Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

TL;DR

This paper analyzes the convergence of a fully discrete finite element method for semilinear stochastic wave equations with additive noise, establishing strong and weak convergence rates and extending previous results to the semilinear case.

Contribution

It extends earlier work by deriving weak convergence rates for semilinear stochastic wave equations using finite element and rational exponential approximations.

Findings

Strong convergence in positive and negative norms.

Weak convergence rate is approximately twice the strong rate.

Numerical simulations confirm theoretical results.

Abstract

We consider the numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise. For the spatial approximation we consider a standard finite element method and for the temporal approximation, a rational approximation of the exponential function. We first show strong convergence of this approximation in both positive and negative order norms. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.