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

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 35:1698–1728, 2015), 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.

An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions

“…Thus in effect evaluating the collocation entries amounts to computing the integrals considered in the present work. (as demonstrated by Gibbs et al (2020)) by oversampling M ≫ N . Thus we introduce additional equispaced sampling points between the spline points t + n,j n,j ∪ t − n,j n,j .…”

“…L1 error = u − utrue L 1 (Γ) utrue L 1 (Γ) , where u(s) = N n=1 V + n (s)e iωs +V − n (s)e −iωs is our approximation to −∂ψ/∂n on Γ and the corresponding reference value was computed using the collocation method provided by Gibbs et al (2020) using 576 degrees of freedom. We observe that for fixed p the error grows at an algebraic rate in k, whilst as we increase p the error decays exponentially, similar to the observations made by Hewett et al (2015) and Gibbs et al (2020). Hence, together with the hybrid basis provided by Hewett et al (2015), our direct Filon method can indeed be used to solve a 2D scattering problem at near uniform cost even for large frequencies.…”

“…To conclude this example, we highlight the conceptual difference between our current Filon-based method, and the numerical steepest descent method used in Gibbs et al (2020). The latter method is highly efficient but requires some knowledge and care in order to split the domain of integration into subintervals on which the basis functions have analytic extensions.…”

A high-frequency boundary element method for scattering by a class of multiple obstacles

“…As suggested in Remark 6.1, sophisticated quadrature rules are required in conjunction with the proposed method, but these rules can be difficult to implement for oscillatory and singular double and triple integrals. Alternatively, the approximation space of §4.3 may be implemented as a collocation BEM (following the approach of Gibbs et al (2019)), which would reduce the dimension of each integral by one, making for easier implementation of oscillatory and singular quadrature rules.…”

An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions

“…Motivated by advances in approximation theory [1,2], recent years have seen the successful application of oversampling as a framework to improve the convergence properties of collocation methods, whilst broadly preserving their conceptual simplicity. Meanwhile, the potential of oversampling has been realized for instance in the context of Trefftz methods [5,14], in hybrid numerical-asymptotic methods for high-frequency wave scattering [10] and for eigenvalue problems involving ordinary or partial differential equations [12].…”