Efficient Fast Multipole Accelerated Boundary Elements via Recursive Computation of Multipole Expansions of Integrals

TL;DR

This paper introduces a novel Quadrature to Expansion (Q2X) method that efficiently computes multipole expansion coefficients for boundary element methods in 3D, significantly accelerating large-scale Laplace problems.

Contribution

The paper presents a new recursive Q2X approach for analytical multipole coefficient generation, enhancing FMM-based boundary element computations with controlled accuracy.

Findings

Q2X reduces computational cost of multipole moments

Method integrates seamlessly with FMM for large problems

Numerical tests confirm accuracy and efficiency

Abstract

In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov subspace methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - {\em they are only needed in terms of their contribution to the multipole expansion coefficients.} We propose a new fast method - \emph{Quadrature to Expansion (Q2X)} - for the analytical generation of the multipole expansion coefficients produced by the integral expressions for single and double layers on surface triangles; charge distributions over line segments and over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on the $O(1)$ per moment cost recursive computation of the moments. The method is developed for boundary element methods involving the Laplace Green's function in ${\mathbb R}^3$. The derived recursions are first compared against classical quadrature algorithms, and then integrated into FMM accelerated boundary element and vortex element methods. Numerical tests are presented and discussed.