  The hybrid numerical-asymptotic (HNA) approach aims to reduce the
computational cost of conventional numerical methods for high frequency wave
scattering problems by enriching the numerical approximation space with
oscillatory basis functions, chosen based on partial knowledge of the high
frequency solution asymptotics. In this paper we propose a new methodology for
the treatment of shadow boundary effects in HNA boundary element methods, using
the classical geometrical theory of diffraction phase functions combined with
mesh refinement. We develop our methodology in the context of scattering by a
class of sound-soft nonconvex polygons, presenting a rigorous numerical
analysis (supported by numerical results) which proves the effectiveness of our
HNA approximation space at high frequencies. Our analysis is based on a study
of certain approximation properties of the Fresnel integral and related
functions, which govern the shadow boundary behaviour.
