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Convergence of infinite element methods for scalar waveguide problems

Abstract

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the perfectly matched layer method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.

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Acknowledgments

The authors dedicate this work to Werner Koch for his inspiration, generosity and enthusiasm concerning the topic of resonances in waveguides. Unfortunately he passed away on August 28, 2012. Moreover, we would like to thank an anonymous referee for detailed and helpful suggestions and corrections. Financial support by the German Science Foundation through grant HO 2551/5 is gratefully acknowledged.

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Correspondence to Thorsten Hohage.

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Communicated by Ralf Hiptmair.

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Hohage, T., Nannen, L. Convergence of infinite element methods for scalar waveguide problems. Bit Numer Math 55, 215–254 (2015). https://doi.org/10.1007/s10543-014-0525-x

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Keywords

  • Transparent boundary conditions
  • Waveguide
  • Perfectly matched layer
  • Hardy space infinite elements

Mathematics Subject Classification

  • 78M10
  • 78A45
  • 65N30
  • 65N12