## Ivan Graham (University of Bath)

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.