Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

TL;DR

This paper introduces a fast, stable hybrid numerical-asymptotic boundary element method for high-frequency scattering problems involving screens and apertures, achieving frequency-independent computational cost through collocation and oversampling techniques.

Contribution

It develops a collocation-based implementation of the high-order hp HNA boundary element method, improving computational efficiency and stability over previous Galerkin approaches for high-frequency scattering.

Findings

The collocation scheme significantly reduces integral evaluation complexity.

Oversampling stabilizes the method with about 25% excess points, independent of frequency.

Numerical results demonstrate effectiveness on fractal-like scatterers.

Abstract

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.