We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. IMA J. Numer. Anal. 35 (2015), pp.1698- 1728, where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices) and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.

中文翻译：

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基于最小二乘搭配的高频屏蔽和孔径问题的快速混合数值-渐近边界元方法

我们提出了一种混合数值渐近 (HNA) 边界元方法 (BEM)，用于二维屏幕和孔径的高频散射，其实现任何规定精度的计算成本仍受频率增加的限制。我们的方法是 Hewett 等人的高阶 hp HNA 近似空间的搭配实现。IMA J. 数字。肛门。35 (2015), pp.1698-1728，其中研究了 Galerkin 实现。当前搭配方案的一个优点是，在 BEM 矩阵条目中出现的一维高度振荡奇异积分比在 Galerkin 情况下出现的二维积分更容易评估，从而导致更快的计算时间。在这里，我们使用最速下降的数值方法以与频率无关的成本计算所需的积分，这涉及复杂的轮廓变形。从 Galerkin 到搭配的变化是非常重要的，因为基于方形线性系统的朴素搭配实现会遭受与 HNA 基础的数值冗余相关的严重数值不稳定性，这会产生高度病态的 BEM 矩阵。在本文中，我们展示了如何通过过采样消除这些不稳定性，并使用截断奇异值分解在加权最小二乘意义上解决由此产生的超定搭配系统。根据我们的数值实验，稳定该方法所需的过采样量是适中的（通常大约 25% 就足够了）并且与频率无关。