Ruisheng Qi

Xiaojie Wang, Ruisheng Qi, Fengze Jiang

This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.

Ruisheng Qi, Xiaojie Wang

This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise. Under appropriate assumptions, we provide error bounds for the proposed full-discrete scheme. It is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem, which allows for higher convergence rate than usual time-stepping schemes. For the space-time white noise case in two or three spatial dimensions, the scheme still exhibits a good convergence performance. Another striking finding is that, even for the velocity with low regularity the scheme always promises first order strong convergence in time. Numerical examples are finally reported to confirm our theoretical findings.

Ruisheng Qi, Xiaojie Wang

The present paper proposes new fully discrete schemes for long-time approximations of stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients in a bounded domain $D \subset \R^d, d =1,2,3 $. A novel family of linearly implicit time-stepping schemes is introduced, based on a standard Galerkin finite element spatial semi-discretization. A distinguishing feature of the schemes is that the proposed finite element fully discrete approximations preserve uniform-in-time moment bounds in a Banach space $L^{r}(D), r >2$, without requiring any restriction on the time-space discretization stepsize ratio. %established... To show it, some non-standard arguments are developed. First, we derive long-time error estimates in the Banach space $L^r(D)$ for finite element fully discrete approximations of the deterministic linear parabolic equation with non-smooth initial value, which is, to our knowledge, new for the literature on numerical PDEs and of independent interest. These error estimates together with the contractive property of the semi-group in $L^{r}(D), r > 2$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy help us establish the desired uniform-in-time moment bounds. Then both strong and weak error bounds of the proposed schemes are carefully analyzed in a setting of low regularity, with uniform-in-time convergence rates obtained for cases of both space-time white and trace-class noises. The analysis is highly nontrivial, due to the finite element discretization, the low regularity and the presence of the super-linearly growing nonlinearity. %in the long-time scenario... the discretization parameters $h$ and $τ$. Finally, numerical results are presented to verify the previous theoretical findings.

Ruisheng Qi, Xiaojie Wang

In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike the usual stochastic wave equation without damping, the underlying problem with space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions. Further, we analyze a spatio-temporal discretization of the problem, performed by a standard finite element method in space and a well-known linear implicit Euler scheme in time. The analysis of the approximation error forces us to significantly enrich existing error estimates of semidiscrete and fully discrete finite element methods for the corresponding linear deterministic equation. The main results show optimal convergence rates in the sense that the orders of convergence in space and in time coincide with the orders of the spatial and temporal regularity of the mild solution, respectively. Numerical examples are finally included to confirm our theoretical findings.