A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem

Abstract : In this paper for two-phase parabolic obstacle-like problem, [\Delta u -u_t=\lambda^+\cdot\chi_{{u>0}}-\lambda^-\cdot\chi_{{u<0}},\quad (t,x)\in (0,T)\times\Omega,] where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain, we will introduce a certain variational form, which allows us to define a notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of discretized scheme to a unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.
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https://hal-ensta-paris.archives-ouvertes.fr//hal-00973802
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Submitted on : Friday, April 4, 2014 - 3:28:33 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:04 AM

Identifiers

• HAL Id : hal-00973802, version 1
• ARXIV : 1111.6287

Citation

Avetik Arakelyan. A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem. 2013. ⟨hal-00973802⟩

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