A high frequency boundary element method for scattering by a class of nonconvex obstacles
A high frequency boundary element method for scattering by a class of nonconvex obstacles
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DOI：
10.1007/s00211-014-0648-7
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发表时间：
2014-01
影响因子：
2.1
通讯作者：
S. Chandler-Wilde;D. Hewett;S. Langdon;A. Twigger
中科院分区：
文献类型：
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作者：
S. Chandler-Wilde;D. Hewett;S. Langdon;A. Twigger
在本文中，我们提出并分析了一种混合数值渐进边界元方法，用于解决一类声音软非凸多边形的高频声散射问题。通过精心选择的振荡基函数丰富了近似空间；这些是通过研究解的高频渐近行为来选择的。我们通过数值例子支持的严格误差分析证明，为了达到任何所需的精度，随着频率的增加，自由度的数量仅与频率的对数成比例增长就足够了，这与传统方法所需的至少线性增长相反。这似乎是针对非凸障碍物散射问题的第一个此类数值分析结果。我们的分析基于边界上解的法向导数的新频率显式界限及其在复平面上的解析延拓。
In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.