Modified representations for the close evaluation problem

TL;DR

This paper introduces modified layer potential representations to improve the accuracy of boundary integral methods near boundaries, addressing the close evaluation problem in solving PDEs.

Contribution

It proposes new modified representations for layer potentials that mitigate the close evaluation issue, enhancing numerical accuracy in boundary integral methods.

Findings

Effective in 2D and 3D numerical examples

Reduces errors in close boundary evaluations

Applicable to Laplace and Helmholtz problems

Abstract

When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem's solution. Similar to Gauss's law used to modify Laplace's double-layer potential, we use modified representations of Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.