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Domain Decomposition Methods in Science and Engineering XXI pp 351–358Cite as

Hybrid Domain Decomposition Solvers for the Helmholtz Equation

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

Abstract

In this work, domain decomposition preconditioners for the Helmholtz and the vector valued wave equation are presented. A mixed hybrid formulation of the underlaying equations provides in a natural way appropriate interface conditions, such that an efficient iterative solution with Krylov subspace methods combined with domain decomposition preconditioners is possible. Apart from a BDDC preconditioner a new Robin type domain decomposition preconditioner with an exact subdomain solver is constructed. The efficiency of these solvers is demonstrated by numerical examples.

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References

  1. Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19(1), 7–32 (1985)

    MATH  MathSciNet  Google Scholar 

  2. Desprès, B.: Méthodes de décomposition de domains pour les problèms de propagation d’ondes en régime harmonique. Ph.D. thesis, Université Paris IX Dauphine (1991)

    Google Scholar 

  3. Dohrmann, C.: A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246–258 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math. 64(5), 697–735 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Erlangga, Y.: Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15(1), 37–66 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Farhat, C., Macedo, A., Lesoinne, M.: A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer. Math. 85(2), 283–308 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Farhat, C., Avery, P., Tezaur, R., Li, J.: FETI-DPH: a dual-primal domain decomposition method for acoustic scattering. J. Comput. Acoust. 13(3), 499–524 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gander, M., Magoulès, F., Nataf, F.: Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24(1), 38–60 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Huber, M.: Hybrid discontinuous galerkin methods for the wave equation. Ph.D. thesis, Vienna University of Technology. http://www.asc.tuwien.ac.at/~mhuber/thesis_huber.pdf

  10. Huber, M., Pechstein, A., Schöberl, J.: Hybrid domain decomposition solvers for the helmholtz and the time harmonic Maxwell’s equation. In: Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, vol. 91, pp. 279–287. Springer, Berlin (2013)

    Google Scholar 

  11. Li, J., Widlund, O.: FETI-DP, BDDC, and block Cholesky methods. Int. J. Numer. Methods Eng. 66(2), 250–271 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Melenk, J.: On generalized finite element methods. Ph.D. thesis, University of Maryland (1995)

    Google Scholar 

  13. Monk, P., Sinwel, A., Schöberl, J.: Hybridizing Raviart-Thomas elements for the Helmholtz equation. Electromagnetics 30(1), 149–176 (2010)

    Article  Google Scholar 

  14. Schöberl, J.: NETGEN - an advanced front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1(1), 41–52 (1997)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, 1040, Wien, Austria

    Martin Huber & Joachim Schöberl

Authors
  1. Martin Huber
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  2. Joachim Schöberl
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Corresponding author

Correspondence to Martin Huber .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

  2. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

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© 2014 Springer International Publishing Switzerland

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Huber, M., Schöberl, J. (2014). Hybrid Domain Decomposition Solvers for the Helmholtz Equation. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_32

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