A frequency-independent boundary element method for scattering by two-dimensional screens and apertures
For computational acoustics, this method overcomes the linear scaling of standard BEM with frequency, enabling efficient high-frequency scattering simulations.
The paper proposes a hybrid numerical-asymptotic boundary element method for scattering by 2D screens, achieving exponential convergence with degrees of freedom scaling as the square of the log of frequency, and numerical results suggest frequency-independent cost for fixed accuracy.
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.