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


Authors: Raphael Kruse and Yue Wu
Journal: Math. Comp. 88 (2019), 2793-2825
MSC (2010): Primary 65C30; Secondary 60H15, 65M12, 65M60
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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Additional Information

Raphael Kruse
Affiliation: Institut für Mathematik, Technische Universität Berlin, Secr. MA 5-3, Straße des 17. Juni 136, DE-10623 Berlin, Germany
Email: kruse@math.tu-berlin.de

Yue Wu
Affiliation: Institut für Mathematik, Technische Universität Berlin, Secr. MA 5-3, Straße des 17. Juni 136, DE-10623 Berlin, Germany
Email: wu@math.tu-berlin.de

DOI: https://doi.org/10.1090/mcom/3421
Keywords: Galerkin finite element method, stochastic evolution equations, randomized Runge--Kutta method, strong convergence, noise approximation
Received by editor(s): January 28, 2018
Received by editor(s) in revised form: December 5, 2018
Published electronically: March 26, 2019
Additional Notes: The authors gratefully acknowledge financial support by the German Research Foundation through the research unit FOR 2402 – Rough paths, stochastic partial differential equations and related topics – at TU Berlin.
Article copyright: © Copyright 2019 American Mathematical Society