Commun. Comput. Phys., 11 (2012), pp. 573-593.

A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions

S. N. Chandler-Wilde 1, S. Langdon 1*, M. Mokgolele 1

1 Department of Mathematics, University of Reading, Whiteknights PO Box 220, Reading RG6 6AX, U.K.

Received 23 December 2009; Accepted (in revised version) 4 January 2011
Available online 24 October 2011


We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

AMS subject classifications: 35J05, 65N38, 65R20

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Key words: Boundary integral equation method, high frequency scattering, convex polygons, impedance boundary conditions.

*Corresponding author.
Email: (S. N. Chandler-Wilde), (S. Langdon), (M. Mokgolele)

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