Asymptotic approximations for the close evaluation of double-layer potentials

TL;DR

This paper develops an asymptotic approximation method for accurately evaluating double-layer potentials near boundaries in boundary integral methods, improving computational efficiency for solving Laplace's equation.

Contribution

It introduces a novel asymptotic approximation for double-layer potentials near boundaries, including nonlocal corrections, with demonstrated numerical efficiency and accuracy.

Findings

Accurately approximates near-boundary double-layer potentials

Requires modest computational resources

Validated through numerical examples

Abstract

When using the boundary integral equation method to solve a boundary value problem, the evaluation of the solution near the boundary is challenging to compute because the layer potentials that represent the solution are nearly-singular integrals. To address this close evaluation problem, we apply an asymptotic analysis of these nearly singular integrals and obtain an asymptotic approximation. We derive the asymptotic approximation for the case of the double-layer potential in two and three dimensions, representing the solution of the interior Dirichlet problem for Laplace's equation. By doing so, we obtain an asymptotic approximation given by the Dirichlet data at the boundary point nearest to the interior evaluation point plus a nonlocal correction. We present numerical methods to compute this asymptotic approximation, and we demonstrate the efficiency and accuracy of the asymptotic approximation through several examples. These examples show that the asymptotic approximation is useful as it accurately approximates the close evaluation of the double-layer potential while requiring only modest computational resources.