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

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 Itô type. The problem is set in a bounded domain of $R^d$ (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 “à 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 $ϵ$, $\Delta t$ 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 $\Delta t=O\left({ϵ}^{2+\theta }\right)$ for a positive $\theta$, the convergence of such a “$(ϵ,\Delta t,h)$” scheme towards the unique weak solution of the initial problem, a priori strongly in $L^2(\Omega ;L^2(0,T;L^2\left(\Lambda \right)))$ and a posteriori also strongly in $L^p(0,T;L^2(\Omega \times \Lambda ))$ for any finite $p\ge 1$.