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Strong approximation errors of both finite element semi-discretization and spatiotemporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive trace-class noise in space dimension d ≤ 3. The full discretization is realized by combining the standard finite element method with the backward Euler timestepping scheme. Distinct from the globally Lipschitz setting, the error analysis becomes rather challenging and demanding, due to the presence of the cubic nonlinearity in the underlying model. By introducing two auxiliary approximation processes, we do appropriate decomposition of the considered error terms and propose a novel approach of error analysis, to successfully recover the expected convergence rates of the numerical schemes. The approach is ingenious and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the full discrete scheme possesses convergence rates of order h γ in space and order τ γ 2 in time, subject to the spatial correlation of the noise process, characterized by AIn particular, a classical convergence rate of order O(h 2 + τ ) is reachable, even in multiple spatial dimensions, when the aforementioned condition is fulfilled with γ = 2. Numerical examples confirm the previous findings.Here · L 2 stands for the Hilbert-Schmidt norm,Ḣ α := D(A α 2 ), α ∈ R and the parameter γ ∈ [1, 2] coming from (1.2) quantifies the spatial regularity of the covariance operator Q of the

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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.

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.< ∞ under the same assumptions [2,23]. This benefits from smoothing effect of the analytic semigroup S(t) generated by the dominant linear operator A. In particular, different from both the stochastic heat equation and the stochastic wave equation, the strongly damped problem driven by space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions (d > 1). For example, the space-time white noise case when d = 2 admits a mild solution u ∈ L ∞ [0, T ]; L 2 (Ω,Ḣ α ) for any α < 1 (consult Remark 2.5 for more details).As the second contribution of this article, we analyze the mean-square approximation errors caused by finite element spatial semi-discretization and space-time full-discretization of (1.1). More precisely,

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.

In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn-Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn-Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.

The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension d ≤ 3. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem with additive noise into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.