A fast hybrid Galerkin method for high-frequency acoustic scattering. Issue 10 (27th July 2017)

• Record Type: Journal Article
• Authors: Li, Song-Hua; Xiang, Shuhuang; Xian, Jun
• Is Part Of: Applicable analysis. Volume 96:Issue 10(2017)
• Page Start: 1698
• Page End: 1712
• Publication Date: 2017-07-27
• Subjects: High-frequency acoustic scattering -- hybrid numerical-asymptotic -- Galerkin method -- Helmholtz equation -- boundary integral equation; 65R20 -- 65N38 -- 35J05; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2016.1265108
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract: We consider the scattering of a time-harmonic acoustic incident plane wave by a smooth convex object. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. Based on the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary. To achieve a good approximation at high frequencies with a relative low number of degrees of freedom, we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes: a polynomial grading on the illuminated side and a geometric grading on the shadow side. Using the asymptotic expansions of the solution, we prove that, as, the number of degree of freedom is able to decrease only very modestly to maintain a fixed absolute error bound ( is a typical behavior). Numerical experiments also show that the method achieves a better accuracy as, for a fixed number of degrees.