Table of Contents
ISRN Computational Mathematics
Volume 2014 (2014), Article ID 745849, 15 pages
http://dx.doi.org/10.1155/2014/745849
Research Article

Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation

Mathematics Department, Idaho State University, 921 S 8th Avenue, Stop 8085, Pocatello, ID 83209-8085, USA

Received 5 August 2013; Accepted 10 November 2013; Published 21 January 2014

Academic Editors: L. Hajdu, S. Manservisi, Y. Peng, and R. Tuzun

Copyright © 2014 Yury Gryazin. 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. Y. Zhuang and X.-H. Sun, “A high-order fast direct solver for singular poisson equations,” Journal of Computational Physics, vol. 171, no. 1, pp. 79–94, 2001. View at Publisher · View at Google Scholar · View at Scopus
  2. J. J. Heys, T. A. Manteuffel, S. F. McCormick, and L. N. Olson, “Algebraic multigrid for higher-order finite elements,” Journal of Computational Physics, vol. 204, no. 2, pp. 520–532, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. S. Orszag, “Spectral methods for problems in complex geometries,” Journal of Computational Physics, vol. 37, pp. 70–92, 1980. View at Google Scholar
  4. I. M. Babuška and S. A. Sauter, “Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?” SIAM Review, vol. 42, no. 3, pp. 451–484, 2000. View at Google Scholar · View at Scopus
  5. A. Bayliss, C. I. Goldstein, and E. Turkel, “On accuracy conditions for the numerical computation of waves,” Journal of Computational Physics, vol. 59, no. 3, pp. 396–404, 1985. View at Google Scholar · View at Scopus
  6. G. Sutmann, “Compact finite difference schemes of sixth order for the Helmholtz equation,” Journal of Computational and Applied Mathematics, vol. 203, no. 1, pp. 15–31, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Nabavi, M. H. K. Siddiqui, and J. Dargahi, “A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation,” Journal of Sound and Vibration, vol. 307, no. 3-5, pp. 972–982, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. I. Singer and E. Turkel, “Sixth-order accurate finite difference schemes for the Helmholtz equation,” Journal of Computational Acoustics, vol. 14, no. 3, pp. 339–351, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Britt, S. Tsynkov, and E. Turkel, “Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes,” Communications in Computational Physics, vol. 9, no. 3, pp. 520–541, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. 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 Google Scholar · View at Scopus
  11. S. Kim, “Parallel multidomain iterative algorithms for the Helmholtz wave equation,” Applied Numerical Mathematics, vol. 17, no. 4, pp. 411–429, 1995. View at Google Scholar · View at Scopus
  12. J. Douglas Jr, J. L. Hensley, and J. E. Roberts, “Alternating-direction iteration method for Helmholtz problems,” Tech. Rep. 214, Mathematics Department, Purdue University, West Lafaette, Ind, USA, 1993. View at Google Scholar
  13. N. Umetani, S. P. MacLachlan, and C. W. Oosterlee, “A multigrid-based shifted Laplacian preconditioner for a fourth-order Helmholtz discretization,” Numerical Linear Algebra with Applications, vol. 16, no. 8, pp. 603–626, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. J. H. Bramble, J. E. Pasciak, and J. Xu, “The analysis of multigrid algorithms for nonsymmetric and indefinite problems,” Mathematics of Computation, vol. 51, pp. 389–414, 1988. View at Google Scholar
  15. O. Ernst and G. H. Golub, “A domain decomposition approach to solving the Helmholtz equation with a radiation boundary condition,” in Domain Decomposition in Science and Engineering, A. Quarteroni, H. Periaux, Y. Kuznetsov, and O. Widdlund, Eds., pp. 177–192, American Mathematical Society, Providence, RI, USA, 1994. View at Google Scholar
  16. H. C. Elman and D. P. O'Leary, “Efficient iterative solution of the three-dimensional Helmholtz equation,” Journal of Computational Physics, vol. 142, no. 1, pp. 163–181, 1998. View at Publisher · View at Google Scholar · View at Scopus
  17. H. C. Elman and D. P. O'Leary, “Eigenanalysis of some preconditioned Helmholtz problems,” Numerische Mathematik, vol. 83, no. 2, pp. 231–257, 1999. View at Google Scholar · View at Scopus
  18. Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “GMRES computation of high frequency electrical field propagation in land mine detection,” Journal of Computational Physics, vol. 158, no. 1, pp. 98–115, 2000. View at Publisher · View at Google Scholar · View at Scopus
  19. J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997.
  20. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2, Birkhäuser, Boston, Mass, USA, 1989.
  21. S. H. Lui, Numerical Analysis of Partial Differential Equations, John Wiley & Sons, New York, NY, USA, 2011.
  22. Y. A. Gryazin, “A Compact sixth order scheme combined with GMRES method for the 3D Helmholtz equation,” in Proceedings of the 10th International Conference on Mathematical and Numerical Aspects of Waves, pp. 339–442, Vancouver, Canada, July, 2011.
  23. G. H. Golub and C. F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 2nd edition, 1989.
  24. Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2003.
  25. E. Heikkola, T. Rossi, and J. Toivanen, “Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition,” International Journal for Numerical Methods in Engineering, vol. 57, no. 14, pp. 2007–2025, 2003. View at Publisher · View at Google Scholar · View at Scopus