# A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

### Abstract

In this paper the numerical solution of nonautonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p(\Omega)$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.

Type
Publication
Mathematics of Computation, vol. 88, 2793-2825
##### Raphael Kruse
###### Professor

Prof. Dr. Raphael Kruse is the head of the working group “Numerik stochastischer Differentialgleichungen” at Martin-Luther-University Halle-Wittenberg. His research interests include numerical methods and stochastic analysis for stochastic evolution equations and Monte Carlo methods.