Dual Iterative Techniques for Solving a Finite Element Approximation of the Biharmonic Equation

Ciarlet, Ph.; Glowinski, R.

Publications des séminaires de mathématiques et informatique de Rennes, no. S4 (1974), article no. 5, 28 p.

[1] Ciarlet, P.G. ; Raviart, P.-A. : A mixed finite element method for the biharmonic equation. To appear in Proceedings of the Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, Mathematics Research Center, University of Wisconsin, Madison, April 01-03, 1974. | MR | Zbl

[2] Argyris, J.H.; Fried, I. ; Scharpf, D.W. : The TUBA family of plate elements for the matrix displacement method. The Aeronautical J.R.Ae. S. 72 (1968), 701-709.

[3] Ciarlet, P.G. : Conforming and nonconforming finite element methods for solving the plate problem, in Conference on the Numerical Solution of Differential Equations (G.A. Watson, Editor), pp. 21-31, Springer-Verlag, New York, 1974. | MR | Zbl

[4] Ciarlet, P.C. : Quelques méthodes d'éléments finis pour le problème d'une plaque encastrée, In Computing Methods In Applied Sciences and Engineering. Part 1 (R. GLOWINSKI and J.L. LIONS, Editors), pp. 156-176, Lecture Notes in Computer Science, Vol. 10, Springer-Verlag, Berlin, 1974. | MR | Zbl

[5] Strang, G. : Variational crimes in the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 689-710, Academic Press, New York, 1972. | MR | Zbl

[6] Glowinski, R. : Approximations externes, par éléments finis de Lagrange d'ordre un et deux, du problème de Dirichlet pour l'opérateur biharmonique. Méthodes itératives de résolution des problèmes approchés, in Topics in Numerical Analysis (J.J.H. Miller, Editor), pp. 123-171, Academic Press, London, 1973. | MR | Zbl

[7] Oden, J.T. : Some contributions to the mathematical theory of mixed finite element approximations, in Theory and Practice in Finite Element Structural Analysis, pp. 3-23, University of Tokyo Press, 1973. | MR | Zbl

[8] Oden, J.T. ; Reddy, J.N. : On dual complementary variational principles in mathematical physics. Int. J. Engng Sci. 12 (1974), 1-29. | MR | Zbl

[9] Reddy, J.N. : A Mathematical Theory of Complementary-Dual Variational Principles and Mixed Finite-Element Approximations of Linear Boundary-Value Problems in Continuum Mechanics, Ph. D. Dissertation, The University of Alabama in Huntsville, Huntsville, 1973.

[10] Bossavit, A. : Une méthode de décomposition de l'opérateur biharmonique. Note HI 585/2, Electricité de France, 1971.

[11] Smith, J. : The coupled equation approach to the numerical solution of the biharmonic equation by finite differences I., SIAM J.Numer, Anal. 5 (1968), 323-339. | MR | Zbl

[12] Smith, J. : On the approximate solution of the first boundary value problem for ${▽}^{4}u=f$, SIAM J. Nuner. Anal. 10 (1973), 967-982. | MR | Zbl

[13] Ehrlich, L.W. : Solving the biharmonic equation as coupled finite difference equations, SIAM J. Numer. Anal. 8 (1971), 278-287. | MR | Zbl

[14] Mclaurin, J.W. : A general coupled equation approach for solving the biharmonic boundary value problem, SIAM J. Numer. Anal. 11 (1974), 14-33. | MR | Zbl

[15] Glowinski, R. ; Lions, J.L. ; Tremolieres, R. : Analyse Numérique des Inéquations Variationnelles (to appear). | Zbl

[16) Ciarlet, P.G. ; Glowinski, R. : Sur la résolution numérique du problème de Dirichlet pour l'opérateur biharmonique. C.R. Acad. Sci. Paris Sér. A (to appear). | Zbl

[17] Ciarlet, P.G. ; Raviart, P.-A. : La Méthode des Eléments Finis pour les Problèmes aux limites Elliptiques. To appear.

[18] Necas, J. : Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967 | Zbl

[19] Kondrat'Ev, V.A. : Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mosk. Mat. Obsc. 16 (1967), 209-292. | MR | Zbl

[20] Ekeland, I.; Temam, R. : Analyse Convexe et Problèmes Variationals. Dunod, Paris, 1974. | Zbl