Summary
We set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the Kirchhoff and to the Mindlin-Reissner formulation of the problem. The analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain. The schemes proceed from the Mindlin-Reissner formulation with modified shear energy.
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References
Arnold, D.N.: Discretization by Finite Elements of a Model Parameter Dependent Problem. Numer. Math.37, 405–421 (1981)
Babuška, I.: Error Bounds for Finite Element Method. Numer. Math.16, 322–333 (1971)
Babuška, I., Aziz, A.: Survey Lectures on the Mathematical Foundations of the Finite Element Method. In: The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations (A.K. Aziz, ed.), pp. 5–539. New York: Academic Press 1973
Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of Mixed Methods Using Mesh Dependent Norms. Math. Comput.35, 1039–1062 (1980)
Bathe, J., Brezzi, F.: On the Convergence of a Four-node Plate Bending Element Based on Mindlin-Reissner Plate Theory and Mixed Interpolation In: MAFELAP V (J.R. Whiteman, ed.), pp. 491–503. London: Academic Press 1985
Batoz, J.L., Bathe, K.-J., Ho, L.-W.: A Study of Three-node Triangular Plate Bending Elements. Int. J. Numer. Methods Eng.15, 1771–1812 (1980)
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)
Brezzi, F.: On the Existence, Uniqueness and Approximation of Saddle-point Problems Arising from Lagrange Multipliers. RAIRO Ser. Rouge8, 129–151 (1974)
Brezzi, F., Fortin, M.: Numerical Approximation of Mindlin-Reissner. Plates. Math. Comput.47, 151–158 (1986)
Brezzi, F., Pitkäranta, J.: On the Stabilization of Finite Element Approximations of the Stokes Equations. In: Efficient Solutions of Elliptic Systems. Notes on Numerical Fluid Mechanics, Vol. 10 (W. Hackbusch, ed.). Braunschweig, Wiesbaden: Vieweg 1984
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North Holland 1978
Destuynder, P.: Méthode d'éléments finis pour le modéle de plaques en flexion de Naghdi-Reissner. RAIRO Anal. Numer.15, 331–369 (1981)
Destuynder, P., Nedelec, J.C.: Approximation numérique du cisaillement transverse dans les plaques minces en flexion. Numer. Math.48, 281–302 (1986)
Fried, F., Yang, S.K.: Triangular, Nine-Degrees-of-Freedom,C 0 Plate Bending Element of Quadratic Accuracy. Quart. Appl. Math.31, 303–312 (1973)
Johnson, C., Pitkäranta, J.: Analysis of Some Mixed Finite Element Methods Related to Reduced Integration. Math. Comput.38, 375–400 (1982)
Kikuchi, F.: On a Finite Element Scheme Based on the Discrete Kirchhoff Assumption. Numer. Math.24, 211–231 (1975)
Kikuchi, F.: On a Mixed Method Related to the Discrete Kirchhoff Assumption. In: Hybrid and Mixed Finite Element Methods (S.N. Atluri, R.H. Gallagher, O.C. Zienkiewicz, eds.), pp. 137–154. Chichester: Wiley 1983
Pitkäranta, J.: On Simple Finite Element Methods for Mindlin Plates. In: Computational Mechanics '86 (G. Yagawa, S.N. Atluri, eds.), Vol. 1, pp. 187–190. Tokyo, Heidelberg, Berlin: Springer 1986
Pitkäranta, J.: A Modified Reissner-Mindlin Formation for Plates with Corners. Report A246, Institute of Mathematics, Helsinki University of Technology 1987
Stenberg, R.: On the Construction of Optimal Mixed Finite Element Methods for the Linear Elasticity Problem. Numer. Math.48, 447–462 (1986)
Stricklin, J.A., Haisler, W.E., Tisdale, P.R., Gunderson, R.: A Rapidly Converging Triangular Plate Element. AIAA J.7, 180–181 (1969).
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Dedicated to Professor Ivo Babuška on the occasion of his 60th birthday
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Pitkäranta, J. Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates. Numer. Math. 53, 237–254 (1988). https://doi.org/10.1007/BF01395887
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DOI: https://doi.org/10.1007/BF01395887