Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equation with Multiplicative Noise

TL;DR

This paper develops and analyzes a fully discrete finite element method for nonlinear stochastic wave equations with multiplicative noise, providing error estimates, stability results, and numerical validation.

Contribution

Introduces a mixed form numerical scheme and two discretization methods for nonlinear terms, enabling error analysis and stability proofs for stochastic wave equations.

Findings

Error estimates in $L^2$ and energy norms established.

Proposed discretization methods ensure stability and high-order moments.

Numerical experiments confirm theoretical convergence and stability.

Abstract

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in [11]. The novelties of this paper are threefold. First, the error estimates cannot not be directly obtained if the numerical scheme in primal form is used. The numerical scheme in mixed form is introduced and several H\"{o}lder continuity results of the strong solution are proved, which are used to establish the error estimates in both $L^2$ norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the $L^2$ stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher order moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms.