Layer potential quadrature on manifold boundary elements with constant densities for Laplace and Helmholtz kernels in $\mathbb{R}^3$

TL;DR

This paper introduces a novel geometric decomposition method for accurately evaluating layer potentials on manifold boundary elements with constant densities, effectively handling singular and nearly singular integrals for Laplace and Helmholtz equations in 3D.

Contribution

It proposes a new two-term decomposition based on differential geometry that reduces singularity and curvature effects, improving numerical evaluation of layer potentials.

Findings

Accurate evaluation of layer potentials demonstrated for Laplace and Helmholtz kernels.

Method effectively handles singular and nearly singular integrals.

Numerical validation confirms high accuracy across various scenarios.

Abstract

A method is proposed for evaluation of single and double layer potentials of the Laplace and Helmholtz equations on piecewise smooth manifold boundary elements with constant densities. The method is based on a novel two-term decomposition of the layer potentials, derived by means of differential geometry. The first term is an integral of a differential 2-form which can be reduced to contour integrals using Stokes' theorem, while the second term is related to the element curvature. This decomposition reduces the degree of singularity and the curvature term can be further regularized by a polar coordinate transform. The method can handle singular and nearly singular integrals. Numerical results validating the accuracy of the method are presented for all combinations of single and double layer potentials, for the Laplace and Helmholtz kernels, and for singular and nearly singular integrals.