Numerische Mathematik

, Volume 46, Issue 4, pp 623–634

A multilevel algorithm for the biharmonic problem

  • Petra Peisker


A finite element discretization of the mixed variable formulation of the biharmonic problem is considered. A multilevel algorithm for the numerical solution of the discrete equations is described. Convergence is proved under the assumption ofH3-regularity.

Subject Classifications

AMS(MOS): 65N30 CR: 1.8 


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  1. 1.
    Babuška, L., Aziz, K.: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), pp. 3–359. New York: Academic Press 1972Google Scholar
  2. 2.
    Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039–1062 (1980)Google Scholar
  3. 3.
    Bank, R., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981)Google Scholar
  4. 4.
    Bank, R.: A comparison of two multilevel iterative methods for nonsymmetric and indefinite finite element equations. Siam J. Numer. Anal.18, 724–744 (1981)CrossRefGoogle Scholar
  5. 5.
    Bjørstad, P.: Fast numerical solution of the biharmonic Dirichlet problem on rectangles. Siam J. Numer. Anal.20, 59–71 (1983)CrossRefGoogle Scholar
  6. 6.
    Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci.2, 556–581 (1980)Google Scholar
  7. 7.
    Braess, D.: The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation. In: Multigrid Methods. Proceedings (W. Hackbusch, U. Trottenberg, eds.), pp. 368–386. Berlin: Springer 1981Google Scholar
  8. 8.
    Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including theV-cycle. Siam J. Numer. Anal.20, 967–975 (1983)Google Scholar
  9. 9.
    Brexzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO8, 129–151 (1974)Google Scholar
  10. 10.
    Brezzi, F., Raviart, P.A.: Mixed finite element methods for 4th order elliptic problems. In: Topics in Numerical Analysis, III (J. Miller, ed.), pp. 33–59. New York: Academic Press 1978Google Scholar
  11. 11.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North Holland 1978Google Scholar
  12. 12.
    Ciarlet, P.G., Glowinski, R.: Dual iterative techniques for solving a finite element approximation of the biharmonic equation. Comput. Methods Appl. Mech.5, 277–295 (1975)CrossRefGoogle Scholar
  13. 13.
    Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), pp. 125–145. New York: Academic Press 1974Google Scholar
  14. 14.
    Glowinski, R., Pirroneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. Siam Rev.21, 167–212 (1979)Google Scholar
  15. 15.
    Hackbusch, W.: Survey of convergence proofs for multigrid iterations. In: Special Topics of Applied Mathematics. (J. Frehse, D. Pallaschke, U. Trottenberg, eds.), pp. 151–164. Amsterdam: North-Holland 1980Google Scholar
  16. 16.
    Hackbusch, W.: Analysis and multigrid solutions of mixed finite element and mixed finite difference equations. Ruhr-Universität Bochum (Preprint 1980)Google Scholar
  17. 17.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems with Applications, I. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  18. 18.
    Verfürth, R.: A multilevel algorithm for mixed problems. Siam J. Numer. Anal.21, 264–271 (1984)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Petra Peisker
    • 1
  1. 1.Abteilung für MathematikRuhr-Universität BochumBochum

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