  In this paper we propose and analyse a hybrid numerical-asymptotic boundary
element method for the solution of problems of high frequency acoustic
scattering by a class of sound-soft nonconvex polygons. The approximation space
is enriched with carefully chosen oscillatory basis functions; these are
selected via a study of the high frequency asymptotic behaviour of the
solution. We demonstrate via a rigorous error analysis, supported by numerical
examples, that to achieve any desired accuracy it is sufficient for the number
of degrees of freedom to grow only in proportion to the logarithm of the
frequency as the frequency increases, in contrast to the at least linear growth
required by conventional methods. This appears to be the first such numerical
analysis result for any problem of scattering by a nonconvex obstacle. Our
analysis is based on new frequency-explicit bounds on the normal derivative of
the solution on the boundary and on its analytic continuation into the complex
plane.
