Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint
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| Format: | Preprint |
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2024
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| _version_ | 1866915473465016320 |
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| author | Bauzet, Caroline Sultan, Cédric Vallet, Guy Zimmermann, Aleksandra |
| author_facet | Bauzet, Caroline Sultan, Cédric Vallet, Guy Zimmermann, Aleksandra |
| contents | 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 $p\geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint Bauzet, Caroline Sultan, Cédric Vallet, Guy Zimmermann, Aleksandra Numerical Analysis Analysis of PDEs 60H15, 35K55, 65M08 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 $p\geq 1$. |
| title | Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint |
| topic | Numerical Analysis Analysis of PDEs 60H15, 35K55, 65M08 |
| url | https://arxiv.org/abs/2407.04399 |