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We propose a new robust method for the computation of scattering of highfrequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L 2 , with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L 2 -coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k 1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k → ∞, for a fixed number of degrees

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form ∫ 1 −1 f (s) exp(iks) ds. We use a Filon approach based on interpolating f at the classical Clenshaw-Curtis points cos(jπ/N), j = 0,. .. , N. The rule may be implemented in O(N log N) operations. We prove error estimates which show explicitly how the error depends both on the parameters k and N and on the Sobolev regularity of f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. We also describe a method for implementating the method and prove its stability both when N ≤ k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.

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We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.