Stochastic exponential integrator for finite element spatial discretization of stochastic elastic equation

Jiang, Fengze; Huang, Chengming; Wang, Xiaojie

doi:10.1016/j.camwa.2015.02.012

Cited by 4 publications

(2 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The deterministic counterpart of (1.1) finds many applications in viscoelastic theory [7,15,16], and its linear version has been numerically studied by [14,21], where a finite element method is used for spatial discretization and rational approximations for analytic semigroup. Particularly when α = 0, the problem (1.1) reduces to a stochastic wave equation (SWE) without damping, numerical approximations of which have been recently studied by many authors [2][3][4][5][8][9][10][11][12][17][18][19][22][23][24]. In contrast to the SWE case (α = 0), the stochastic strongly damped wave equations (α > 0) are much less well-understood, from both theoretical and numerical point of view.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

2018

IMA Journal of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

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,

show abstract

Section: Introductionmentioning

confidence: 99%

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

Wang

2018

IMA Journal of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Great attention has been devoted in the last decades to numerical approximations of evolutionary stochastic partial differential equations (SPDEs) (see, e.g. [2,4,9,11,12,21,23,25] and references therein). In the present work, we concentrate on a class of semi-linear SPDEs of second order with damping, described by    du t = αLu t dt + Lu dt + F (u) dt + dW (t), in D × (0, T ], u(·, 0) = u 0 , u t (·, 0) = v 0 , in D, u = 0, on ∂D × (0, T ], The initial data u 0 , v 0 are assumed to be F 0 -measurable random variables.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

2017

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Exponential discrete gradient schemes for a class of stochastic differential equations

Ruan

Wang

2022

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Stochastic exponential integrator for finite element spatial discretization of stochastic elastic equation

Cited by 4 publications

References 17 publications

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

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

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

Exponential discrete gradient schemes for a class of stochastic differential equations

Contact Info

Product

Resources

About