In this talk we discuss the numerical solution of the problem of acoustic plane wave scattering by a 2D convex smooth sound-soft object using hybrid numerical-asymptotic methods.
In recent joint work with Víctor Domínguez (Pamplona) and Valery Smyshlyaev (Bath) we developed Galerkin methods with oscillatory basis functions for this problem and proved that the resulting discretisations are almost uniformly accurate as the wave number k increases.
The key components of the analysis are: (i) Estimates for the continuity and coercivity of the boundary integral operators explicitly in terms of k. (ii) A proper description of the asymptotic behaviour of the solution in a format suitable for numerical analysis, by further development of the classical asymptotics results for this problem. (iii) Design of suitable ansatz spaces for use in the Galerkin method and the analysis of their consistency error. (iv) Construction of quadrature methods for the highly oscillatory Galerkin integrals.
In the talk we will describe recent results on this programme of work and some remaining open problems.