Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise

TL;DR

This paper establishes optimal error estimates for Galerkin finite element methods applied to semilinear stochastic PDEs with multiplicative noise, covering both spatial and spatio-temporal discretizations.

Contribution

It provides the first comprehensive analysis of strong convergence errors for various Galerkin methods applied to nonlinear SPDEs with multiplicative noise.

Findings

Optimal error estimates for spatial semidiscrete Galerkin methods.

Optimal error bounds for spatio-temporal discretizations using Euler-Maruyama.

Results applicable to finite element and spectral Galerkin methods.

Abstract

We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler-Maruyama method. In both cases we obtain optimal error estimates. The proofs are based on sharp integral versions of well-known error estimates for the corresponding deterministic linear homogeneous equation together with optimal regularity results for the mild solution of the SPDE. The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin approximations.