Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

We consider the numerical approximation of a general second-order semilinear parabolic stochastic partial differential equation driven by multiplicative and additive space–time noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in a trace class and give a convergence proof in the root-mean-square L² norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction–diffusion equation in two dimensions as well as a nonlinear example of a two-dimensional stochastic advection–diffusion–reaction equation motivated from realistic porous media flow.