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.

翻译：在$\mathbb{R}^3$中的边界元方法（BEM）中，矩阵元素和右端项通常通过对层势函数与线、三角形和四面体体积单元上某函数乘积的解析或数值求积计算得出。当问题规模增大时，所得线性系统常通过Krylov子空间方法迭代求解，并使用快速多极方法（FMM）加速所需的矩阵向量乘积。采用FMM加速时，矩阵的大部分元素无需显式计算——仅需以多极展开系数的贡献形式参与计算。本文提出一种新的快速方法——\emph{求积到展开（Q2X）}——用于解析生成由表面积分表达式（单层和双层势）产生的多极展开系数；以及线单元和体积四面体上的电荷分布产生的系数。该方法与FMM高度集成，并具有可控误差。其核心是基于每矩量$O(1)$成本的矩递归计算。该方法针对涉及$\mathbb{R}^3$中Laplace格林函数的边界元方法而开发。首先将推导的递归公式与经典求积算法进行比较，随后将其集成至FMM加速的边界元和涡元方法中。最后呈现并讨论了数值实验结果。