Abstract

We consider high-frequency multiple-scattering problems in the exterior of two-dimensional smooth scatterers consisting of finitely many compact, disjoint, and strictly convex obstacles. To deal with this problem, we propose Galerkin boundary element methods, namely the frequency-adapted Galerkin boundary element methods and Galerkin boundary element methods generated using frequency-dependent changes of variables. For both of these new algorithms, in connection with each multiple-scattering iterate, we show that the number of degrees of freedom needs to increase as \(\mathcal {O}(k^{\epsilon })\) (for any \(\epsilon >0\)) with increasing wavenumber k to attain frequency-independent error tolerances. We support our theoretical developments by a variety of numerical implementations.

中文翻译：

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高频多重散射问题的Galerkin边界元方法

摘要

我们考虑二维光滑散射体外部的高频多重散射问题，该散射体由有限多个紧凑，不相交和严格凸出的障碍组成。为了解决这个问题，我们提出了Galerkin边界元方法，即频率自适应的Galerkin边界元方法和使用变量随频率变化的Galerkin边界元方法。对于这两个新算法，结合每个多重散射迭代，我们表明自由度的数量需要增加为\（\ mathcal {O}（k ^ {\ epsilon}）\）（对于任何\ （\ epsilon> 0 \））随着波数k的增加获得与频率无关的误差容限。我们通过各种数值实现来支持我们的理论发展。