A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

TL;DR

This paper introduces a hybrid boundary element method for high-frequency wave scattering by 2D screens and apertures, achieving exponential convergence and frequency-independent computational cost through oscillatory basis functions and specialized quadrature.

Contribution

The paper presents a novel frequency-independent boundary element method with exponential convergence for 2D scattering problems, improving efficiency at high frequencies.

Findings

Converges exponentially with degrees of freedom

Achieves fixed accuracy at high frequencies with constant cost

Uses Filon quadrature for oscillatory integral computation

Abstract

We propose and analyse a hybrid numerical-asymptotic $hp$ boundary element method for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom $N$ increases, and that to achieve any desired accuracy it is sufficient to increase $N$ in proportion to the square of the logarithm of the frequency as the frequency increases (standard boundary element methods require $N$ to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary "breakwater" problem of propagation through an aperture in an infinite sound-hard screen.