Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation

TL;DR

This paper introduces a finite difference analog of boundary integral equations to solve Dirichlet problems for elliptic PDEs, providing a practical, accurate, and theoretically justified numerical method that preserves key mathematical properties.

Contribution

It proves the validity and unique solvability of a simplified finite difference method for boundary integral equations in general domains, with $O(h^2)$ accuracy.

Findings

Method produces $O(h^2)$ accuracy for harmonic functions.

Ensures unique solvability with bounds in energy norms.

Maintains mathematical structure of classical integral equations.

Abstract

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation can be solved by replacing the integral operator with a finite difference calculation on a regular grid. A practical method of this type has been developed by the second author. In this paper we prove the validity of a simplified version of this method for the Dirichlet problem in a general domain in $R^2$ or $R^3$. Given a boundary value, we solve for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley-Weller solution for the unknown harmonic function with accuracy $O(h^2)$. We prove the unique solvability for the density, with bounds in norms based on the energy or Dirichlet norm, using techniques which mimic those of exact potentials. The analysis reveals that this crude method maintains much of the mathematical structure of the classical integral equation. Examples are included.