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Two scale Hardy space infinite elements for scalar waveguide problems

Abstract

We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds.

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Acknowledgments

Open access funding provided by Austrian Science Fund (FWF). The first author acknowledges support from the Austrian Science Fund (FWF) grant W1245-N25.

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Correspondence to Martin Halla.

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Communicated by: Ivan Graham

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Halla, M., Nannen, L. Two scale Hardy space infinite elements for scalar waveguide problems. Adv Comput Math 44, 611–643 (2018). https://doi.org/10.1007/s10444-017-9549-5

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Keywords

  • Waveguide
  • Cut-off frequency
  • Wood’s anomaly
  • Pole condition
  • Hardy space infinite element method

Mathematics Subject Classification (2010)

  • 65H17
  • 65N12
  • 65N30
  • 78M10