On Using the Delta-Trigonometric Method to Solve the 2-D Neumann Potential Problem

Abstract

The Neumann problem for Laplace’s equation is often solved by means of the single layer potential representation, leading to a Fredholm integral equation of the second kind. We propose to solve this integral equation using a Petrov-Galerkin method with trigonometric polynomials as test functions, and a span of delta distributions centered at the boundary points as trial functions. For the exterior boundary value problem, the approximate potential converges exponentially away from the boundary and algebraically up to the boundary. We show that these convergence results hold even when the discretization matrices are computed via trapezoidal integration and present numerical examples to confirm our theory. For the interior boundary value problem, we suggest that the approximate potential also converges exponentially away from the boundary and algebraically up to the boundary, and present numerical examples to confirm our conjecture.