We address the convergence analysis of a numerical scheme for an Allen-Cahn problem with constraint and with a stochastic external force given by a multiplicative noise of Itô type. The problem is set up in a bounded spatial domain of dimension \(2\) or \(3\) and homogeneous Neumann boundary conditions are considered.
We propose a time-space discretization, of semi-implicit Euler-Maruyama type with respect to time and a Two-Point Flux Approximation (TPFA) with respect to space for a regularized version of the constrained problem. Under the assumption \(\Delta t=\mathcal{O}(\epsilon^{2+\theta})\) for a positive \(\theta\) on the time parameter \(\Delta t\) and the regularization parameter \(\epsilon\) we show the convergence our scheme towards the unique variational solution of the problem.

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