Numerische Mathematik

, Volume 118, Issue 3, pp 457–483 | Cite as

Smoothing factor, order of prolongation and actual multigrid convergence

  • Artem NapovEmail author
  • Yvan Notay


We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V-cycle multigrid. We derive a two-sided bound that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow and away from 1 when both the smoothing factor and an additional parameter are small enough. Besides the smoothing factor, the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix associated with small eigenvalues. Nice V-cycle convergence is guaranteed if the tangent of this angle has an upper bound proportional to the eigenvalue, whereas nice two-grid convergence requires a bound proportional to the square root of the eigenvalue. We also discuss the well-known rule which relates the order of the prolongation to that of the differential operator associated to the problem. We first define a frequency based order which in most cases amounts to the so-called high frequency order as defined in Hemker (J Comput Appl Math 32:423–429, 1990). We give a firmer basis to the related order rule by showing that, together with the requirement of having the smoothing factor away from one, it provides necessary and sufficient conditions for having the two-grid convergence rate away from 1. A stronger condition is further shown to be sufficient for optimal convergence with the V-cycle. The presented results apply to rigorous Fourier analysis for regular discrete PDEs, and also to local Fourier analysis via the discussion of semi-positive systems as may arise from the discretization of PDEs with periodic boundary conditions.

Mathematics Subject Classification (2000)

65F10 65N55 65N12 


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  1. 1.
    Ben Israel A., Greville T.N.E.: Generalized Inverses: Theory and Applications. Wiley, New York (1974)zbMATHGoogle Scholar
  2. 2.
    Brandt A.: Guide to multigrid development. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lectures Notes in Mathematics No. 960, pp. 220–312. Springer, Berlin (1982)Google Scholar
  3. 3.
    Brandt A.: Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L2-norm. SIAM J. Numer. Anal. 31, 1695–1730 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brezina M., Cleary A.J., Falgout R.D., Henson V.E., Jones J.E., Manteuffel T.A., McCormick S.F., Ruge J.W.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22, 1570–1592 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chartier T., Falgout R.D., Henson V.E., Jones J., Manteuffel T., McCormick S., Ruge J., Vassilevski P.S.: Spectral AMGe (ρAMGe). SIAM J. Sci. Comput. 25, 1–26 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Donatelli M.: An algebraic generalization of local fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices. Numer. Linear Algebra Appl. 17, 179–197 (2010)MathSciNetGoogle Scholar
  7. 7.
    Hackbusch W.: Multi-grid Methods and Applications. Springer, Berlin (1985)zbMATHGoogle Scholar
  8. 8.
    Hemker P.: On the order of prolongations and restrictions in multigrid procedures. J. Comput. Appl. Math. 32, 423–429 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Mandel J., McCormick S.F., Ruge J.W.: An algebraic theory for multigrid methods for variational problems. SIAM J. Numer. Anal. 25, 91–110 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    McCormick S.F.: Multigrid methods for variational problems: general theory for the V-cycle. SIAM J. Numer. Anal. 22, 634–643 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Muresan A.C., Notay Y.: Analysis of aggregation-based multigrid. SIAM J. Sci. Comput. 30, 1082–1103 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Napov, A.: Algebraic analysis of V-cycle multigrid and aggregation-based two-grid methods. PhD thesis, Service de Métrologie Nucléaire, Université Libre de Bruxelles, Brussels, Belgium (2010)Google Scholar
  13. 13.
    Napov A., Notay Y.: Comparison of bounds for V-cycle multigrid. Appl. Numer. Math. 60, 176–192 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Napov A., Notay Y.: When does two-grid optimality carry over to the V-cycle?. Numer. Linear Algebra Appl. 17, 273–290 (2010)MathSciNetGoogle Scholar
  15. 15.
    Ruge J.W., Stüben K.: Algebraic multigrid (AMG). In: McCormick, S.F. (eds) Multigrid Methods. Frontiers in Applied Mathematics, vol. 3, pp. 73–130. SIAM, Philadelphia (1987)Google Scholar
  16. 16.
    Serra-Capizzano S.: Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numerische Mathematik 92, 433–465 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Stevenson, R.: On the validity of local mode analysis of multi-grid methods. PhD thesis, Utrecht University (1990)Google Scholar
  18. 18.
    Stevenson R.: More on the order of prolongations and restrictions. Appl. Numer. Math. 58, 1875–1880 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stüben K., Trottenberg K.U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lectures Notes in Mathematics No. 960, pp. 1–176. Springer, Berlin (1982)Google Scholar
  20. 20.
    Trottenberg U., Oosterlee C.W., Schüller A.: Multigrid. Academic Press, London (2001)zbMATHGoogle Scholar
  21. 21.
    Vassilevski P.S.: Multilevel Block Factorization Preconditioners. Springer, New York (2008)zbMATHGoogle Scholar
  22. 22.
    Wesseling P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)zbMATHGoogle Scholar
  23. 23.
    Wienands R., Joppich W.: Practical Fourier Analysis for Multigrid Methods. Chapman & Hall/CRC Press, Boca Raton (2005)zbMATHGoogle Scholar
  24. 24.
    Wienands R., Oosterlee C.W.: On three-grid Fourier analysis for multigrid. SIAM J. Sci. Comput. 23, 651–671 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Yavneh I.: Coarse-grid correction for nonelliptic and singular perturbation problems. SIAM J. Sci. Comput. 19, 1682–1699 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Service de Métrologie NucléaireUniversité Libre de Bruxelles (C.P. 165/84)BrusselsBelgium

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