The adjoint double layer potential on smooth surfaces in $\mathbb{R}^3$ and the Neumann problem

TL;DR

This paper introduces a new regularization method for computing the adjoint double layer potential in 3D Laplace problems, achieving high accuracy and convergence without special coordinate systems.

Contribution

The authors develop a simple, accurate regularization technique for the adjoint double layer potential that improves error bounds and enables efficient numerical solutions of the Neumann problem.

Findings

Error of regularization is $O( ext{}\delta^3)$, improved to $O( ext{}}\delta^5)$ with modification.

Achieves about $O(h^4)$ convergence in numerical examples.

Method does not require special coordinates for integral evaluation.

Abstract

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. We then regularize the Green's function, with a radial parameter $\delta$. We show that a natural regularization has error $O(\delta^3)$, and a simple modification improves the error to $O(\delta^5)$. The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem and evaluate the solution on the boundary. Choosing $\delta = ch^{4/5}$, we find about $O(h^4)$ convergence in our examples, where $h$ is the spacing in a background grid.