Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering
This work addresses a known bottleneck in computational wave scattering for acoustics and electromagnetics, offering an incremental improvement by combining geometrical theory of diffraction with mesh refinement.
The paper proposes a new methodology for handling shadow boundary effects in hybrid numerical-asymptotic boundary element methods for high-frequency scattering, achieving provable effectiveness through rigorous numerical analysis and numerical results.
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.