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 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 $p\geq 1$.