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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 365124, 9 pages
http://dx.doi.org/10.1155/2012/365124
Research Article

A Modified SSOR Preconditioning Strategy for Helmholtz Equations

1College of Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
2School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, China

Received 22 August 2011; Revised 7 November 2011; Accepted 18 November 2011

Academic Editor: Kok Kwang Phoon

Copyright © 2012 Shi-Liang Wu and Cui-Xia 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. C.-H. Guo, “Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the Helmholtz equation,” Applied Numerical Mathematics, vol. 19, no. 4, pp. 495–508, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, “On a class of preconditioners for solving the Helmholtz equation,” Applied Numerical Mathematics, vol. 50, no. 3-4, pp. 409–425, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A. Bayliss, C. I. Goldstein, and E. Turkel, “An iterative method for the Helmholtz equation,” Journal of Computational Physics, vol. 49, no. 3, pp. 443–457, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. G. Bao and W. W. Sun, “A fast algorithm for the electromagnetic scattering from a large cavity,” SIAM Journal on Scientific Computing, vol. 27, no. 2, pp. 553–574, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Y. Wang, K. Du, and W. W. Sun, “Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity,” Numerical Linear Algebra with Applications, vol. 16, no. 5, pp. 345–363, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, Boston, Mass, USA, 1996.
  7. J. Gozani, A. Nachshon, and E. Turkel, “Conjugate gradient comipled with multi-grid for an indefinite problem,” in Advances in Computer Methods for Partial Differential Equations, R. Vichnevestsky and R. S. Tepelman, Eds., vol. 5, pp. 425–427, IMACS, New Brunswick, NJ, USA, 1984. View at Google Scholar
  8. A. L. Laird, “Preconditioned iterative solution of the 2D Helmholtz equation,” First Year's Report 02/12, Hugh's College, Oxford, UK, 2002. View at Google Scholar
  9. Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, “Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation,” Applied Numerical Mathematics, vol. 56, no. 5, pp. 648–666, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Gander, “AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization,” Journal of Computational Acoustics, vol. 9, no. 4, pp. 1499–1506, 2001. View at Google Scholar
  11. R. E. Plessix and W. A. Mulder, “Separation-of-variables as a preconditioner for an iterative Helmholtz solver,” Applied Numerical Mathematics, vol. 44, no. 3, pp. 385–400, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. M. M. Made, R. Beauwens, and G. Warzée, “Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix,” Communications in Numerical Methods in Engineering, vol. 16, no. 11, pp. 801–817, 2000. View at Google Scholar · View at Zentralblatt MATH
  13. M. Benzi, “Preconditioning techniques for large linear systems: a survey,” Journal of Computational Physics, vol. 182, no. 2, pp. 418–477, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. F. Mazzia and R. Alan McCoy, “Numerical experimetns with a shifted SSOR preconditioner for symmetric matrices,” type TR/PA/98/12, CERFACS, 1998. View at Google Scholar
  15. Z.-Z. Bai, “Mofidied block SSOR preconditioners for symmetric positive definite linear systems,” Annals of Operations Research, vol. 103, pp. 263–282, 2001. View at Publisher · View at Google Scholar
  16. X. Chen, K. C. Toh, and K. K. Phoon, “A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations,” International Journal for Numerical Methods in Engineering, vol. 65, no. 6, pp. 785–807, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Y.-H. Ran and L. Yuan, “On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3050–3068, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. O. Axelsson, “A generalized SSOR method,” BIT, vol. 18, pp. 443–467, 1972. View at Google Scholar
  19. A. Hadjidimos, “Successive overrelaxation (SOR) and related methods,” Journal of Computational and Applied Mathematics, vol. 123, no. 1-2, pp. 177–199, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. O. Axelsson, Iterative Solution Methods, Cambridge University Press, New York, NY, USA, 1995.
  21. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436, 1952. View at Google Scholar · View at Zentralblatt MATH
  22. H. A. van der Vorst and J. B. M. Melissen, “A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex,” IEEE Transactions on Magnetics, vol. 26, pp. 706–708, 1990. View at Google Scholar
  23. R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM Journal on Scientific Computing, vol. 13, no. 1, pp. 425–448, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM Journal on Scientific Computing, vol. 7, no. 3, pp. 856–869, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. S.-L. Wu, T.-Z. Huang, L. Li, and L.-L. Xiong, “Positive stable preconditioners for symmetric indefinite linear systems arising from Helmholtz equations,” Physics Letters A, vol. 373, no. 29, pp. 2401–2407, 2009. View at Publisher · View at Google Scholar