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

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: Delta u - u(t) = lambda(+) center dot chi({u>0}) - lambda(-) center dot chi{u<0}, (t,x) is an element of(0,T) x Omega, where T < infinity, lambda(+) , lambda(-) > 0 are Lipschitz continuous functions, and Omega subset of 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.