Discretization of the Interior Neumann Problem using Lusin Wavelets

Abstract

We prove the existence of Hilbert space frames for the complex Hardy subspaces of L2(T), consisting of simple rational functions whose poles are arranged according to a Whitney partition. We also present parts of the classical existence theory for the Dirichlet and Neumann problems based on layer potentials. We use our frame to construct new methods—the Casazza-Christensen method (CC method) and the Whitney method of fundamental solutions (WMFS)—for numerically solving Laplace’s equation with Neumann boundary data on the unit disk. Our goal is to resolve problems in computing the solution to high accuracy near the boundary, that is typical for the boundary integral equation method (BIE method). The methods are implemented in MATLAB and their performances are analyzed. Both methods converge exponentially when the exact solution is a polynomial, but when the exact solution is a rational function with a pole just outside of the boundary, the convergence is considerably slower. For polynomial data, the accuracy of the new methods near the boundary is much better than for a simple implementation of the BIE method. Some partial theoretical results related to the convergence and conditioning of the method of fundamental solutions and the WMFS are proved.