论文标题
快速多极的方法，用于评估具有局部校正四倍的层电位
Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures
论文作者
论文摘要
虽然快速多极方法（FMM）广泛用于对由拉普拉斯（Laplace），helmholtz，Maxwell或Stokes方程控制的潜在磁场的快速评估，但它们与高阶四相结合以评估层势的耦合仍然是活跃研究的领域。在三个维度中，需要解决许多问题，包括将表面作为高阶贴片的结合，合并精确的正交规则，用于在此类斑块上整合奇异或弱的green功能，以及它们与FMM近距离近距离相互作用的OCT-Tree数据结构的耦合。尽管后者对于点分布很简单，但贴片的近场是由其物理尺寸确定的，而不是表面上离散点的分布。 在这里，我们提出了一个通用框架，用于将局部校正的四元素与FMM有效耦合，主要依赖于所谓的通用高斯四肢规则，并补充了自适应整合。但是，该方法非常通用，并且很容易适用于其他方案，例如扩展的正交（QBX）。我们还引入了许多加速度以降低正交生成本身的成本，并提供了几个数值的声学散射示例，这些散射证明了该方案的准确性，鲁棒性和计算效率。在Intel I5 2.3GHz处理器的单个核心上，该方案的实现可以在每秒1000至10,000点之间生成近场正交校正，具体取决于准确性和所需的精度。本工作中描述的算法的福特实现可在https://gitlab.com/fastalgorithms/fmm3dbie上获得。
While fast multipole methods (FMMs) are in widespread use for the rapid evaluation of potential fields governed by the Laplace, Helmholtz, Maxwell or Stokes equations, their coupling to high-order quadratures for evaluating layer potentials is still an area of active research. In three dimensions, a number of issues need to be addressed, including the specification of the surface as the union of high-order patches, the incorporation of accurate quadrature rules for integrating singular or weakly singular Green's functions on such patches, and their coupling to the oct-tree data structures on which the FMM separates near and far field interactions. Although the latter is straightforward for point distributions, the near field for a patch is determined by its physical dimensions, not the distribution of discretization points on the surface. Here, we present a general framework for efficiently coupling locally corrected quadratures with FMMs, relying primarily on what are called generalized Gaussian quadratures rules, supplemented by adaptive integration. The approach, however, is quite general and easily applicable to other schemes, such as Quadrature by Expansion (QBX). We also introduce a number of accelerations to reduce the cost of quadrature generation itself, and present several numerical examples of acoustic scattering that demonstrate the accuracy, robustness, and computational efficiency of the scheme. On a single core of an Intel i5 2.3GHz processor, a Fortran implementation of the scheme can generate near field quadrature corrections for between 1000 and 10,000 points per second, depending on the order of accuracy and the desired precision. A Fortran implementation of the algorithm described in this work is available at https://gitlab.com/fastalgorithms/fmm3dbie.
