Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation
Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation
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声学中组合势积分算子的条件数估计及其边界元离散化
DOI：
10.1002/num.20643
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发表时间：
2010
影响因子：
3.9
通讯作者：
Betcke T
中科院分区：
文献类型：
--
作者：
Betcke T
我们考虑经典的耦合，组合场积分方程公式的时间谐波声散射的声音软有界障碍。在最近的工作中，我们已经证明了这些公式的L2条件数的上下界，以及经典声学单层和双层势算子的范数。这些界限在一定程度上明确了条件数对波数k、散射体几何形状和耦合参数的依赖性。例如，在通常选择耦合参数的情况下，当散射体为圆形或球形时，条件数的增长速度为k1/3ask→∞，而对于一类“捕获”障碍物，条件数的增长速度为k7/5。在这篇文章中，我们证明了进一步的界限，锐化和扩展我们以前的结果。特别地，我们证明了存在条件数增长与exp（γk）一样快的陷阱障碍，当γ> 0时，通过某种序列，ask→∞.这一结果取决于指数本地化界的拉普拉斯特征函数在椭圆，我们证明在附录中。我们还阐明了低k时二维耦合参数的正确选择。在文章的第二部分中，我们重点讨论了这些算子的边界元离散化。我们讨论了在何种程度上的连续运营商的界限也满足其离散的同行，通过数值实验，我们提供了一些理论结果，定量和渐近的支持证据，进一步表明的上限和下限可能会更尖锐。© 2010 Wiley Periodicals，Inc. Numer Methods Partial Differential Eq，2010
We consider the classical coupled, combined‐field integral equation formulations for time‐harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on theL2condition numbers for these formulations and also on the norms of the classical acoustic single‐ and double‐layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave numberk, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows likek1/3ask→∞, when the scatterer is a circle or sphere, it can grow as fast ask7/5for a class of “trapping” obstacles. In this article, we prove further bounds, sharpening and extending our previous results. In particular, we show that there exist trapping obstacles for which the condition numbers grow as fast as exp(γk), for someγ> 0, ask→∞through some sequence. This result depends on exponential localization bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for lowk. In the second part of the article, we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010
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