Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise

The aim of this article is to provide further strong convergence results for a spatio-temporal discretization of semilinear parabolic stochastic partial differential equations driven by additive noise. The approximation in space is performed by a standard finite element method and in time by a linear implicit Euler method. It is revealed how exactly the strong convergence rate of the full discretization relies on the regularity of the driven process. In particular, the full discretization attains an optimal convergence rate of order O(h2+τ) as the driven noise process is of trace class and satisfies certain regularity assumptions. Numerical examples corroborate the claimed strong orders of convergence.