Numerische Mathematik

, Volume 63, Issue 1, pp 315–344 | Cite as

Multilevel preconditioning

  • Wolfgang Dahmen
  • Angela Kunoth


This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.

Mathematics Subject Classification (1991)

65F35 65N30 41A63 41A17 46E35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] Adams, R.A. (1978): Sobolev Spaces. Academic Press, New YorkGoogle Scholar
  2. [Awa] Aware Inc., Cambridge, Mass. (1990): Wavelet analysis and the numerical solutions of partial differential equations. Progress ReportGoogle Scholar
  3. [Bä] Bänsch, E. (1991): Local mesh refinement in two and three dimensions. IMPACT (to appear)Google Scholar
  4. [BSW] Bank, R.E., Sherman, A.H., Weiser, A. (1983): Refinement algorithms and data structures for regular local mesh refinement. In: R. Stepleman et al. (eds.), Scientific Computing. Amsterdam IMACS, North-Holland, pp. 3–17Google Scholar
  5. [BH] de Boor, C., Höllig, K. (1982): B-splines from parallelepipeds. J. Anal. Math.42, 99–115Google Scholar
  6. [B] Bornemann, F.A. (1991): A sharpened condition number estimate for the BPX preconditioner of elliptic finite element problems on highly nonuniform triangulations. Preprint SC91-9, ZIBGoogle Scholar
  7. [BPX] Bramble, J.H., Pasciak, J.E., Xu, J. (1990): Parallel multilevel preconditioners. Math. Comput.55, 1–22Google Scholar
  8. [CW] Cai, Z., Weinan, E. (1991): Hierarchical method for elliptic problems using wavelets. ManuscriptGoogle Scholar
  9. [CDM] Cavaretta, A.S., Dahmen, W., Micchelli, C.A. (1991): Stationary Subdivision, Memoirs of Amer. Math. Soc., Vol. 93, #453Google Scholar
  10. [CSW] Chui, C.K., Stöckler, J., Ward, J.D. (1991): Compactly supported box spline wavelets. PreprintGoogle Scholar
  11. [CDF] Cohen, A., Daubechies I., Feauveau, J.-C. (1990): Biorthogonal bases of compactly supported wavelets. PreprintGoogle Scholar
  12. [DDS] Dahmen, W., De Vore, R.A., Scherer, K. (1980): Multidimensional spline approximation. SIAM J. Numer. Anal.17, 380–402Google Scholar
  13. [DM1] Dahmen W., Micchelli, C.A. (1983): Recent progress in multivariate splines. In: C.K. Chui, L.L. Schumaker, J.D. Ward, eds., Approximation Theory IV. Academic Press, New York, pp. 27–121Google Scholar
  14. [DM2] Dahmen, W., Micchelli, C.A. (1983): Translates of multivariate splines. LAA52/53, 217–234Google Scholar
  15. [DM3] Dahmen W., Micchelli, C.A. (1991): Using the refinement equation for evaluating integrals of wavelets. Siam J. Numer. Anal. (to appear)Google Scholar
  16. [DM4] Dahmen, W., Micchelli, C.A.: Dual wavelet expansions for general scalings. In preparationGoogle Scholar
  17. [DOS] Dahmen, W., Oswald, P., Shi, X.Q. (1991):C 1-Hierarchical bases. J. Comp. Appl. Math. (to appear)Google Scholar
  18. [DPS] Dahmen, W., Prößdorf, S., Schneider, R. (1992): Wavelet approximation methods for pseudodifferential equations. I. Stability and Convergence. Preprint of the Institute of Applied Analysis and Stochastics, No. 7, BerlinGoogle Scholar
  19. [Dau] Daubechies, I. (1987): Orthonormal bases of wavelets with compact support. Commun. Pure Appl. Math.41, 909–996Google Scholar
  20. [DLY] Deuflhard, P., Leinen, P., Yserentant, H. (1989): Concepts of an adaptive finite element code. IMPACT Comput. Sci. Engin.1, 3–35Google Scholar
  21. [DJP] DeVore, R.A., Jawerth, B., Popov, V.A. (1990): Compression of wavelet decompositions. Preprint, University of South CarolinaGoogle Scholar
  22. [DP1] DeVore, R.A., Popov, V.A. (1988): Interpolation of Besov spaces, Trans. Amer. Math. Soc.305, 397–414Google Scholar
  23. [DP2] DeVore R.A., Popov, V.A. (1987): Free multivariate splines. Const. Approx.3, 239–248Google Scholar
  24. [DS] DeVore R.A., Sharpley, R.C. (1983): Maximal Functions Measuring Smoothness. Memoirs of the Amer. Math. Soc. 293, ProvidenceGoogle Scholar
  25. [GLRT] Glowinski, R., Lawton, W.M., Ravachol, M., Tenenbaum, E. (1989): Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Preprint, Aware Inc., Cambridge, Mass.Google Scholar
  26. [J] Jaffard, S. (1990): Wavelet methods for fast resolution of elliptic problems. PreprintGoogle Scholar
  27. [JM] Jia, R.Q., Micchelli, C.A. (1991): Using the refinement equation for the construction of pre-wavelets II: Powers of two. In: P.J. Laurent, A. Le Méhauté, L.L. Schumaker, eds., Curves and Surfaces. Academic Press, New YorkGoogle Scholar
  28. [JS] Johnen, H., Scherer, K. (1977): On the equivalence of theK-functional and the moduli of continuity and some applications, Constructive Theory of Functions of Several Variables. Springer Lecture Notes in Mathematics 571. Springer, Berlin Heidelberg New York, pp. 119–140Google Scholar
  29. [JW] Jonsson A., Wallin, H. (1984): Function Spaces on Subsets of ℝn. Harwood Academic Publishers. Mathematical Reports, Vol. 2Google Scholar
  30. [L] Leinen, P. (1990): Ein schneller adaptiver Löser für elliptische Randwertprobleme auf Seriell- und Parallelrechnern, Thesis, Universität DortmundGoogle Scholar
  31. [Mal] Mallat, S. (1989): Multiresolution approximation and wavelet orthonormal bases ofL 2. Trans. Amer. Math. Soc.315, 69–88Google Scholar
  32. [M] Meyer, Y. (1990): Ondelettes. Hermann, ParisGoogle Scholar
  33. [N] Nikolskii, S.M. (1977): Approximation of Functions of Several Variables and Imbedding Theorems, 2nd ed. Nauka, MoscowGoogle Scholar
  34. [O1] Oswald, P. (1990): On function spaces related to finite element approximation theory. Z. Anal. Anwendungen9, 43–64Google Scholar
  35. [O2] Oswald, P. (1992): Hierarchical conforming finite element methods for the biharmonic equation. SIAM J. Numer. Anal. (to appear)Google Scholar
  36. [O3] Oswald, P. (1991); On discrete norm estimates related to multilevel preconditioners in the finite element method. PreprintGoogle Scholar
  37. [PP] Popov, V.A., Petrushev, P. (1987): Rational approximation of real valued functions. Encyclopedia Math. Appl., Vol. 28. Cambridge University Press, CambridgeGoogle Scholar
  38. [RS] Riemenschneider, S., Shen, Z. (1991): Wavelets and pre-wavelets in low dimensions. PreprintGoogle Scholar
  39. [Sh] Sharpley, R.C. (1983): Cone conditions and the modulus of continuity. In: Proceedings of the 2nd Conference on Approx. Theory, Vol. 3. Edmonton, Canadian Math. Soc., Amer. Math. Soc., pp. 341–351Google Scholar
  40. [SO] Storozhenko E.A., Oswald, P. (1978): Jackson's theorem in the spacesL p(ℝk), 0<p<1. Siberian Math.19, 630–639Google Scholar
  41. [S] Stein, E.M. (1970): Singular Integrals and Differentiability Properties of Functions. Princeton University Press, PrincetonGoogle Scholar
  42. [T] Triebel, H. (1978): Interpolation Theory, Function Spaces, Differential Operators. Dt. Verl. Wiss., BerlinGoogle Scholar
  43. [Y1] Yserentant, H. (1986): On the multilevel splitting of finite element spaces. Numer. Math.49, 379–412Google Scholar
  44. [Y2] Yserentant, H. (1990): Two preconditioners based on the multilevel splitting of finite element spaces. Numer. Math.58, 163–184Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Wolfgang Dahmen
    • 1
  • Angela Kunoth
    • 2
  1. 1.Institut für Geometrie und Praktische MathematikAachenGermany
  2. 2.Institut für Mathematik IFreie Universität BerlinBerlin 33Germany

Personalised recommendations