Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise

We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyse 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.