Fully discrete finite element methods for nonlinear stochastic elastic wave equations with multiplicative noise

TL;DR

This paper develops and analyzes fully discrete finite element methods for nonlinear stochastic elastic wave equations with multiplicative noise, proving strong convergence and validating results through numerical experiments.

Contribution

It introduces a new fully discrete finite element scheme for nonlinear stochastic elastic wave equations with multiplicative noise, including rigorous convergence analysis.

Findings

Proved strong convergence in the energy norm with rate O(k+h^r)

Validated theoretical error estimates through numerical experiments

Demonstrated efficiency of the proposed numerical methods

Abstract

This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution is carried out and a fully discrete finite element method is proposed. Strong convergence in the energy norm with rate $\cal{O}(k+h^r)$ is proved, where $k$ and $h$ denote respectively the temporal and spatial mesh sizes, and $r(\geq 1)$ is the order of the finite element. Numerical experiments are provided to test the efficiency of proposed numerical methods and to validate the theoretical error estimate results.