Numerical analysis of the stochastic Stefan problem

TL;DR

This paper develops a convergence analysis for the gradient discretisation method applied to a stochastic Stefan problem with multiplicative noise, demonstrating its applicability to various numerical schemes.

Contribution

It provides the first convergence proof for GDM applied to stochastic Stefan problems, covering multiple numerical methods within a unified framework.

Findings

Convergence of GDM solutions proved using compactness and stochastic analysis.

Applicable to diverse numerical schemes like finite elements and finite volume methods.

Numerical tests validate theoretical convergence results.

Abstract

The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorohod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.