Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise

TL;DR

This paper analyzes the convergence of stochastic exponential integrators combined with finite element discretization for second-order semi-linear parabolic SPDEs with both multiplicative and additive noise, including practical implementation results.

Contribution

It provides a convergence proof in the mean square norm for these integrators applied to SPDEs with trace class noise, incorporating finite element and finite volume methods.

Findings

Convergence in mean square norm established for the proposed methods.

Implementation results for linear and nonlinear 2D SPDEs.

Finite element and finite volume methods both effectively discretize the SPDEs.

Abstract

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDEs) driven by space-time noise, for multiplicative and additive noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in trace class and give a convergence proof in the mean square $L^{2}$ norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.