Fast algorithms for Quadrature by Expansion I: Globally valid expansions
Fast algorithms for Quadrature by Expansion I: Globally valid expansions
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通过扩展求积的快速算法 I：全局有效的扩展
DOI：
10.1016/j.jcp.2017.04.062
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
M. O’Neil
中科院分区：
文献类型：
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作者：
M. Rachh;A. Klöckner;M. O’Neil
使用积分方程方法求解偏微分方程边值问题的有效数值解需要两个主要工具：用于评价具有奇异核的层势积分算子的求积规则，以及用于求解由此产生的稠密线性系统的快速算法。传统上，这些工具是单独开发的。在这项工作中，我们提出了一个统一的数值方案的基础上耦合正交扩展，最近的正交方法，定制的快速多极子方法（FMM）的二维亥姆霍兹方程。该方法允许在线性时间复杂度的层电位的评估，在空间中的任何地方，与一个统一的，用户选择的准确度水平作为一个黑盒计算方法。提供这种能力需要超出标准的几何和算法的考虑，以及仔细考虑多极翻译的准确性。我们说明了我们的方法的速度和精度与各种数值例子。
The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on couplingQuadrature by Expansion, a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method.Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples.