Numerische Mathematik

, Volume 129, Issue 4, pp 647–689

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

  • S. N. Chandler-Wilde
  • D. P. Hewett
  • S. Langdon
  • A. Twigger
Article

DOI: 10.1007/s00211-014-0648-7

Cite this article as:
Chandler-Wilde, S.N., Hewett, D.P., Langdon, S. et al. Numer. Math. (2015) 129: 647. doi:10.1007/s00211-014-0648-7

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.

Mathematics Subject Classification

65N38 65R20 78A45 78M15 78M35 

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • S. N. Chandler-Wilde
    • 1
  • D. P. Hewett
    • 1
    • 2
  • S. Langdon
    • 1
  • A. Twigger
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK