Advances in Computational Mathematics

, Volume 44, Issue 3, pp 611–643 | Cite as

Two scale Hardy space infinite elements for scalar waveguide problems

  • Martin HallaEmail author
  • Lothar Nannen
Open Access


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.


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

Mathematics Subject Classification (2010)

65H17 65N12 65N30 78M10 



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


  1. 1.
    Bécache, É. , Bonnet-BenDhia, A.-S., Legendre, G.: Perfectly matched layers for the convected Helmholtz equation. SIAM J. Numer. Anal. 42, 409–433 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bermúdez, A., Hervella-Nieto, L., Prieto, A., Rodrí guez, R.: An exact bounded perfectly matched layer for time-harmonic scattering problems. SIAM J. Sci. Comput. 30(1), 312–338 (2007/08)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonnet-BenDhia, A.-S., Chambeyron, C., Legendre, G.: On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves. Wave Motion 51(2), 266–283 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonnet-BenDhia, A.-S., Chesnel, L., Ciarlet, P.: T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: Mathematical Modelling and Numerical Analysis 46(6), 1363–1387 (2012). cited By 22MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonnet-BenDhia, A.-S., Ciarlet, Jr, P., Zwölf, C. M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234(6), 1912–1919 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)CrossRefGoogle Scholar
  7. 7.
    Chew, W.C., Weedon, W.H.: A 3d perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw. Opt. Technol. Lett. 7, 590–604 (1994)CrossRefGoogle Scholar
  8. 8.
    Ciarlet, P.G.: Studies in Mathematics and its Applications, Vol. 4: The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
  9. 9.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators. Vol. I, volume 49 of Operator Theory: Advances and Applications. Basel, Birkhäuser Verlag (1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gohberg, I., Leiterer, J.: Methods from complex analysis in several variables: Holomorphic Operator Functions of One Variable and Applications, Volume 192 of Operator Theory Advances and Applications. Birkhäuser Verlag, Basel (2009)Google Scholar
  12. 12.
    Halla, M.: Convergence of Hardy space infinite elements for Helmholtz scattering and resonance problems. SIAM J. Numer. Anal. 54(3), 1385–1400 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Halla, M.: Regular Galerkin Approximation of Holomorphic T-Garding Operator Eigenvalue Problems. Report 04/2016, Institute for Analysis and Scientific Computing, TU Wien (2016)Google Scholar
  14. 14.
    Halla, M., Hohage, T., Nannen, L., Schöberl, J.: Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs. Numer. Math. 133(1), 103–139 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Halla, M., Nannen, L.: Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems. Wave Motion 59, 94–110 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hein, S., Hohage, T., Koch, W., Schöberl, J.: Acoustic resonances in high lift configuration. J Fluid Mech. 582, 179–202 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hohage, T., Nannen, L.: Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47(2), 972–996 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hohage, T., Nannen, L.: Convergence of infinite element methods for scalar waveguide problems. BIT Numer. Math. 55(1), 215–254 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karma, O.: Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17(3-4), 365–387 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Karma, O.: Approximation in eigenvalue problems for holomorphic F redholm operator functions. II. (Convergence rate). Numer. Funct. Anal. Optim. 17(3-4), 389–408 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kress, R.: Linear Integral Equations, Volume 82 of Applied Mathematical Sciences, 2nd edn. Springer-Verlag, New York (1999)CrossRefGoogle Scholar
  22. 22.
    Levitin, M., Marletta, M.: A simple method of calculating eigenvalues and resonances in domains with infinite regular ends. Proc. Roy. Soc. Edinburgh Sect. A 138(5), 1043–1065 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nannen, L.: Software module ngs-waves. addon to the mesh generator Netgen and the high order finite element code NGSolve (2014)
  24. 24.
    Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems with Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)CrossRefzbMATHGoogle Scholar
  25. 25.
    Netrusov, Y., Safarov, Y.: Weyl asymptotic formula for the Laplacian on domains with rough boundaries. Commun. Math. Phys. 253(2), 481–509 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Racec, P.N., Racec, E.R., Neidhardt, H.: Evanescent channels and scattering in cylindrical nanowire heterostructures. Phys. Rev. B 79, 155305 (2009)CrossRefGoogle Scholar
  27. 27.
    Rotter, S., Libisch, F., Burgdörfer, J., Kuhl, U., Stöckmann, H.-J.: Tunable Fano resonances in transport through microwave billiards. Phys. Rev. E 69, 046208 (2004)CrossRefGoogle Scholar
  28. 28.
    Schöberl, J.: Netgen - an advancing front 2d/3d-mesh generator based on abstract rules. Comput. Visual. Sci. 1, 41–52 (1997)CrossRefzbMATHGoogle Scholar
  29. 29.
    Schöberl, J.: C++11 implementation of finite elements in ngsolve. Preprint 30/2014, Institute for Analysis and Scientific Computing, TU Wien (2014)Google Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienWienAustria

Personalised recommendations