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

TL;DR

This paper introduces a new method to handle shadow boundary effects in high frequency wave scattering problems using hybrid numerical-asymptotic boundary element methods, combining diffraction theory with mesh refinement for improved accuracy.

Contribution

It presents a novel approach integrating geometrical diffraction theory with mesh refinement to effectively treat shadow boundary effects in HNA boundary element methods for nonconvex polygon scattering.

Findings

Proves the effectiveness of the new HNA approximation space at high frequencies.

Provides rigorous numerical analysis supported by numerical results.

Analyzes approximation properties of Fresnel integrals related to shadow boundary behavior.

Abstract

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.