Error estimates of finite element method for semilinear stochastic strongly damped wave equation

Abstract

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,

An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noise

“…From both the theoretical and numerical point of view, the deterministic problem has been extensively studied (e.g., [10,14,22]). However, the corresponding stochastic strongly damped wave equations are theoretically and numerically far from well-understood from existing literature [5,18]. In the classical monograph [5], a stochastic strongly damped wave equation with multiplicative noise was discussed and a unique mild solution was established.…”

“…In the classical monograph [5], a stochastic strongly damped wave equation with multiplicative noise was discussed and a unique mild solution was established. In our recent publication [18], we analyzed the regularity properties of the mild solution to (1.1) and examine error estimates of a full discretization, done by the finite element spatial approximation together with the well-known linear implicit Euler temporal discretization. It was shown there that, for a certain class of stochastic SDWEs, the convergence rates of the usual full discretization coincide with the space-time regularity properties of the mild solution (see Theorem 2.4 in [18]).…”

“…In our recent publication [18], we analyzed the regularity properties of the mild solution to (1.1) and examine error estimates of a full discretization, done by the finite element spatial approximation together with the well-known linear implicit Euler temporal discretization. It was shown there that, for a certain class of stochastic SDWEs, the convergence rates of the usual full discretization coincide with the space-time regularity properties of the mild solution (see Theorem 2.4 in [18]). In this article, we aim to introduce a so-called accelerated exponential time integrator for the problem (1.1), which, as we will show later, promises higher convergence order in time than the order of temporal regularity of the underlying problem.…”

“…This means, even for the velocity v with very low regularity, the numerical approximation shows a first order strong convergence in time. Recall from [18] that, the finite element spatial discretization requires (1.2) to be satisfied with γ ≥ 0 and this excludes the space-time white noise case in two or three dimensions. Furthermore, the convergence rates of numerical approximations obtained in [18] coincide exactly with the orders of the regularity.…”

“…Recall from [18] that, the finite element spatial discretization requires (1.2) to be satisfied with γ ≥ 0 and this excludes the space-time white noise case in two or three dimensions. Furthermore, the convergence rates of numerical approximations obtained in [18] coincide exactly with the orders of the regularity. As one can see from earlier discussions, the newly proposed scheme shows significantly improved convergence rates in time.…”