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On solving a mixed finite element approximation of the dirichlet problem for the biharmonic operator by a "quasi-direct" method and various iterative methods

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Part of the Lecture Notes in Mathematics book series (LNM,volume 606)

Keywords

  • Iterative Method
  • Dirichlet Problem
  • Gradient Method
  • Conjugate Gradient Method
  • Finite Element Approximation

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Bibliography

  1. BOSSAVIT, A. Une méthode de décomposition de l’opérateur biharmonique. Note HI 585/2, Electricité de France, (1971).

    Google Scholar 

  2. BOURGAT, J.F. Numerical study of a dual iterative method for solving a finite element approximation of the biharmonic equation, LABORIA Report 156, and to appear in Comp. Meth. Applied Mech. Eng.

    Google Scholar 

  3. BOURGAT, J.F., GLOWINSKI, R., PIRONNEAU, O. Numerical methods for the Dirichlet problem for the biharmonic equation and applications (to appear)

    Google Scholar 

  4. CEA,J. Optimisation. Théorie et Algorithmes, Dunod, 1971.

    Google Scholar 

  5. CIARLET, P.G., GLOWINSKI, R. Dual iterative techniques for solving a finite element approximation of the biharmonic equation, Comp. Meth. Applied Mech. Eng. 5, (1975), pp. 277–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. 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. Acad. Press, (1974), pp. 125–145.

    Google Scholar 

  7. Interpolation theory over curved element with application to finite element methods. Comp. Meth. Applied Mech. Eng. 1, (1972), pp. 217–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. CONCUS, P., GOLUB G.H. Monography on conjugate gradient (to appear).

    Google Scholar 

  9. DANIEL, J.W. The approximate minimization of functionals. Prentice Hall (1970).

    Google Scholar 

  10. EHRLICH, L.W. Solving the biharmonic equation as coupled difference equations Siam. J. Num. Anal. 8 (1971), pp. 278–287

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Coupled harmonic equations, SOR and Chebyshef acceleration, Math. Comp. 26, (1972), pp. 335–343.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Solving the biharmonic equation in a square. Comm. ACM, 16 (1973), pp. 711–714.

    CrossRef  MATH  Google Scholar 

  13. EHRLICH, L.W., GUPTA, M.M. Some difference schemes for the biharmonic equation Siam J. Num. Anal. 12, (1975), pp. 773–790.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. GLOWINSKI,R. Approximations externes par éléments finis d’ordre un et deux du problème de Dirichlet pour Δ2. In Topics in Numerical Analysis, J.J.H. Miller Ed, Academic Press, (1973), pp. 123–171.

    Google Scholar 

  15. Sur l’équation biharmonique dans un domaine multi-connexe C.R.A.S. Paris. (to appear).

    Google Scholar 

  16. GLOWINSKI, R., LIONS, J.L., TREMOLIERES, R. Analyse Numérique des Inéquations variationnelles (Tome 2), Dunod-Bordas, (1976).

    Google Scholar 

  17. GLOWINSKI, R., PIRONNEAU O. Sur la résolution numérique du problème de Dirichlet pour l’opérateur biharmonique par une méthode "quasi-directe". C.R.A.S. Paris, t. 281 A, pp. 223–226, (1976).

    MathSciNet  MATH  Google Scholar 

  18. Sur la résolution numérique du problème de Dirichlet pour Δ2 par la méthode du gradient conjugué. Applications. C.R.A.S. Paris, t. 282 A, p. 1315–1318 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Sur la résolution par une méthode "quasi directe", et par diverses méthodes itératives, d’une approximation par éléments finis mixtes du problème de Dirichlet pour Δ2.Report 76010, Laboratoire d’Analyse Numérique, Université Paris6, (1976).

    Google Scholar 

  20. Stanford University report (to appear).

    Google Scholar 

  21. LIONS, J.L., MAGENES, E. Problèmes aux limites non homogènes, (T. 1), Dunod, 1968.

    Google Scholar 

  22. MARCHOUK, G.I., KUZNETSOV, J.A. Méthodes Itératives et Fonctionnelles Ouadratiques dans: Sur les Méthodes Numériques en Sciences Physiques et Economiques, LIONS J.L., MARCHOUK G.I., Ed. Dunod, 1974, pp. 1–132.

    Google Scholar 

  23. Mc LAURIN, J.W. A genral coupled equation approach for solving the biharmonic boundary value problem, Siam J. Num. Anal. 11, (1974) pp. 14–33.

    CrossRef  MathSciNet  Google Scholar 

  24. POLAK, E. Computational Methods in Optimization, Acad. Press, 1971.

    Google Scholar 

  25. SHOLZ, R. Approximation Von sattelpunkten mit finiten elementen (to appear).

    Google Scholar 

  26. VARGA,R.S. Matrix iterative Analysis, Prentice-Hall, 1962.

    Google Scholar 

  27. YOUNG, D.M. Iterative solution of large linear systems, Acad. Press, 1971.

    Google Scholar 

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Glowinski, R., Pironneau, O. (1977). On solving a mixed finite element approximation of the dirichlet problem for the biharmonic operator by a "quasi-direct" method and various iterative methods. In: Galligani, I., Magenes, E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol 606. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064462

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  • DOI: https://doi.org/10.1007/BFb0064462

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