Well-posed boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains
Well-posed boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains
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用于解决二维 Lipschitz 域中亥姆霍兹传输问题的适定边界积分方程公式和 Nystr"om 离散化
DOI：
10.1216/jie-2016-28-3-395
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
C. Turc
中科院分区：
文献类型：
--
作者：
V. Domínguez;M. Lyon;C. Turc
我们提出了一个比较的性能求解器的基础上Nystrom离散的几个适定的边界积分方程制剂的亥姆霍兹传输问题在二维Lipschitz域。具体地说，我们集中讨论了Helmholtz传输问题的四类边界积分公式：（1）传输问题的经典第一类积分方程，（2）传输问题的经典第二类积分方程，（3）单积分方程，（4）最近引入的广义组合源积分方程的某些直接对应。前两个公式是唯一的公式，其适定性在Lipschitz域严格建立。在Lipschitz域上的Helmholtz透射问题的解的边界迹的适当的函数空间中，我们建立了后两个公式的适定性。我们给出了充分的数值证据，证明基于公式（3）和（4）的Nystrom求解器在计算上比基于经典公式（1）和（2）的求解器更有优势，特别是在高频高对比度传输问题的情况下。
We present a comparison between the performance of solvers based on Nystrom discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems (1) the classical first kind integral equations for transmission problems, (2) the classical second kind integral equations for transmission problems, (3) the {\em single} integral equation formulations, and (4) certain direct counterparts of recently introduced Generalized Combined Source Integral Equations. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystrom solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.