Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint

TL;DR

This paper provides a theoretical convergence analysis of a numerical scheme for a stochastic Allen-Cahn problem with constraints, demonstrating strong convergence under specific discretization conditions.

Contribution

It introduces a novel convergence proof for a finite-volume scheme applied to a constrained stochastic Allen-Cahn equation with multiplicative noise.

Findings

Proves convergence of the scheme to the unique weak solution.

Establishes strong convergence in relevant function spaces.

Identifies conditions on time step relative to regularization parameter.

Abstract

The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of It\^o type. The problem is set in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and homogeneous Neumann boundary conditions are considered. The employed strategy consists in building a numerical scheme on a regularized version \`a la Moreau-Yosida of the constrained problem, and passing to the limit simultaneously with respect to the regularization parameter and the time and space steps, denoted respectively by $\epsilon$, $\Delta t$ and $h$. Combining a semi-implicit Euler-Maruyama time discretization with a Two-Point Flux Approximation (TPFA) scheme for the spatial variable, one is able to prove, under the assumption $\Delta t=\mathcal{O}(\epsilon^{2+\theta})$ for a positive $\theta$, the convergence of such a $(\epsilon, \Delta t, h)$ scheme towards the unique weak solution of the initial problem, \textit{ a priori} strongly in $L^2(\Omega;L^2(0,T;L^2(\Lambda)))$ and \textit{a posteriori} also strongly in $L^{p}(0,T; L^2(\Omega\times \Lambda))$ for any finite $p\geq 1$.