Multigrid methods: from geometrical to algebraic versions

  • Gundolf Haase
  • Ulrich Langer
Part of the NATO Science Series book series (NAII, volume 75)


Nowadays the multigrid technique is one of the most efficient methods for solving a large class of problems including elliptic boundary value problems for partial differential equations (PDEs) or systems of PDEs. Our lecture notes start with a motivation for the multigrid idea and a brief review of the multigrid history, its situation and the perspectives. The next part of our paper is devoted to the algorithmical aspects of the construction of multigrid methods including also algebraic approaches and parallelization techniques. Algebraic multigrid methods are now quite popular in practical applications because they can be included in conventional finite element packages without changing the data structure of the package. In the theoretical part of our lecture, we present some general approaches to the convergence and efficiency analysis of multigrid methods. Finally, we discuss implementation issues and present some numerical results obtained from the application of our algebraic multigrid package PEBBLES to large-scale problems in medicine and engineering.


Coarse Grid Fine Grid Multigrid Method Finite Element Equation Multigrid Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Anwander, M. Kuhn, S. Reitzinger, and C. Wolters, A parallel algebraic multigrid solver for the finite element method source localization in the human brain,Computing and Visualization in Science, 2002 (to appear).Google Scholar
  2. [2]
    G. P. Astrachancev, An iterative method for solving elliptic net problems, USSR Computational Math, and Math. Phys. 11 (2) (1971), 171–182.CrossRefGoogle Scholar
  3. [3]
    O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.zbMATHCrossRefGoogle Scholar
  4. [4]
    N. S. Bachvalov, On the convergence of a relaxation method with natural constraints on the elliptic operator,USSR Computational Math, and Math. Phys. 6 (5) (1966), 101–135.CrossRefGoogle Scholar
  5. [5]
    R. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, SIAM, Philadelphia, PA, 1994.zbMATHGoogle Scholar
  6. [6]
    B. Bastian, Parallele Adaptive Mehrgitterverfahren, Teubner Skr. Numer., B.G. Teubner, Stuttgart, 1996.zbMATHGoogle Scholar
  7. [7]
    P. Bastian, UG version 2.0 - short manual, Preprint 92–14, IWR Heidelberg, 1992.Google Scholar
  8. [8]
    F. A. Bornemann and P. Deuflhard, The cascadic multigrid method for elliptic problems, Numer. Math. 75 (1996), 135–152.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    D. Braess, Towards algebraic multigrid for elliptic problems of second order, Computing 55 (1995), 379–393.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 1997.zbMATHGoogle Scholar
  11. [11]
    D. Braess, M. Dryja, and W. Hackbusch, A multigrid method for nonconforming fe-discretisations with application to non-matching grids, Computing 63 (1999), 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the V-cycle, SIAM J. Numer. Anal. 20 (1983), 967–975.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J. H. Bramble, Multigrid Methods, Pitman Res. Notes Math. Ser. 294, Longman Sci. Tech., Harlow, 1993.zbMATHGoogle Scholar
  14. [14]
    J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comput 49 (180) (1987), 311–329.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    J. H. Bramble and J. E. Pasciak, New estimates for multilevel algorithms including the F-cycle, Math. Comput 60 (1993), 447–471.MathSciNetzbMATHGoogle Scholar
  16. [16]
    A. Brandt, Multi-level adaptive techniques (MLAT) for fast numerical solution to boundary value problems, in: Proc. 3rd Internat. Conf. on Numerical Methods in Fluid Mechanics, Paris, 1972, Lecture Notes in Phys., Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
  17. [17]
    A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput. 31 (1977), 333–390.zbMATHCrossRefGoogle Scholar
  18. [18]
    A. Brandt, Algebraic multigrid theory: The symmetric case, Appl. Math. Comput. 19 (1986), 23–56.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    A. Brandt, Multiscale scientific computation: Review 2001, in: Multiscale and Multiresolution Methods (T. Barth, R. Haimes, and T. Chan, eds.), Springer-Verlag, Berlin-Heidelberg-New York, 2001.Google Scholar
  20. [20]
    A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations, in: Sparsity and Its Applications (D. J. Evans, ed.), Cambridge University Press, Cambridge, 1985, 257–284.Google Scholar
  21. [21]
    A. Brandt, S. F. McCormick, and J. W. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with application in geodetic computations, Technical Report CO POB 1852, Inst. Comp. Studies State Univ., 1982.Google Scholar
  22. [22]
    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.zbMATHGoogle Scholar
  23. [23]
    M. Brezina, A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick, and J. Ruge, Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput. 22 (5) (2000), 1570–1592.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed., SIAM, Philadelphia, PA, 2000.zbMATHCrossRefGoogle Scholar
  25. [25]
    P. Deuflhard, Cascadic conjugate gradient methods for elliptic partial differential equations I: Algorithm and numerical results, Preprint SC 93–23, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1993.Google Scholar
  26. [26]
    E. Dick, K. Riemslagh, and J. Vierendeels, eds., Multigrid Methods VI. Proceedings Sixth European Multigrid Conference, Springer-Verlag, Berlin, 2000, 27–30.Google Scholar
  27. [27]
    C. Douglas, G. Haase, and U. Langer,A Tutorial on Elliptic PDEs and Parallel Solution Methods, SIAM, 2002, in preparation.Google Scholar
  28. [28]
    R. P. Fedorenko, A relaxation method for elliptic difference equations, USSR Computational Math, and Math. Phys. 1 (5) (1961), 1092–1096.MathSciNetzbMATHGoogle Scholar
  29. [29]
    R. P. Fedorenko, The speed of convergence of one iterative process, USSR Computational Math, and Math. Phys. 4 (3) (1964), 227–235.CrossRefGoogle Scholar
  30. [30]
    T. Grauschopf, M. Griebel, and H. Regler, Additive multilevel-preconditioners based on bilinear interpolation, matrix dependent geometric coarsening and algebraic-multigrid coarsening for second order elliptic PDEs, SFB-Bericht Nr. 342/02/96, Technische Uni-versität, München, 1996.Google Scholar
  31. [31]
    G. Haase, A parallel AMG for overlapping and non-overlapping domain decomposition, Electron. Trans. Numer. Anal. 10 (2000), 41–55.MathSciNetzbMATHGoogle Scholar
  32. [32]
    G. Haase, M. Kuhn, and U. Langer, Parallel multigrid 3d Maxwell solvers, Parallel Comput.6 (27) (2001), 761–775.MathSciNetCrossRefGoogle Scholar
  33. [33]
    G. Haase, M. Kuhn, U. Langer, S. Reitzinger, and J. Schöberl, Parallel Maxwell solvers, in: Scientific Computing in Electrical Engineering (U. van Rienen, M. Günther, and D. Hecht, eds.), Springer-Verlag, Berlin-Heidelberg-New York, 2000, 71–78.Google Scholar
  34. [34]
    G. Haase, M. Kuhn, and S. Reitzinger, Parallel AMG on distributed memory computers, SIAM SISC (2002), to appear.Google Scholar
  35. [35]
    G. Haase, U. Langer, and A. Meyer, The approximate Dirichlet decomposition method. I, II, Computing 47 (1991), 137–167.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    G. Haase, U. Langer, S. Reitzinger, and J. Schöberl, Algebraic multigrid methods based on element preconditioning, Internat. J. Computer Math. 80 (3–4) (2001).Google Scholar
  37. [37]
    W. Hackbusch, Implementation of the multi-grid method for solving partial differential equations, Technical Report RA 82, IBM T. J. Watson Research Centre, 1976.Google Scholar
  38. [38]
    W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985.Google Scholar
  39. [39]
    W. Hackbusch and U. Trottenberg, eds., First European Conference on Multigrid Methods, Lecture Notes in Math. 960, Springer-Verlag, Berlin-Heidelberg-New York, 1982.Google Scholar
  40. [40]
    W. Hackbusch and U. Trottenberg, eds, Second European Conference on Multigrid Methods, Lecture Notes in Math. 1228, Springer-Verlag, Berlin-Heidelberg-New York, 1986.Google Scholar
  41. [41]
    W. Hackbusch and U. Trottenberg, eds, Third European Conference on Multigrid Methods, Internat. Ser. Numer. Math. 98, Birkhäuser, Basel, 1991.zbMATHGoogle Scholar
  42. [42]
    W. Hackbusch and G. Wittum, eds., Multigrid Methods V. Proceedings of the Fifth European Multigrid Conference, Springer-Verlag, Berlin, 1998.Google Scholar
  43. [43]
    P. W. Hemker and P. Wesseling, eds., Multigrid Methods IV. Proceedings of the Fourth European Multigrid Conference, Birkhäuser, Basel, 1994.zbMATHGoogle Scholar
  44. [44]
    V. Henson and P. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23 (2) (2001), 629–650.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    V. E. Henson and U. M. Yang, BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Technical Report UCRL-JC-141495, Lawrence Livermore National Laboratory, 2000.Google Scholar
  46. [46]
    R. Hiptmair, Multigrid methods for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), 204–225.MathSciNetCrossRefGoogle Scholar
  47. [47]
    J. Jones and P. Vassilevski, AMGe based on element agglomeration, SIAM J. Sci. Comput. 23 (1) (2001), 109–133.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    M. Jung, On the parallelization of multi-grid methods using a non-overlapping domain decomposition data structure, Appl. Numer. Math. 23 (1) (1997), 119–137.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    M. Jung and U. Langer, Applications of multilevel methods to practical problems, Surveys Math. Industry 1 (1991), 217–257.MathSciNetzbMATHGoogle Scholar
  50. [50]
    M. Jung, U. Langer, A. Meyer, W. Queck, and M. Schneider, Multigrid preconditioners and their applications, in: Proc. 3rd Multigrid Seminar, Biesenthal, GDR, 1989 (G. Telschow, ed.), Report-Nr. R-MATH-03/89, Karl-Weierstrass-Institute of the Academy of Science of the GDR, Berlin, 1989, 11–52.Google Scholar
  51. [51]
    M. Kaltenbacher, S. Reitzinger, and J. Schöberl, Algebraic multigrid for solving 3D nonlinear electrostatic and magnetostatic field problems, IEEE Trans. Magnetics 36 (4) (2000), 1561–1564.CrossRefGoogle Scholar
  52. [52]
    F. Kickinger, Algebraic multigrid for discrete elliptic second-order problems, in: Multigrid Methos V. Proc. 5th European Multigrid Conf. (W. Hackbusch, ed.), Lecture Notes in Comput. Sci. Engrg. 3, Springer-Verlag, New York, 1998, 157–172.Google Scholar
  53. [53]
    V. G. Korneev, Finite Element Schemes of Higher Order of Accuracy, Leningrad University Press, Leningrad, 1977, (Russian).Google Scholar
  54. [54]
    A. Krechel and K. Stüben, Parallel algebraic multigrid based on subdomain blocking, Parallel Comput.8 (27) (2001), 1009–1031.CrossRefGoogle Scholar
  55. [55]
    M. Kuhn, U. Langer, and J. Schöberl, Scientific computing tools for 3d magnetic field prpblems, in: The Mathematics of Finite Elements and Applications (J. R. Witheman, ed.), Elsevier, Amsterdam, 2000, 239–258.Google Scholar
  56. [56]
    U. Langer, On the choice of iterative parameters in the relaxation method on a sequence of meshes, USSR Computational Math, and Math. Phys. 22 (5) (1982), 98–114.zbMATHCrossRefGoogle Scholar
  57. [57]
    K. H. Law, A parallel finite element solution method, Comput. & Structures 23 (6) (1989), 845–858.CrossRefGoogle Scholar
  58. [58]
    S. McCormick, Multilevel Adaptive Methods for Partial Differential Equations, Frontiers Appl. Math. 6, SIAM, Philadelphia, PA, 1989.zbMATHCrossRefGoogle Scholar
  59. [59]
    G. Meurant, Computer Solution of Large Systems, North-Holland, Amsterdam, 1999.zbMATHGoogle Scholar
  60. [60]
    S. Reitzinger, Algebraic Multigrid Methods for Large Scale Finite Element Equations, Universitätsverlag Rudolf Trauner, Linz, 2001.Google Scholar
  61. [61]
    S. Reitzinger and J. Schöberl, An algebraic multigrid method for finite element discretization with edge elements, Numer. Linear Algebra Appl. (2002), to appear.Google Scholar
  62. [62]
    U. Rüde, Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers Appl. Math. 13, SIAM, Philadelphia, PA, 1993.zbMATHCrossRefGoogle Scholar
  63. [63]
    J. W. Ruge and K. Stüben, Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG), in: Multigrid Methods for Integral and Differential Equations (D. J. Paddon and H. Holstein, eds.), Inst. Math. Appl. Conf. Ser., Clarendon Press, Oxford, 1985, 169–212.Google Scholar
  64. [64]
    J. W. Ruge and K. Stüben, Algebraic multigrid (AMG), in: Multigrid Methods (S. F. McCormick, ed.), Frontiers Appl. Math. 3, SIAM, Philadelphia, PA, 1987, 73–130.CrossRefGoogle Scholar
  65. [65]
    J. Schöberl, Multigrid methods for a parameter dependent problem in primal variables, Numer. Math. 84 (1999), 97–119.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [66]
    V. V. Shaidurov, Multigrid Methods for Finite Elements, Kluwer, Dordrecht, 1995.zbMATHGoogle Scholar
  67. [67]
    K. Stüben, Algebraic multigrid: An introduction with applications, in: Multigrid (U. Trottenberg, C. Oosterlee, and A. Schüller, eds.), Academic Press, 2000, 413–532.Google Scholar
  68. [68]
    K. Stüben, A review of algebraic multigrid, J. Comput Appl. Math. 128 (2001), 281– 309.MathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    U. Trottenberg, C. Oosterlee, and A. Schüller, eds., Multigrid, Academic Press, 2000.Google Scholar
  70. [70]
    St. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems, Teubner, Stuttgart, 1993.Google Scholar
  71. [71]
    P. Vaněk, Acceleration of convergence of a two level algorithm by smoothing transfer operators, Appl. Math. 37 (1992), 265–274.MathSciNetzbMATHGoogle Scholar
  72. [72]
    P. Vaněk, J. Mandel, and M. Brezina, Algebraic multigrid based on smoothed aggregation for second and fourth order problems, Computing 56 (1996), 179–196.MathSciNetzbMATHCrossRefGoogle Scholar
  73. [73]
    P. Wesseling, An Introduction to Multigrid Methods, Wiley, Chichester, 1992.zbMATHGoogle Scholar
  74. [74]
    H. Yserentant, Old and new convergence proofs for multigrid methods, in: Acta Numerica, Cambridge University Press, 1993, 285–326.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Gundolf Haase
    • 1
  • Ulrich Langer
    • 1
  1. 1.Institute of Computational MathematicsJohannes Kepler University of LinzLinzAustria

Personalised recommendations