Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014, Article ID 490156, 10 pages
http://dx.doi.org/10.1155/2014/490156
Research Article

A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method

1Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
2School of Mathematics and Computer, Dali University, Dali, Yunnan 671003, China

Received 24 June 2014; Accepted 26 August 2014; Published 25 September 2014

Academic Editor: Kim M. Liew

Copyright © 2014 Xingwen Zhu and Lixiang Zhang. 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. W. L. Briggs, V. E. Henson, and S. McCormick, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, 2nd edition, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  2. W. Hackbusch, Multigrid Methods and Applications, Springer, Berlin, Germany, 1985.
  3. U. Trottenberg, C. W. Oosterlee, and A. schuller, Multigrid, Academic Press, New York, NY, USA, 2001. View at MathSciNet
  4. P. Wesseling, An Introduction to Multigrid Methods, John Wiley, Chichester , UK, 1992. View at MathSciNet
  5. K. Stüben and U. Trottenberg, “Multigrid methods: fundamental algorithms, model problem analysis and applications,” in Multigrid Methods, W. Hackbusch and U. Trottenberg, Eds., vol. 960 of Lecture Notes in Mathematics, pp. 1–176, Springer, Berlin, Germany, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. Brandt and O. E. Livne, 1984 Guide to Multigrid Development in Multigrid Methods, Society for Industrial and Applied Mathematics, 2011. View at Publisher · View at Google Scholar
  7. A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Mathematics of Computation, vol. 31, no. 138, pp. 333–390, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Boonen, J. van lent, and S. Vandewalle, “Local Fourier analysis of multigrid for the curl-curl equation,” SIAM Journal on Scientific Computing, vol. 30, no. 4, pp. 1730–1755, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Vandewalle and G. Horton, “Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods,” Computing, vol. 54, no. 4, pp. 317–330, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  10. R. Wienands and C. W. Oosterlee, “On three-grid Fourier analysis for multigrid,” SIAM Journal on Scientific Computing, vol. 23, no. 2, pp. 651–671, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  11. R. Wienands and W. Joppich, Practical Fourier Analysis for Multigrid Methods, CRC Press, 2005.
  12. A. Brandt, “Rigorous quantitative analysis of multigrid—I: constant coefficients two-level cycle with &-norm,” SIAM Journal on Numerical Analysis, vol. 31, pp. 1695–1730, 1994. View at Google Scholar
  13. C. Rodrigo, P. Salinas, F. J. Gaspar, and F. J. Lisbona, “Local Fourier analysis for cell-centered multigrid methods on triangular grids,” Journal of Computational and Applied Mathematics, vol. 259, pp. 35–47, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. F. J. Gaspar, J. L. Gracia, and F. J. Lisbona, “Fourier analysis for multigrid methods on triangular grids,” SIAM Journal on Scientific Computing, vol. 31, no. 3, pp. 2081–2102, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. G. Zhou and S. R. Fulton, “Fourier analysis of multigrid methods on hexagonal grids,” SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 1518–1538, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. B. Gmeiner, T. Gradl, F. Gaspar, and U. Rüde, “Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis,” Computers & Mathematics with Applications, vol. 65, no. 4, pp. 694–711, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. P. MacLachlan and C. W. Oosterlee, “Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs,” Numerical Linear Algebra with Applications, vol. 18, no. 4, pp. 751–774, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Cools and W. Vanroose, “Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems,” Numerical Linear Algebra with Applications, vol. 20, no. 4, pp. 575–597, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. Rodrigo, F. J. Gaspar, C. W. Oosterlee, and I. Yavneh, “Accuracy measures and Fourier analysis for the full multigrid algorithm,” SIAM Journal on Scientific Computing, vol. 32, no. 5, pp. 3108–3129, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. O. E. Livne and A. Brandt, “Local mode analysis of multicolor and composite relaxation schemes,” Computers & Mathematics with Applications, vol. 47, no. 2-3, pp. 301–317, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. J. Kuo and T. F. Chan, “Two-color Fourier analysis of iterative algorithms for elliptic problems with red/black ordering,” SIAM Journal on Scientific and Statistical Computing, vol. 11, no. 4, pp. 767–793, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  22. I. Yavneh, “On red-black SOR smoothing in multigrid,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 180–192, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  23. C. J. Kuo and B. C. Levy, “Two-color Fourier analysis of the multigrid method with red-black Gauss-Seidel smoothing,” Applied Mathematics and Computation, vol. 29, no. 1, pp. 69–87, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  24. I. Yavneh, “Multigrid smoothing factors for red-black Gauss-Seidel relaxation applied to a class of elliptic operators,” SIAM Journal on Numerical Analysis, vol. 32, no. 4, pp. 1126–1138, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  25. C. Rodrigo, F. J. Gaspar, and F. J. Lisbona, “Multicolor Fourier analysis of the multigrid method for quadratic FEM discretizations,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 11182–11195, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. T. N. Venkatesh, V. R. Sarasamma, S. Rajalakshmy, K. C. Sahu, and R. Govindarajan, “Super-linear speed-up of a parallel multigrid Navier-Stokes solver on Flosolver,” Current Science, vol. 88, no. 4, pp. 589–593, 2005. View at Google Scholar
  27. K. C. Sahu and R. Govindarajan, “Stability of flow through a slowly diverging pipe,” Journal of Fluid Mechanics, vol. 531, pp. 325–334, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. M. Adams and H. F. Jordan, “Is SOR color-blind?” SIAM Journal on Scientific and Statistical Computing, vol. 7, no. 2, pp. 490–506, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Landon Boyd, Solving the Poisson Problem in Parallel with S.O.R., http://www.cs.ubc.ca/~blandon/cpsc521/cpsc521boyd.pdf.