Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains
Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains
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解析域亥姆霍兹问题基本解法的稳定性和收敛性
DOI：
10.1016/j.jcp.2008.04.008
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发表时间：
2008
影响因子：
4.1
通讯作者：
Barnett A
中科院分区：
文献类型：
--
作者：
Barnett A
基本解法 (MFS) 是解决拉普拉斯和亥姆霍兹边值问题的常用工具。它的主要缺点是它经常导致病态方程组。在本文中，我们研究了解析域上的内亥姆霍兹问题，如何选择 MFS 基函数的奇点（电荷点），以便可以通过 MFS 基以数值稳定的方式表示近似解。对于单位圆盘上的亥姆霍兹问题，我们给出了完整的分析，其中包括高频（短波长）限制。对于更困难和非凸的域（例如新月形），我们演示了电荷点的正确选择如何与复杂平面中边界值问题的解决方案可以分析继续的距离相关联，而这又取决于域形状和边界数据。利用这一点，我们开发了一种定位电荷点的方法，使我们能够在各种分析领域达到通常为 10-11 的误差标准。在高频率下，每个波长仅需要 3 个点，这与边界积分方法相比非常有利。
The method of fundamental solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper, we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10-11on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods.