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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 617390, 13 pages
http://dx.doi.org/10.1155/2012/617390
Research Article

New Preconditioners with Two Variable Relaxation Parameters for the Discretized Time-Harmonic Maxwell Equations in Mixed Form

1Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu, Nanjing 210046, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guangxi, Guilin 541004, China

Received 4 May 2012; Revised 3 July 2012; Accepted 4 July 2012

Academic Editor: Ion Zaballa

Copyright © 2012 Yuping Zeng and Chenliang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. Chen, Q. Du, and J. Zou, “Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients,” SIAM Journal on Numerical Analysis, vol. 37, no. 5, pp. 1542–1570, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. G.-H. Cheng, T.-Z. Huang, and S.-Q. Shen, “Block triangular preconditioners for the discretized time-harmonic Maxwell equations in mixed form,” Computer Physics Communications, vol. 180, no. 2, pp. 192–196, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. C. Greif and D. Schötzau, “Preconditioners for the discretized time-harmonic Maxwell equations in mixed form,” Numerical Linear Algebra with Applications, vol. 14, no. 4, pp. 281–297, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Q.-Y. Hu and J. Zou, “Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions,” Mathematics of Computation, vol. 73, no. 245, pp. 35–61, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. I. Perugia, D. Schötzau, and P. Monk, “Stabilized interior penalty methods for the time-harmonic Maxwell equations,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 41-42, pp. 4675–4697, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. D. N. Arnold, R. S. Falk, and R. Winther, “Multigrid in H(div) and H(curl),” Numerische Mathematik, vol. 85, no. 2, pp. 197–217, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. Gopalakrishnan, J. E. Pasciak, and L. F. Demkowicz, “Analysis of a multigrid algorithm for time harmonic Maxwell equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 1, pp. 90–108, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. R. Hiptmair, “Multigrid method for Maxwell's equations,” SIAM Journal on Numerical Analysis, vol. 36, no. 1, pp. 204–225, 1999. View at Publisher · View at Google Scholar
  9. S. Reitzinger and J. Schöberl, “An algebraic multigrid method for finite element discretizations with edge elements,” Numerical Linear Algebra with Applications, vol. 9, no. 3, pp. 223–238, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Alonso and A. Valli, “An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations,” Mathematics of Computation, vol. 68, no. 226, pp. 607–631, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Q.-Y. Hu and J. Zou, “A nonoverlapping domain decomposition method for Maxwell's equations in three dimensions,” SIAM Journal on Numerical Analysis, vol. 41, no. 5, pp. 1682–1708, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Toselli, “Overlapping Schwarz methods for Maxwell's equations in three dimensions,” Numerische Mathematik, vol. 86, no. 4, pp. 733–752, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. A. Toselli, O. B. Widlund, and B. I. Wohlmuth, “An iterative substructuring method for Maxwell's equations in two dimensions,” Mathematics of Computation, vol. 70, no. 235, pp. 935–949, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. Hiptmair and J. Xu, “Nodal auxiliary space preconditioning in H(curl) and H(div) spaces,” SIAM Journal on Numerical Analysis, vol. 45, no. 6, pp. 2483–2509, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. T.-Z. Huang, L.-T. Zhang, T.-X. Gu, and X.-Y. Zuo, “New block triangular preconditioner for linear systems arising from the discretized time-harmonic Maxwell equations,” Computer Physics Communications, vol. 180, no. 10, pp. 1853–1859, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Z.-Z. Bai, B. N. Parlett, and Z.-Q. Wang, “On generalized successive overrelaxation methods for augmented linear systems,” Numerische Mathematik, vol. 102, no. 1, pp. 1–38, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Z.-Z. Bai and Z.-Q. Wang, “On parameterized inexact Uzawa methods for generalized saddle point problems,” Linear Algebra and Its Applications, vol. 428, no. 11-12, pp. 2900–2932, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Z.-Z. Bai, “Structured preconditioners for nonsingular matrices of block two-by-two structures,” Mathematics of Computation, vol. 75, no. 254, pp. 791–815, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Z.-Z. Bai and G. H. Golub, “Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems,” IMA Journal of Numerical Analysis, vol. 27, no. 1, pp. 1–23, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z.-Z. Bai, G. H. Golub, and C.-K. Li, “Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices,” Mathematics of Computation, vol. 76, no. 257, pp. 287–298, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. Z.-Z. Bai, G. H. Golub, and M. K. Ng, “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 3, pp. 603–626–626, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Z.-Z. Bai, G. H. Golub, and J.-Y. Pan, “Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems,” Numerische Mathematik, vol. 98, no. 1, pp. 1–32, 2004. View at Publisher · View at Google Scholar
  23. Z.-Z. Bai, M. K. Ng, and Z.-Q. Wang, “Constraint preconditioners for symmetric indefinite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 2, pp. 410–433, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Z.-Z. Bai and Z.-Q. Wang, “Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems,” Journal of Computational and Applied Mathematics, vol. 187, no. 2, pp. 202–226, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. P. B. Monk, “A mixed method for approximating Maxwell's equations,” SIAM Journal on Numerical Analysis, vol. 28, no. 6, pp. 1610–1634, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. Monk, Finite Elements For Maxwell's Equations, Oxford University Press, New York, NY, USA, 2003.
  27. J.-C. Nédélec, “Mixed finite elements in ℝ3,” Numerische Mathematik, vol. 35, no. 3, pp. 315–341, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. C. Grief and D. Schötzau, “Preconditioers for saddle point linear systems with highly singular (1, 1) blocks, ENTA,” Special Volume on Saddle Point Systems, vol. 22, pp. 114–121, 2006.
  29. D. Li, C. Grief, and D. Schotzau, “Parallel numerical solution of the timeharmonic Maxwell equations in mixed form,” Linear Algebra and Its Applications, vol. 19, no. 3, pp. 525–539, 2012. View at Publisher · View at Google Scholar
  30. A. Greenbaum, Iterative Methods for Solving Linear Ystems, vol. 17, SIAM, Philadelphia, Pa, USA, 1997. View at Publisher · View at Google Scholar