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

Kruse, Raphael; Wu, Yue

doi:10.1090/mcom/3421

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
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by Raphael Kruse and Yue Wu;

Math. Comp. 88 (2019), 2793-2825

DOI: https://doi.org/10.1090/mcom/3421

Published electronically: March 26, 2019

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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 $L^p$-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.

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