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Pré-Publication, Document De Travail Année : 2024

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

Résumé

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 per- turbed 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, ∆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 ∆t = O(ε$^{2+θ}$) for a positive θ, the convergence of such a “(ε,∆t,h)” scheme towards the unique weak solution of the initial problem, a priori strongly in $L^2$(Ω; L$^2$(0, T ; L$^2$(Λ))) and a posteriori also strongly in $L^p$(0, T ; $L^2$(Ω × Λ)) for any finite p ≥ 1.
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Dates et versions

hal-04636670 , version 1 (05-07-2024)

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  • HAL Id : hal-04636670 , version 1

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Caroline Bauzet, Cédric Sultan, Guy Vallet, Aleksandra Zimmermann. Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint. 2024. ⟨hal-04636670⟩
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