Error bounds for full space-time splitting discretizations of semi-linear SPDEs -- with a focus on dG domain decompositions

TL;DR

This paper develops error bounds for a fully discretized scheme solving semi-linear SPDEs using domain decomposition with discontinuous Galerkin methods, enabling efficient parallel computations.

Contribution

It introduces a non-iterative domain decomposition approach with error analysis for semi-linear SPDEs using space-time splitting and DG discretization.

Findings

Established strong convergence results for the discretized scheme.

Validated theoretical error bounds with numerical experiments.

Demonstrated the efficiency of the domain decomposition method.

Abstract

We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.