Efficient Exact Quadrature of Regular Solid Harmonics Times Polynomials Over Simplices in $\mathbb{R}^3$

TL;DR

This paper extends a recursive numerical method for exact integration of regular solid harmonics over simplices in three dimensions, supporting polynomial densities while maintaining optimal complexity, useful for boundary element and vortex methods.

Contribution

It generalizes the Quadrature to Expansion (Q2X) method to handle polynomial density functions over simplices in D, preserving optimal asymptotic complexity.

Findings

Supports arbitrary polynomial densities over simplices.

Maintains optimal asymptotic complexity in the generalized method.

Applicable to boundary element and vortex methods with FMM.

Abstract

A generalization of a recently introduced recursive numerical method for the exact evaluation of integrals of regular solid harmonics and their normal derivatives over simplex elements in $\mathbb{R}^3$ is presented. The original Quadrature to Expansion (Q2X) method achieves optimal per-element asymptotic complexity, however, it considered only constant density functions over the elements. Here, we generalize this method to support arbitrary degree polynomial density functions, which is achieved in an extended recursive framework while maintaining the optimality of the complexity. The method is derived for 1- and 2- simplex elements in $\mathbb{R}^3$ and can be used for the boundary element method and vortex methods coupled with the fast multipole method.