A High Frequency hp Boundary Element Method for Scattering by Convex Polygons
A High Frequency hp Boundary Element Method for Scattering by Convex Polygons
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DOI：
10.1137/110856812
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发表时间：
2013-02
期刊：
影响因子：
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通讯作者：
D. Hewett;S. Langdon;J. Melenk
中科院分区：
文献类型：
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作者：
D. Hewett;S. Langdon;J. Melenk
在本文中，我们提出并分析了一种混合$HP$边界元法的解决方案的高频声散射问题的声音软凸多边形，在近似空间中丰富的振荡基函数，有效地捕捉高频渐近的解决方案。我们证明，无论是理论上和通过数值例子，指数收敛的多项式的顺序，而且提供严格的误差估计，我们的近似的解决方案和远场图案，其中依赖于频率的所有常数是明确的。重要的是，这些估计证明，为了在这些量的计算中实现任何期望的精度，随着频率的增加，与常规方法所需的至少线性增长相比，与频率的对数成比例地增加自由度的数量就足够了。
In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.