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.

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces H s (Ω) and H s (Ω), for s ∈ R and an open Ω ⊂ R n . We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.A main result of the paper is to exhibit one-and two-dimensional counterexamples that show that H s (Ω) and H s (Ω) are not in general interpolation scales. It is well-known that these Sobolev spaces are interpolation scales for all s ∈ R when Ω is Lipschitz. In that case we demonstrate, via a number of counterexamples that, in general (we suspect, in fact, whenever Ω R n ), H s (Ω) and H s (Ω) are not exact interpolation scales. Indeed, we exhibit simple examples where the ratio of interpolation norm to intrinsic Sobolev norm may be arbitrarily large. Along the way we give explicit formulas for some of the interpolation norms arising that may be of interest in their own right. We remark that our investigations, which are inspired by applications arising in boundary integral equation methods (see [9]), in particular are inspired by McLean [18], and by its appendix on interpolation of Banach and Sobolev spaces. However a result of §4 is that one result claimed by McLean ( [18, Theorem B.8]) is false.Much of the Hilbert space Section 3 builds strongly on previous work. In particular, our result that, with the right normalisations, the norms in the K-and J-methods of interpolation coincide in the Hilbert space case is a (corrected version of) an earlier result of Ameur [2] (the normalisations proposed and the definition of the J-method norm seem inaccurate in [2]). What is new in our Theorem 3.3 is the method of proof-all of our proofs in this section are based on the spectral theorem that every bounded normal operator is unitarily equivalent to a multiplication operator on L 2 (X , M, µ), for some measure space (X , M, µ), this coupled with an elementary explicit treatment of interpolation on weighted L 2 spaceswhich deals seamlessly with the general Hilbert space case without an assumption of separability or that H 0 ∩ H 1 is dense in H 0 and H 1 . Again, our result in Theorem 3.5 that there is only one (geometric) interpolation space of exponent θ, when interpolating Hilbert spaces, is a version of McCarthy's [17] uniqueness theorem. What is new is that we treat the general Hilbert space case by a method of proof based on the aforementioned spectral theorem. O...

Abstract. In this paper we propose and analyse a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.Key words. High frequency scattering, Boundary Element Method, hp-method AMS subject classifications. 65N12, 65N38, 65R201. Introduction. Conventional numerical schemes for time-harmonic acoustic scattering problems, with piecewise polynomial approximation spaces, become prohibitively expensive in the high frequency regime where the scatterer is large compared to the wavelength of the incident wave. For two-dimensional problems the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to frequency. On the other hand, approximation via high frequency asymptotics alone is often insufficiently accurate when the frequency lies within ranges of practical interest. These issues are very well understood; see e.g. [9,10,37,30,12] and the many references therein.The problem of 'bridging the gap' between conventional numerical methods and fully asymptotic approaches has received a great deal of attention in recent years. Significant progress has been made in developing numerical methods which can achieve a prescribed level of accuracy at high frequencies with fewer degrees of freedom than conventional approaches. A key idea underpinning much recent work is to express the scattered field as a sum of products of known oscillatory functions, selected using knowledge of the high frequency asymptotics, with slowly oscillating amplitude functions, and to approximate just the amplitudes by piecewise polynomials (we call this the hybrid numerical-asymptotic approach). Applying this idea within a boundary element method (BEM) context is particularly attractive since in this case one need only understand the high frequency behaviour on the boundary of the scatterer, rather than throughout the whole propagation domain. Computational methods implementing this approach have been applied successfully to problems of scattering by both smooth [8, 21, 22, 26] and non-smooth [15, 20, 2, 29, 16] convex scatterers, the latter building on previous work on hybrid h-version BEM methods for the special problem of acoustic scat...

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.