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A high frequency boundary element method for scattering by a class of nonconvex obstacles

Numerische Mathematik volume 129, pages 647–689 (2015)Cite this article

Abstract

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

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Acknowledgments

The authors thank the editor and the anonymous reviewers for their helpful comments, and acknowledge funding support from EPSRC (EP/F067798/1 to S.N.C-W., D.P.H. and S.L., and DTG studentship to A.T.).

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Author notes

  1. D. P. Hewett

    Present address: Mathematical Institute, University of Oxford, Oxford, UK

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Reading, Reading, UK

    S. N. Chandler-Wilde, D. P. Hewett, S. Langdon & A. Twigger

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  1. S. N. Chandler-Wilde
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  2. D. P. Hewett
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  3. S. Langdon
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  4. A. Twigger
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Correspondence to D. P. Hewett.

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Supported by EPSRC grant EP/F067798/1.

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Chandler-Wilde, S.N., Hewett, D.P., Langdon, S. et al. A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numer. Math. 129, 647–689 (2015). https://doi.org/10.1007/s00211-014-0648-7

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  • DOI: https://doi.org/10.1007/s00211-014-0648-7

Mathematics Subject Classification

  • 65N38
  • 65R20
  • 78A45
  • 78M15
  • 78M35