=l^{-3}$ for $l=1,2,\dots$, for the one-dimensional model discussed in Section 6 of [22].

$\mathit{\alpha }$	$1-\mathit{\alpha }$	$\mathit{\tau }=1/4$	$1/8$	$1/16$	$1/32$	Order
$0.5$	0.5	3.69 × 10−3	1.89 × 10−3	1.43 × 10−3	1.01 × 10−3
			0.81	0.32	0.49	0.54 (0.50)
$0.6$	0.4	4.70 × 10−3	2.49 × 10−3	1.80 × 10−3	1.03 × 10−3
			0.84	0.40	0.65	0.63 (0.60)
$0.7$	0.3	6.01 × 10−3	3.15 × 10−3	2.20 × 10−3	1.08 × 10−3
			0.89	0.46	0.81	0.72 (0.70)
$0.8$	0.2	7.21 × 10−3	3.61 × 10−3	2.38 × 10−3	1.11 × 10−3
			0.94	0.54	0.96	0.81 (0.80)
$0.9$	0.1	8.26 × 10−3	3.97 × 10−3	2.53 × 10−3	1.11 × 10−3
			1.00	0.59	1.04	0.88 (0.90)

, we solve this model using the same parameters as in Table 1. We observe that the temporal convergence rates of our scheme are $O({\tau }^{\alpha })$, which are notably higher than the $O({\tau }^{\alpha /2})$ order reported in [22].

Conclusions

In this work, we propose a Milstein scheme for solving the stochastic semilinear time-fractional subdiffusion equation with fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established using the Banach fixed point theorem. The temporal and spatial regularity of the mild solution are derived through the semigroup approach. For numerical approximation, the standard Galerkin finite element method is employed for spatial discretization, the Grünwald–Letnikov method is used for temporal discretization, and the Milstein scheme is used to approximate the multiplicative noise. Detailed error estimates for the proposed scheme are also provided.