Error estimates of finite element method for semi-linear stochastic strongly damped wave equation
For researchers working on numerical methods for stochastic partial differential equations, this work extends error analysis to a new class of damped wave equations, showing that optimal rates are achievable even with space-time white noise in multiple dimensions.
This paper proves existence, uniqueness, and space-time regularity of mild solutions to semi-linear stochastic strongly damped wave equations with additive Gaussian noise, and establishes optimal convergence rates for finite element spatial discretization combined with linear implicit Euler time stepping, matching the regularity orders of the solution.
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.