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

Published online by Cambridge University Press:  20 August 2015

S. N. Chandler-Wilde ,
S. Langdon  and
M. Mokgolele
S. N. Chandler-Wilde*
Affiliation:
Department of Mathematics, University of Reading, Whiteknights PO Box 220, Reading RG6 6AX, U.K
S. Langdon*
Affiliation:
Department of Mathematics, University of Reading, Whiteknights PO Box 220, Reading RG6 6AX, U.K
M. Mokgolele*
Affiliation:
Department of Mathematics, University of Reading, Whiteknights PO Box 220, Reading RG6 6AX, U.K
*
Email address:s.n.chandler-wilde@reading.ac.uk
*Corresponding author.Email:s.langdon@reading.ac.uk
Email address:m_mokgolele@yahoo.co.uk
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Abstract

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.

Keywords

35J0565N3865R20Boundary integral equation methodhigh frequency scatteringconvex polygonsimpedance boundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 573 - 593
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Bernard, J. M. L., A spectral approach for scattering by impedance polygons, Quart. J. Mech. Appl. Math., 59 (2006), 517–550.Google Scholar
[2]Burton, A. J. and Miller, G. F., The applications of integral equations to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. Lond. Ser. A, 323 (1971), 201–210.Google Scholar
[3]Chandler-Wilde, S. N. and Graham, I. G., Boundary integral methods in high frequency scattering, in “Highly Oscillatory Problems”, Engquist, B., Fokas, T., Hairer, E., Iserles, A., editors, CUP (2009), 154–193.CrossRefGoogle Scholar
[4]Chandler-Wilde, S. N. and Hothersall, D. C., On the Green Function for Two-Dimensional Acoustic Propagation above a Homogeneous Impedance Plane, Research Report, Dept. of Civil Engineering, University of Bradford, UK, 1991.Google Scholar
[5]Chandler-Wilde, S. N. and Hothersall, D. C., A uniformly valid far field asymptotic expansion of the Green function for two-dimensional propagation above a homogeneous impedance plane, J. Sound Vibration, 182 (1995), 665–675.CrossRefGoogle Scholar
[6]Chandler-Wilde, S. N. and Langdon, S., A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J. Numer. Anal., 45 (2007), 610–640.Google Scholar
[7]Chandler-Wilde, S. N., Langdon, S. and Ritter, L., A high-wavenumber boundary-element method for an acoustic scattering problem, Phil. Trans. R. Soc. Lond. A, 362 (2004), 647–671.Google Scholar
[8]Chandler-Wilde, S. N. and Zhang, B., Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. R. Soc. Lond. A, 454 (1998), 519–542.Google Scholar
[9]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, John Wiley, 1983.Google Scholar
[10]Dominguez, V., Graham, I. G., and Smyshlyaev, V. P., A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering, Numer. Math., 106 (2007), 471–510.CrossRefGoogle Scholar
[11]Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Springer, 1977.Google Scholar
[12]Herman, M. I. and Volakis, J. L., High frequency scattering from polygonal impedance cylinders and strips, IEEE Trans. Antennas and Propagation, 36 (1988), 679–689.Google Scholar
[13]Howarth, C., Integral Equation Formulations for Scattering Problems, MSc thesis, University of Reading, 2009.Google Scholar
[14]Langdon, S. and Chandler-Wilde, S. N., A wavenumber independent boundary element method for an acoustic scattering problem, SIAM J. Numer. Anal., 43 (2006), 2450–2477.Google Scholar
[15]McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, CUP, 2000.Google Scholar
[16]Mokgolele, M., Numerical Solution of High Frequency Acoustic Scattering Problems, PhD Thesis, University of Reading, 2009.Google Scholar
[17]Osipov, A., Hongo, K. and Kobayashi, H., High-frequency approximations for electromagnetic field near a face of an impedance wedge, IEEE Trans. Antennas and Propagation, 50 (2002), 930–940.CrossRefGoogle Scholar
[18]Osipov, A. V. and Norris, A. N., The Malyuzhinets theory for scattering from wedge boundaries: A review, Wave Motion, 29 (1999), 313–340.CrossRefGoogle Scholar
[19]Perrey-Debain, E., Trevelyan, J. and Bettess, P., On wave boundary elements for radiation and scattering problems with piecewise constant impedance, IEEE Trans. Antennas and Propagation, 53 (2005), 876–879.CrossRefGoogle Scholar
[20]Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential Equations, Springer, 1999.Google Scholar
[21]Schenck, H. A., Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Am., 44 (1968), 41–58.Google Scholar
[22]Ylä-Oijala, P. and Järvenpää, S., Iterative solution of high-order boundary element method for acoustic impedance boundary value problems, J. Sound Vibration, 291 (2006), 824–843.CrossRefGoogle Scholar