Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference Method

TL;DR

This paper develops a finite difference numerical method for the two-phase obstacle problem, establishing existence, uniqueness, and convergence of the solution, and demonstrating its effectiveness through numerical simulations.

Contribution

It introduces a new finite difference scheme for the two-phase obstacle problem, including a convergence-proof projected Gauss-Seidel algorithm.

Findings

Proved existence and uniqueness of the discrete solution.

Established convergence of the proposed algorithm.

Validated the method with numerical simulations.

Abstract

In this paper we consider the numerical approximation of the two-phase membrane (obstacle) problem by finite difference method. First, we introduce the notion of viscosity solution for the problem and construct certain discrete nonlinear approximation system. The existence and uniqueness of the solution of the discrete nonlinear system is proved. Based on that scheme, we propose projected Gauss-Seidel algorithm and prove its convergence. At the end of the paper we present some numerical simulations.