Recursive Analytical Quadrature of Laplace and Helmholtz Layer Potentials in $\mathbb{R}^3$

TL;DR

This paper introduces a recursive analytical method for efficiently evaluating layer potentials in boundary element methods for Laplace and Helmholtz equations, handling singularities within a unified framework.

Contribution

It develops a novel recursive scheme (RIPE) that simplifies the evaluation of various layer potentials for flat boundary elements, improving accuracy and efficiency.

Findings

Supports nearly singular, singular, and hypersingular integrals without modifications

Demonstrates high accuracy and computational efficiency

Applicable to both Laplace and Helmholtz kernels

Abstract

A method for the analytical evaluation of layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for piecewise flat boundary elements with polynomial shape functions. The method is based on dimension-reduction via the divergence theorem and a Recursive scheme for evaluating the resulting line Integrals for Polynomial Elements (RIPE). It is used to evaluate single layer, double layer, adjoint double layer, and hypersingular potentials, for both the Laplace and the Helmholtz kernels. It naturally supports nearly singular, singular, and hypersingular integrals under a single framework without separate modifications. The developed framework exhibits accuracy and efficiency.