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

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

2011-11-27

arxiv.org/abs/1111.6287v1

Mathematics

Numerical Analysis

Scientific paper

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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