混合数值-渐近方法中阴影边界效应
Shadow boundary effects in hybrid numerical-asymptotic methods for high frequency scattering
AI总结：
本文提出了一种处理混合数值-渐近方法中阴影边界效应的新方法，结合几何衍射理论和网格细化，通过研究菲涅耳积分的近似性质，证明了在高频情况下HNA近似空间的有效性。
摘要：
混合数值-渐近（HNA）方法旨在通过在数值近似空间中加入振荡基函数来降低传统数值方法在高频波散射问题中的计算成本。本文提出了一种新方法，用于处理HNA边界元方法中的阴影边界效应，结合经典几何衍射理论的相位函数和网格细化。我们在此背景下研究了由一类声软非凸多边形散射问题，通过研究菲涅耳积分及相关函数的近似性质，证明了在高频情况下HNA近似空间的有效性。
英文摘要：
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.
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