Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint
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 $\mathbb{R}^d$ (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 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 $ε$, $Δt$ 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 $Δt=\mathcal{O}(ε^{2+θ})$ for a positive $θ$, the convergence of such a $(ε, Δt, h)$ scheme towards the unique weak solution of the initial problem, \textit{ a priori} strongly in $L^2(Ω;L^2(0,T;L^2(Λ)))$ and \textit{a posteriori} also strongly in $L^{p}(0,T; L^2(Ω\times Λ))$ for any finite $p\geq 1$.