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

Abstract

In this paper the numerical solution of non-autonomous 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 $L^p$-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.

31 pages, 1 figure

References in corpus (2)

• A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients
• On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients

Cited by in corpus (2)

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• Space-time approximation of stochastic $p$-Laplace systems