Numerical Algorithms

, Volume 26, Issue 3, pp 219–234 | Cite as

Analysis of tensor product multigrid

  • S. Börm
  • R. Hiptmair


We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced.

The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semi-coarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at one-dimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid V-cycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction.

Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof.

robust multigrid methods anisotropic elliptic problems semi-coarsening 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Barros, The Poisson equation on the unit disk: A multigrid solver using polar coordinates, Appl. Math. Comput. 25 (1988) 123–135.Google Scholar
  2. [2]
    S. Barros, Multigrid methods for two-and three-dimensional Poisson-type equations on the sphere, J. Comput. Phys. 92 (1991) 313–348.Google Scholar
  3. [3]
    J. Bey and G. Wittum, Downwind numbering: Robust multigrid for convection-diffusion problems, Appl. Numer. Math. 23 (1997) 177–192.Google Scholar
  4. [4]
    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.Google Scholar
  5. [5]
    J. Bramble, Multigrid Methods, Pitman Research Notes in Mathematics Series, Vol. 294 (Longman, London, 1993).Google Scholar
  6. [6]
    A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977) 333–390.Google Scholar
  7. [7]
    A. Brandt, Multigrid Techniques: 1984 Guide with Applications, GMD-Studien, Vol. 85 (GMD, Bonn, 1984).Google Scholar
  8. [8]
    P. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4 (North-Holland, Amsterdam, 1978).Google Scholar
  9. [9]
    J. Dendy and C. Tazartes, Grandchild of the frequency decomposition multigrid method, SIAM J. Sci. Comput. 2 (1995) 307–319.Google Scholar
  10. [10]
    P. DeZeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math. 33 (1990) 1–27.Google Scholar
  11. [11]
    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, Appl. Numer. Math. 23 (1997) 63–95.Google Scholar
  12. [12]
    W. Hackbusch, Multi-grid Methods and Applications (Springer-Verlag, Berlin, 1985).Google Scholar
  13. [13]
    W. Hackbusch, The frequency decomposition multi-grid method, Numer. Math. 56 (1989) 229–245.Google Scholar
  14. [14]
    W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, Vol. 95 (Springer-Verlag, New York, 1993).Google Scholar
  15. [15]
    W. Hackbusch, The frequency decomposition multi-grid method, in: Multigrid Methods IV. Proc. of the 4th European Multigrid Conf., ed. P. Hemker, Amsterdam, Netherlands, 6–9 July 1993 (Birkhäuser, Basel, 1994) pp. 43–56.Google Scholar
  16. [16]
    W. Hackbusch and T. Probst, Downwind Gauss–Seidel smoothing for convection dominated problems, Numer. Linear Algebra Appl. 4 (1997) 85–102.Google Scholar
  17. [17]
    P. Lancaster, Theory of Matrices (Academic Press, New York, 1969).Google Scholar
  18. [18]
    P. Lötstedt, Convergence analysis of iterative methods by pseudodifference operators, Technical Report, Department of Scientific Computing, Uppsala University, Uppsala, Sweden (1999).Google Scholar
  19. [19]
    D. Mavriplis, Directional coarsening and smoothing for anisotropic Navier–Stokes problems, Electron. Trans. Numer. Anal. 6 (1997) 182–197.Google Scholar
  20. [20]
    W. Mulder, A new multigrid approach to convection problems, J. Comput. Phys. 83 (1989) 303–323.Google Scholar
  21. [21]
    C. Oosterlee, F. Gaspar, T. Washio and R. Wienands, Multigrid line smoothers for higher order upwind discretizations of convection dominated problems, J. Comput. Phys. 139 (1998) 274–307.Google Scholar
  22. [22]
    C. Pflaum, A robust multilevel algorithm for anisotropic elliptic equations, Technical Report 224, Mathematisches Institut, Universität Würzburg, Würzbug, Germany (1998).Google Scholar
  23. [23]
    C. Pflaum, Fast and robust multilevel algorithms, Habilitationsschrift, Mathematische Institute, Universität Würzburg, Würzburg, Germany (1999).Google Scholar
  24. [24]
    A. Reusken, On a robust multigrid solver, Computing 56 (1996) 303–322.Google Scholar
  25. [25]
    J. Ruge and K. Stüben, Algebraic multigrid, in: Multigrid Methods, ed. S. McCormick, Frontiers in Applied Mathematics (SIAM, Philadelphia, PA, 1987) ch. 4, pp. 73–130.Google Scholar
  26. [26]
    R. Stevenson, New estimates of the contraction number of V-cycle multigrid with applications to anisotropic equations, in: Incomplete Decompositions, Proc. of the 8th GAMM Seminar, Kiel, eds. W. Hackbusch and G. Wittum, Notes on Numerical FluidMechanics, Vol. 41 (Vieweg, Braunschweig, 1993) pp. 159–167.Google Scholar
  27. [27]
    R. Stevenson, Robustness of multi-grid applied to anisotropic equations on convex domains and on domains with re-entrant corners, Numer. Math. 66 (1993) 373–398.Google Scholar
  28. [28]
    R. Stevenson, Modified ILU as a smoother, Numer. Math. 68 (1994) 295–309.Google Scholar
  29. [29]
    K. Stüben and U. Trottenberg, MultigridMethods: Fundamental Algorithms, Model Problem Analysis and Applications, Lecture Notes in Mathematics, Vol. 960 (Springer-Verlag, Berlin, 1982).Google Scholar
  30. [30]
    T. Washio and C. Oosterlee, Flexible multiple semicoarsening for three-dimensional singularly perturbed problems, SIAM J. Sci. Comput. 19 (1998) 1646–1666.Google Scholar
  31. [31]
    P. Wesseling, An Introduction to Multigrid Methods (Wiley, Chichester, 1992).Google Scholar
  32. [32]
    G. Wittum, On the robustness of ILU-smoothing, SIAM J. Sci. Statist. Comput. 10 (1989) 699–717.Google Scholar
  33. [33]
    J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992) 581–613.Google Scholar
  34. [34]
    H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numer. (1993) 285–326.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • S. Börm
    • 1
  • R. Hiptmair
    • 2
  1. 1.Institut für Praktische MathematikUniversität KielGermany
  2. 2.Universität TübingenGermany

Personalised recommendations