Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

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.