Numerische Mathematik

, Volume 57, Issue 1, pp 15–38 | Cite as

On the convergence of multi-grid methods with transforming smoothers

Theory with applications to the Navier-Stokes equations
  • Gabriel Wittum


In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].

Subject Classifications

AMS(MOS): 65N20 CR: G1.8 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.17, 35–92 (1964)Google Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo21, 337–344 (1984)Google Scholar
  3. 3.
    Becker, C., Ferziger, J.H. Peric, M., Scheuerer, G.: Finite-volume multigrid solutions of the two-dimensional incompressible Navier-Stokes equations. In: Hackbusch W. (ed.) Robust multi-grid methods. Proceedings of the fourth GAMM-Seminar Kiel, Jan 1988Google Scholar
  4. 4.
    Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studien Nr. 85, Bonn (1984)Google Scholar
  5. 5.
    Brandt, A., Dinar, N.: Multigrid solutions to elliptic flow problems. ICASE Report Nr. 79-15 (1979)Google Scholar
  6. 6.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO, Modelisation Math. Anal. Numer.8, 129–151 (1974)Google Scholar
  7. 7.
    Ciminno, G.: La ricerca scientifica, ser II, vol 1, pp. 326–333. In: Pubblicazioni dell' Insituto per le Applicazioni del Calcolo,34, (1938)Google Scholar
  8. 8.
    Euler, L.: Inventio summae cuiusque seriei ex dato termino generali. Comment. Acad. Sci. Petropolitanae8, 9–22 (1741)Google Scholar
  9. 9.
    Fuchs, L.: Multi-grid schemes for incompressible flows. In: Hackbusch W ((ed.) Efficient solvers for elliptic systems. Notes on numerical fluid mechanics, vol. 10. Braunschweig: Vieweg 1984Google Scholar
  10. 10.
    Fuchs, L., Zhao, H.-S.: Solution of three-dimensional viscous incompressible flows by a multigrid method. Int. J. Numer. Methods Fluids4, 539–555 (1984)Google Scholar
  11. 11.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Series in Computational Mathematics. Berlin-Heidelberg-New York: Springer 1986Google Scholar
  12. 12.
    Hackbusch, W.: Analysis and multi-grid solutions of mixed finite element and mixed difference equations. Report, Ruhr-Universität Bochum (1980)Google Scholar
  13. 13.
    Hackbusch, W.: Multi-grid methods and applications. Berlin-Heidelberg-New York: Springer 1985Google Scholar
  14. 14.
    Hackbusch, W.: On the regularity of difference schemes. Ark. Mat.19, 71–95 (1981)Google Scholar
  15. 15.
    Hackbusch, W.: On the regularity of difference schemes—part II: regularity estimates for linear and nonlinear problems. Ark. Math.21, 3–28 (1983)Google Scholar
  16. 16.
    Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Stuttgart: Teubner 1986Google Scholar
  17. 17.
    Hackbusch, W.: Convergence of multi-grid iterations applied to difference equations. Math. Comput.34, 425–440 (1980)Google Scholar
  18. 18.
    Harlow, F.H., Welch, J.E.: Numerical Calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids8, 12, 2182–2189 (1965)Google Scholar
  19. 19.
    Kaczmarz, S.: Angenäherte Auflösung von Systemen linearer Gleichungen. Bull. Acad. Polon. Sci. Lett.A35, 355–357 (1937)Google Scholar
  20. 20.
    Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. 1 Berlin-Heidelberg-New York: Springer 1972Google Scholar
  21. 21.
    Lonsdale, G.: Solution of a rotating Navier-Stokes problem by a nonlinear multigrid algorithm. Report Nr. 105, Manchester University 1985Google Scholar
  22. 22.
    Pätsch, J.: Ein Mehrgitterverfahren zur Berechnung der Eigenwerte und der Eigenvektoren der Oberflächenschwingungen eines abgeschlossenen Wasserbeckens. Master thesis, Institut für Informatik und Praktische Mathematik, CAU Kiel. (1987)Google Scholar
  23. 23.
    Patankar, S.V., Spalding, D.B.: A calculation procedure for heat and mass transfer in threedimensional parabolic flows. Int. J. Heat Mass Transfer15, 1787–1806 (1972)Google Scholar
  24. 24.
    Pau, V., Lewis, E.: Application of the multigird technique to the pressure-correction equation for the SIMPLE algorithm. In: Hackbusch, W., Trottenberg, U. (eds.), Multigrid methods, special topics and applications GMD-Studien Nr. 110. St. Augustin (1986)Google Scholar
  25. 25.
    Shaw, G.J., Sivalonganathan, S.: On the smoothing properties of the simple pressure-correction algorithm. Int. J. Numer. Methods Fluids8, 441–461 (1988)Google Scholar
  26. 26.
    Sivalonganathan, S., Shaw G.J.: A multigrid method for recincirculating flows. Int. J. Numer. Methods Fluids8, 417–440 (1988)Google Scholar
  27. 27.
    Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using finite element technique. Comput. Fluids1, 73–100 (1973)Google Scholar
  28. 28.
    Varga, R.S.: Matrix iterative analysis. Englewood Cliffs: Prentice Hall 1962Google Scholar
  29. 29.
    Wesseling, P.: Theoretical and practical aspects of a multigrid method. SIAM J. Sci. Stat. Comput.3, 387–407 (1982)Google Scholar
  30. 30.
    Wittum, G.: Distributive Iterationen für indefinite Systeme. Thesis, Universität Kiel 1986Google Scholar
  31. 31.
    Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers—algorithms and numerical results. Numer. Math.54, 543–563 (1989)Google Scholar
  32. 32.
    Wittum, G.: On the robustness of ILU-smoothing. SIAM Journal of Sci. Stat. Comput.10, 699–717 (1989)Google Scholar
  33. 33.
    Wittum, G.: Linear iterations as smoothers in multi-grid methods. Imp. Comput. Sci. Engeneering1, 180–212 (1989)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Gabriel Wittum
    • 1
  1. 1.Sonderforschungsbereich 123Universität HeidelbergHeidelbergFederal Republic of Germany

Personalised recommendations