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Parallel Multigrid Methods and Coarse Grid LDLT Solver for Maxwell’s Eigenvalue Problem

  • Daniel Maurer
  • Christian Wieners
Conference paper

Abstract

We consider efficient numerical solution methods for Maxwell’s eigenvalue problem in bounded domains. A suitable finite element discretization with Nédélec elements on tetrahedra leads to large linear systems of equations, where the resulting matrices are symmetric but singular and where the kernel corresponds to divergence-free vector fields. The discretized eigenvalue problem is solved by a preconditioned iterative eigenvalue solver, using a modification of the LOBPCGmethod extended by a projection onto the divergence-free vector fields, where we use a multigrid preconditioner for a regularized Maxwell problem as well as for the solution of the projection problem. In both cases, a new parallel direct block LDL T decomposition is used for the solution of the coarse grid problem. LDL T

Keywords

Parallel computing finite elements block LDLT decomposition Maxwell’s eigenvalue problem LOBPCG method multigrid preconditioner 

Notes

Acknowledgements

The authors acknowledge the financial support from BMBF grant 01IH08014A within the joint research project ASIL (Advanced Solvers Integrated Library). We thank W. Koch (CST) for the construction of the accelerator configuration.

References

  1. 1.
    Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.-Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hiptmair, R.: Multigrid method for maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–436 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Knayzev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Scientific Comput. 23(2), 517–541 (2001)CrossRefGoogle Scholar
  4. 4.
    Maurer, D., Wieners, C.: A parallel block LU decomposition method for distributed finite element matrices. Parallel Comput. http://dx.doi.org/10.1016/j.parco.2011.05.007 (2011)
  5. 5.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon, Oxford (2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. Sloot, P.M.A. et al. (eds.) Computational science – ICCS 2002. 2nd international conference, Amsterdam, the Netherlands, April 21–24, 2002. Proceedings. Part 2. Berlin, Springer. Lect. Notes Comput. Sci. 2330, 355–363 (2002)Google Scholar
  7. 7.
    Schenk, O., Gärtner, K., Fichtner, W., Stricker, A.: PARDISO: A high-performance serial and parallel sparse linear solver in semiconductor device simulation. FGCS. Future Generat. Comput. Syst. 18(1), 69–78 (2001)zbMATHCrossRefGoogle Scholar
  8. 8.
    Wieners, C.: Distributed point objects. A new concept for parallel finite elements. Kornhuber, R., et al. (eds.) Domain decomposition methods in science and engineering. Selected papers of the 15th international conference on domain decomposition, Berlin, Germany, July 21–25, 2003. Berlin, Springer. Lect. Notes Comput. Sci. Eng. 40, 175–182 (2005)Google Scholar
  9. 9.
    Wieners, C.: A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Vis. Sci. 13(4), 161–175 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Zaglmayr, S.: High order finite element methods for electromagnetic field computation. PhD thesis, Johannes Kepler Universität Linz (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Institute for Applied and Numerical Mathematics 3Karlsruhe Institute of TechnologyKarlsruheGermany

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