Volume 57, Number 2, March-April 2023
|Page(s)||745 - 783|
|Published online||27 March 2023|
Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise
Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France
2 CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France
3 Universität Duisburg-Essen, Fakultät für Mathematik, Essen, Germany
* Corresponding author: email@example.com
Accepted: 20 October 2022
We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod’s representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.
Mathematics Subject Classification: 60H15 / 35K05 / 65M08
Key words: Stochastic heat equation / multiplicative Lipschitz noise / finite-volume method / stochastic compactness method / variational approach / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2023
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