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Armenian Journal of Mathematics=Հայկական մաթեմատիկական հանդես

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Electronic Open Access Publication of the National Academy of Sciences of Armenia

#### Title:

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

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#### Subject:

Partial differential equations ; Numerical analysis

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#### Abstract:

In this paper we treat the numerical approximation of the 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 introduce a certain variation form, which allows us to define a notion of viscosity solution. We use defined viscosity solutions framework to apply Barles-Souganidis theory. The numerical projected 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 the discretized scheme to the unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.

#### Publisher:

National Academy of Sciences of Armenia