Convergence of Finite Difference Methods for Poisson's Equation with Interfaces

TL;DR

This work addresses convergence of a finite difference scheme for the Poisson equation with discontinuous coefficients across an interface using the Ghost Fluid Method. It recasts the problem in a weak formulation with a uniformly elliptic bilinear form $B[u,v] = ∫_Ω β ∇u · ∇v$ and constructs an abstract convergence framework based on uniform boundedness, uniformly bounded extension operators $T^h$ with strong approximation, and weak consistency, which together yield convergence of the discrete solutions to the continuous weak solution. The numerical method is derived by discretizing the weak problem on a rectangle and recasting the scheme into the Ghost Fluid Method form; convergence is proved without a rate due to the lack of full $H^2$ regularity. Overall, the results justify the use of GFM for interface problems and show a principled path from weak formulations to convergent finite difference methods.

Abstract

In this paper, a weak formulation of the discontinuous variable coefficient Poisson equation with interfacial jumps is studied. The existence, uniqueness and regularity of solutions of this problem are obtained. It is shown that the application of the Ghost Fluid Method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the Ghost Fluid Method is proven to be convergent.

Theorems & Definitions (27)

• Definition 2.1
• Theorem 2.1
• proof
• Definition 2.2
• Theorem 2.2
• proof
• Remark 1
• Lemma 3.1
• proof
• Remark 2