Numerical Analysis of equivariant evolution equations
11-031 Raphael Kruse.
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 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.
