Numerische Mathematik

, Volume 133, Issue 1, pp 103–139 | Cite as

Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs

  • Martin Halla
  • Thorsten Hohage
  • Lothar Nannen
  • Joachim Schöberl
Article

Abstract

We consider time harmonic wave equations in cylindrical wave-guides with physical solutions for which the signs of group and phase velocities differ. The perfectly matched layer methods select modes with positive phase velocity, and hence they yield stable, but unphysical solutions for such problems. We derive an infinite element method for a physically correct discretization of such wave-guide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are Hardy spaces of a curved domain. The Hardy space is constructed such that it contains a simple and convenient Riesz basis with small condition numbers. In this paper the new method is only discussed for a one-dimensional fourth order model problem. Exponential convergence is shown. The method does not use a modal separation and works on an interval of frequencies. Numerical experiments confirm exponential convergence.

Mathematics Subject Classification

65N30 35J50 30H10 

References

  1. 1.
    Achenbach, J.D.: Wave propagation in elastic solids (North-Holland Series in Applied Mathematics and Mechanics). North-Holland Series in Applied Mathematics and Mechanics, vol. 16. North Holland (1987)Google Scholar
  2. 2.
    Bécache, É., Bonnet-BenDhia, A.S., Legendre, G.: Perfectly matched layers for the convected Helmholtz equation. SIAM J. Numer. Anal. 42, 409–433 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bell, S.R.: The Cauchy Transform, Potential Theory, and Conformal Mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)Google Scholar
  4. 4.
    Bonnet-BenDhia, A.S., Chambeyron, C., Legendre, G.: On the use of perfectly matched layers in the presence of long or backward guided elastic waves. Wave Motion, (2015). doi:10.1016/j.wavemoti.2013.08.001
  5. 5.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer-Verlag, New York (1999). doi:10.1007/978-1-4612-1426-7 CrossRefMATHGoogle Scholar
  6. 6.
    Duren, P.L.: Theory of \(H^{p}\) Spaces. Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)Google Scholar
  7. 7.
    Graff, K.: Wave Motion in Elastic Solids. Oxford Engineering Science Series. Clarendon Press, Oxford (1975)Google Scholar
  8. 8.
    Halla, M.: Modeling and numerical simulation of wave propagation in elastic wave guides. Master’s thesis, Vienna UT (2012)Google Scholar
  9. 9.
    Halla, M., Hohage, T., Nannen, L., Schöberl, J.: Efficient and robust approximation of the Helmholtz equation: Hardy space method for waveguides. Report 55/2012, Mathematisches Forschungsinstitut Oberwolfach (2012)Google Scholar
  10. 10.
    Halla, M., Nannen, L.: Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems. (2015). arXiv:1506.04781
  11. 11.
    Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall Inc., Englewood Cliffs (1962)Google Scholar
  12. 12.
    Hohage, T., Nannen, L.: Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47(2), 972–996 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hohage, T., Nannen, L.: Convergence of infinite element methods for scalar waveguide problems. Preprint 31/2013, Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3 (2013)Google Scholar
  14. 14.
    Kress, R.: Linear Integral Equations, Applied Mathematical Sciences, vol. 82, 2nd edn. Springer-Verlag, New York (1999)CrossRefGoogle Scholar
  15. 15.
    Nannen, L., Schöberl, J.: Software module ngs-waves. http://sourceforge.net/projects/ngs-waves/ . Addon to the mesh generator Netgen and the high order finite element code NGSolve (2014)
  16. 16.
    Pommerenke, C.: Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer-Verlag, Berlin (1992)Google Scholar
  17. 17.
    Schöberl, J.: Netgen: an advancing front 2d/3d-mesh generator based on abstract rules. Comput. Visual. Sci 1, 41–52 (1997)CrossRefMATHGoogle Scholar
  18. 18.
    Schöberl, J.: C++11 implementation of finite elements in NGSolve. Preprint 30, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014)Google Scholar
  19. 19.
    Skelton, E.A., Adams, S.D.M., Craster, R.V.: Guided elastic waves and perfectly matched layers. Wave Motion 44(7–8), 573–592 (2007). doi:10.1016/j.wavemoti.2007.03.001 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Martin Halla
    • 1
  • Thorsten Hohage
    • 2
  • Lothar Nannen
    • 1
  • Joachim Schöberl
    • 1
  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienViennaAustria
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August Universität GöttingenGöttingenGermany

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