Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise
Provides rigorous error bounds for numerical solutions of SPDEs, which is important for computational scientists and engineers solving stochastic PDEs.
The paper proves optimal strong error estimates for Galerkin finite element methods applied to semilinear SPDEs with multiplicative noise, achieving optimal convergence rates for both spatial and spatio-temporal discretizations.
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.