In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods. * Colour online for monochrome figures available at journals.cambridge.org/anu.

Abstract. In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.Key words. Galerkin boundary element method, high frequency scattering, convex polygons, Helmholtz equation, large wave number, Lipschitz domains AMS subject classifications. 35J05, 65R201. Introduction. The scattering of time-harmonic acoustic waves by bounded obstacles is a classical problem that has received much attention in the literature over the years. Much effort has been put into the development of efficient numerical schemes, but an outstanding question yet to be fully resolved is how to achieve an accurate approximation to the scattered wave with a reasonable computational cost in the case that the scattering obstacle is large compared to the wavelength of the incident field.The standard boundary or finite element method approach is to seek an approximation to the scattered field from a space of piecewise polynomial functions. However, due to the oscillatory nature of the solution, such an approach suffers from the limitation that a fixed number of degrees of freedom K are required per wavelength in order to achieve a good level of accuracy, with the accepted guideline in the engineering literature being to take K = 10 (see for example [53] and the references therein). A further difficulty, at least for the finite element method, is the presence of "pollution errors", phase errors in wave propagation across the domain, which can lead to even more severe restrictions on the value of K when the wavelength is short [9,39].Let L be a linear dimension of the scattering obstacle, and set k = 2π/λ, where λ is the wavelength of the incident wave, so that k is the wave number, proportional to the frequency of the incident wave. Then a consequence of fixing K is that the number of degrees of freedom will be proportional to (kL) d , where d = N in the case of the finite element method (FEM), d = N − 1 in the case of the boundary element method (BEM), and N = 2...

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single-and double-layer potential operators.

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.

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the L 2 condition numbers for these formulations and also on the norms of the classical acoustic single-and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number k, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like k 1/3 as k → ∞, when the scatterer is a circle or sphere, it can grow as fast as k 7/5 for a class of "trapping" obstacles. In this article, we prove further bounds, sharpening and extending our previous results. In particular, we show that there exist trapping obstacles for which the condition numbers grow as fast as exp(γ k), for some γ > 0, as k → ∞ through some sequence. This result depends on exponential localization bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low k. In the second part of the article, we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.