DeepAI AI Chat
Log In Sign Up

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

by   Andrew Gibbs, et al.

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 application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.


page 3

page 12

page 13

page 15

page 19


Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

This paper is concerned with the design of two different classes of Gale...

Asymptotic expansions of high-frequency multiple scattering iterations for sound hard scattering problems

We consider the two-dimensional high-frequency plane wave scattering pro...

Almost complete analytical integration in Galerkin BEM

In this work, semi-analytical formulae for the numerical evaluation of s...

Calderón Preconditioning for Acoustic Scattering at Multi-Screens

We propose a preconditioner for the Helmholtz exterior problems on multi...

Wavelet Galerkin Method for an Electromagnetic Scattering Problem

The Helmholtz equation is challenging to solve numerically due to the po...

Efficient function approximation on general bounded domains using wavelets on a cartesian grid

Fourier extension is an approximation method that alleviates the periodi...