Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

TL;DR

This paper introduces a high-order, recursive reduction quadrature scheme for efficiently evaluating Laplace layer potentials and their derivatives on smooth surfaces in three dimensions, achieving high accuracy and significant speedup.

Contribution

The novel recursive reduction quadrature method reduces the need for adaptive integration, enabling fast, high-accuracy evaluation of surface integrals in three dimensions.

Findings

Achieves twelve-digit accuracy for close evaluations.

Provides at least five times speedup in quadrature matrix construction.

Largely eliminates the need for adaptive integration in high-order quadratures.

Abstract

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.