Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics  Project B3Numerical Analysis of equivariant evolution equations11031 Raphael Kruse. 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 spatiotemporal discretization which is based on a linear implicit EulerMaruyama method. In both cases we obtain optimal error estimates. The proofs are based on sharp integral versions of wellknown 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.
