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: \[Δ u -u_t=λ^+·χ_{u>0 -λ^-·χ_{u<0 ,(t,x)(0,T)×Ω,\] where T < ∞, λ^+ ,λ^- > 0 are Lipschitz continuous functions, and ΩR^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. , number=1, journal=Armenian Journal of Mathematics, author=Arakelyan, Avetik, year=2015, month=May, pages=32–49