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

Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering