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 NannenEmail author
  • Joachim Schöberl


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 



We would like to thank Timo Weidl for suggesting the model problem to us. Moreover, the authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Martin Halla
    • 1
  • Thorsten Hohage
    • 2
  • Lothar Nannen
    • 1
    Email author
  • 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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