A fast hybrid Galerkin method for high-frequency acoustic scattering

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.

Keywords:

• High-frequency acoustic scattering
• hybrid numerical-asymptotic
• Galerkin method
• Helmholtz equation
• boundary integral equation

AMS Subject Classifications:

• 65R20
• 65N38
• 35J05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported partly by NSF of China [grant number 11371376], [grant number 11422102]; the Hunan Provincial Natural Science Foundation of China [grant number 2016JJ4037]; the Scientific Research Fund of Hunan Provincial Education Department [grant number 15A077]; the Fundamental Research Funds for the Central Universities [grant number 15lgzd07]; the Construct Program of the Key Discipline in Hunan Province; the Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.