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High Order Transparent Boundary Conditions for the Helmholtz Equation

  • Lothar NannenEmail author
Chapter
Part of the Geosystems Mathematics book series (GSMA)

Abstract

We consider finite element simulations of the Helmholtz equation in unbounded domains. For computational purposes, these domains are truncated to bounded domains using transparent boundary conditions at the artificial boundaries. We present here two numerical realizations of transparent boundary conditions: the complex scaling or perfectly matched layer method and the Hardy space infinite element method. Both methods are Galerkin methods, but their variational framework differs. Proofs of convergence of the methods are given in detail for one dimensional problems. In higher dimensions radial as well as Cartesian constructions are introduced with references to the known theory.

Notes

Acknowledgements

Support from the Austrian Science Fund (FWF) through grant P26252 is gratefully acknowledged.

References

  1. 1.
    É. Bécache, A.-S. Bonnet-BenDhia, and G. Legendre, Perfectly matched layers for the convected Helmholtz equation, SIAM Journal on Numerical Analysis, 42 (2004), pp. 409–433.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185–200.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, An exact bounded perfectly matched layer for time-harmonic scattering problems, SIAM J. Sci. Comput., 30 (2007/08), pp. 312–338.Google Scholar
  4. 4.
    J. H. Bramble and J. E. Pasciak, Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems, Math. Comp., 76 (2007), pp. 597–614 (electronic).Google Scholar
  5. 5.
    J. H. Bramble and J. E. Pasciak,, Analysis of a Cartesian PML approximation to acoustic scattering problems in \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), J. Comput. Appl. Math., 247 (2013), pp. 209–230.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, third ed., 2008.Google Scholar
  7. 7.
    Z. Chen, C. Liang, and X. Xiang, An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number, Inverse Problems and Imaging, 7 (2013), pp. 663–678.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    W. C. Chew and W. H. Weedon, A 3d perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microwave Optical Tech. Letters, 7 (1994), pp. 590–604.CrossRefGoogle Scholar
  9. 9.
    P. G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4.Google Scholar
  10. 10.
    F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), pp. 2061–2090 (electronic).Google Scholar
  11. 11.
    D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 of Applied Mathematical Sciences, Springer-Verlag, Berlin, second ed., 1998.Google Scholar
  12. 12.
    L. Demkowicz and K. Gerdes, Convergence of the infinite element methods for the Helmholtz equation in separable domains, Numer. Math., 79 (1998), pp. 11–42.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Demkowicz and F. Ihlenburg, Analysis of a coupled finite-infinite element method for exterior Helmholtz problems, Numer. Math., 88 (2001), pp. 43–73.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. L. Duren, Theory of H p spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970.Google Scholar
  15. 15.
    D. Givoli, High-order local non-reflecting boundary conditions: a review, Wave Motion, 39 (2004), pp. 319–326.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Halla, Convergence of Hardy space infinite elements for Helmholtz scattering and resonance problems, Preprint 10/2015, Institute for Analysis and Scientific Computing; TU Wien, 2013, ISBN: 978-3-902627-05-6, 2015.Google Scholar
  17. 17.
    M. Halla, T. Hohage, L. Nannen, and J. Schöberl, Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs, Numerische Mathematik, (2015), pp. 1–37.Google Scholar
  18. 18.
    M. Halla and L. Nannen, Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems, Wave Motion, 59 (2015), pp. 94 – 110.MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. D. Hislop and I. M. Sigal, Introduction to spectral theory, vol. 113 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. With applications to Schrödinger operators.Google Scholar
  20. 20.
    T. Hohage and L. Nannen, Hardy space infinite elements for scattering and resonance problems, SIAM J. Numer. Anal., 47 (2009), pp. 972–996.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P. D. Hislop and I. M. Sigal,, Convergence of infinite element methods for scalar waveguide problems, BIT Numerical Mathematics, 55 (2014), pp. 215–254.MathSciNetGoogle Scholar
  22. 22.
    T. Hohage, F. Schmidt, and L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition. I. Theory, SIAM J. Math. Anal., 35 (2003), pp. 183–210.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    P. D. Hislop and I. M. Sigal,,, Solving time-harmonic scattering problems based on the pole condition. II. Convergence of the PML method, SIAM J. Math. Anal., 35 (2003), pp. 547–560.MathSciNetCrossRefGoogle Scholar
  24. 24.
    F. Ihlenburg, Finite element analysis of acoustic scattering, vol. 132 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998.Google Scholar
  25. 25.
    S. Kim and J. E. Pasciak, The computation of resonances in open systems using a perfectly matched layer, Math. Comp., 78 (2009), pp. 1375–1398.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    P. D. Hislop and I. M. Sigal,,,, Analysis of a Cartesian PML approximation to acoustic scattering problems in \(\mathbb{R}^{2}\), J. Math. Anal. Appl., 370 (2010), pp. 168–186.MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. Kress, Linear integral equations, vol. 82 of Applied Mathematical Sciences, Springer-Verlag, New York, second ed., 1999.Google Scholar
  28. 28.
    R. Kress, Chapter 1.2.1 - specific theoretical tools, in Scattering, R. P. Sabatier, ed., Academic Press, London, 2002, pp. 37 – 51.CrossRefGoogle Scholar
  29. 29.
    M. Lassas and E. Somersalo, On the existence and the convergence of the solution of the PML equations, Computing, 60 (1998), pp. 229–241.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M. Lassas and E. Somersalo, Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1183–1207.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    N. Moiseyev, Quantum theory of resonances: Calculating energies, width and cross-sections by complex scaling, Physics reports, 302 (1998), pp. 211–293.CrossRefGoogle Scholar
  32. 32.
    L. Nannen, T. Hohage, A. Schädle, and J. Schöberl, Exact Sequences of High Order Hardy Space Infinite Elements for Exterior Maxwell Problems, SIAM J. Sci. Comput., 35 (2013), pp. A1024–A1048.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    L. Nannen and A. Schädle, Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities, Wave Motion, 48 (2010), pp. 116–129.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation, Computers Math. Appl., 29 (1995), pp. 53–76.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M. E. Taylor, Partial differential equations. II, vol. 116 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. Qualitative studies of linear equations.Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Technische Universität WienWienAustria

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