Summary
The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.
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References
Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. M2AN19, 7–32 (1985)
Axelsson, O., Barker, V.: Finite element solution of boundary value problems: Theory and computation. New York: Academic Press 1984
Babuška, J., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039–1062 (1980)
Bank, R., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35–51 (1981)
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)
Bourgat, J.F.: Numerical study of a dual iterative method for solving a finite element approximation of the biharmonic equation. Comput. Methods Appl. Mech. Eng.9, 203–218 (1976)
Braess, D., Hackbusch, W.: A new convergence proof for the multigrid method including theV-cycle. SIAM J. Numer. Anal.20, 967–975 (1983)
Braess, D., Peisker, P.: On the numerical solution of the biharmonic equation and the role of squaring matrices for preconditioning. IMA J. Numer. Anal.6, 393–404 (1986).
Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978
Hackbusch, W.: Multi-grid methods and applications. Berlin-Heidelberg-New York-Tokyo: Springer 1985
Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numér.9, 9–53 (1975)
Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications 1. Berlin, Heidelberg, New York: Springer 1972
Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart.19, 149–169 (1968)
Rannacher, R.: On nonconforming and mixed finite element methods for plate bending problems. The linear case. RAIRO Anal. Numér.13, 369–387 (1979)
Stoer, J.: Solution of large systems of equations by conjugate type methods. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of the Art, pp. 540–565, Berlin-Heidelberg-New York: Springer 1983
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Peisker, P., Braess, D. A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation. Numer. Math. 50, 567–586 (1987). https://doi.org/10.1007/BF01408577
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DOI: https://doi.org/10.1007/BF01408577