Skip to main content
SpringerLink
Log in

Block Preconditioning Strategies for High Order Finite Element Discretization of the Time-Harmonic Maxwell Equations

  • Chapter
  • First Online:
Book cover Scientific Computing in Electrical Engineering SCEE 2010

Part of the book series: Mathematics in Industry ((TECMI,volume 16))

  • 1393 Accesses

Abstract

We study block preconditioning strategies for the solution of large sparse complex coefficients linear systems resulting from the discretization of the time-harmonic Maxwell equations by a high order discontinuous finite element method formulated on unstructured simplicial meshes. The proposed strategies are based on principles from incomplete factorization methods. Moreover, a complex shift is applied to the diagonal entries of the underlying matrices, a technique that has recently been exploited successfully in similar contexts and in particular for the multigrid solution of the scalar Helmholtz equation. Numerical results are presented for 2D and 3D electromagnetic wave propagation problems in homogeneous and heterogeneous media.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amestoy, P., Duff, I., L’Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Meth. App. Mech. Engng. 184, 501–520 (2000)

    Article  MATH  Google Scholar 

  2. Bollhöfer, M., Grote, M., Schenk, O.: Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media. SIAM J. Sci. Comput. 31(5), 3781–3805 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bollhöfer, M., Saad, Y.: Multilevel preconditioners constructed from inverse–based ILUs. SIAM J. Sci. Comput. 27(5), 1627–1650 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cockburn, B., Karniadakis, G., Shu, C. (eds.): Discontinuous Galerkin methods. Theory, computation and applications. In: Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)

    Google Scholar 

  5. Cockburn, B., Shu, C. (eds.): Special issue on discontinuous Galerkin methods. of J. Sci. Comput. 22-23 (2005)

    Google Scholar 

  6. Dawson, C. (ed.): Special issue on discontinuous Galerkin methods. Comput. Meth. App. Mech. Engng. 195 (2006)

    Google Scholar 

  7. Dolean, V., Fol, H., Lanteri, S., Perrussel, R.: Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods. J. Comp. Appl. Math. 218(2), 435–445 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dolean, V., Lanteri, S., Perrussel, R.: A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. J. Comput. Phys. 227(3), 2044–2072 (2007)

    Article  MathSciNet  Google Scholar 

  9. Erlangga, Y., Oosterlee, C., Vuik, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27, 1471–1492 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Erlangga, Y., Vuik, C., Oosterlee, C.: Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56, 648–666 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs systems I. General theory. SIAM J. Numer. Anal. 44(2), 753–778 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. van Gijzen, M., Erlangga, Y., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29, 1942–1958 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Magolu monga Made, M.: Incomplete factorization based preconditionings for solving the Helmholtz equation. Int J. Numer. Meth. Eng. 50, 1077–1101 (2001)

    Google Scholar 

  14. Monk, P.: Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics, TU Braunschweig, D-38106, Braunschweig, Germany

    Matthias Bollhöfer

  2. NACHOS project-team, INRIA Sophia Antipolis - Méditerranée research center, 2004 Route des Lucioles, 93, F-06902, Sophia Antipolis Cedex, France

    Stéphane Lanteri

Authors
  1. Matthias Bollhöfer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stéphane Lanteri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Bollhöfer .

Editor information

Editors and Affiliations

  1. ONERA, avenue Édouard Belin 2, Toulouse, 31055, France

    Bastiaan Michielsen

  2. LAPLACE-ENSEEIHT, rue Charles Camichel 2, Toulouse, 31071, France

    Jean-René Poirier

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bollhöfer, M., Lanteri, S. (2012). Block Preconditioning Strategies for High Order Finite Element Discretization of the Time-Harmonic Maxwell Equations. In: Michielsen, B., Poirier, JR. (eds) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry(), vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22453-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation