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 study, via variational methods, the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface. The boundary ∂D is assumed to lie within a finite distance of a flat plane and the incident wave is that arising from an inhomogeneous term in the Helmholtz equation whose support lies within some finite distance of the boundary ∂D. Via analysis of an equivalent variational formulation, we provide the first proof of existence of a unique solution to a three-dimensional rough surface scattering problem for an arbitrary wave number. Our method of analysis does not require any smoothness of the boundary which can, for example, be the graph of an arbitrary bounded continuous function. An attractive feature is that all constants in a priori bounds, for example the inf-sup constant of the sesquilinear form, are bounded by explicit functions of the wave number and the maximum surface elevation. 1. Introduction. This paper is concerned with the development and analysis of a variational formulation for scattering by unbounded surfaces, in particular, with the study of what are termed rough surface scattering problems in the engineering literature. We shall use the phrase rough surface to denote surfaces which are a (usually nonlocal) perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. Such problems arise frequently in applications, for example in modeling acoustic and electromagnetic wave propagation over outdoor ground and sea surfaces, and are the subject of intensive studies in the engineering literature, with a view to developing both rigorous methods of computation and approximate, asymptotic, or statistical methods (see, e.g., the reviews and monographs by Ogilvy In this paper we will focus on a particular, typical problem of the class, which models time harmonic acoustic scattering by a sound soft rough surface. In particular, we seek to solve the Helmholtz equation with wave number k > 0, Δu + k 2 u = g, in the perturbed half-plane or half-space D ⊂ R n , n = 2, 3. We suppose that the homogeneous Dirichlet boundary condition u = 0 holds on ∂D, and a suitable radiation condition is imposed to select a unique solution to this problem. We shall give in the next section complete details about our assumptions on D and on the radiation condition, but we now note that the inhomogeneous term g might be in L 2 (D) with bounded support, or be a more general distribution. In addition the boundary ∂D may or may not be the graph of a function.

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598-618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

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...

In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichletto-Neumann map applied on the boundary. The second formulation is as a second kind boundary integral equation in which the solution is sought as a combined single-and double-layer potential. For the variational formulation we obtain, in the case when the obstacle is starlike, explicit upper and lower bounds which show that the inf-sup constant decreases like k −1 as the wave number k increases. We also give an example where the obstacle is not starlike and the inf-sup constant decreases at least as fast as k −2. For the boundary integral equation formulation, if the boundary is also Lipschitz and piecewise smooth, we show that the norm of the inverse boundary integral operator is bounded independently of k if the coupling parameter is chosen correctly. The methods we use also lead to explicit bounds on the solution of the scattering problem in the energy norm when the obstacle is starlike in which the dependence of the norm of the solution on the wave number and on the geometry are made explicit.