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arXiv · 1801.08531

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

Raphael Kruse·Yue Wu
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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.

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Raphael Kruse, Yue Wu. 2018-01-25. A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations. https://doi.org/10.1090/mcom%2F3421

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