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A multigrid method for the membrane problem

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Abstract

A multigrid method for the membrane problem is studied. It turns out that some extra devices are needed in problems of elasticity if locking is present on the auxiliary grids. Symmetric successive relaxation is recommended for the smoothing steps whenever the matrices can be kept in the fast storage. Otherwise one may choose Jacobi relaxation. Finally some comments on preconditioned conjugate gradient methods are given.

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References

  • Axelsson, O.; Barker, V.A. (1984): Finite element solution of boundary value problems. Theory and Computation. New York: Academic Press.

    Google Scholar 

  • Braess, D. (1986): On the combination of multigrid methods and conjugate gradients

  • Braess, D.; Hackbush, W. (1983): A new convergence proof for the multigrid method including the ν-cycle. SIAM J. Numer. Anal. 20, 967–975

    Google Scholar 

  • Brand G. (1985): Multigrid methods in finite element analysis of plane elastic structures

  • Brandt, A. (1977): Multi-level adaptive solution to boundary-value problems. Math. Comp. 31, 333–390

    Google Scholar 

  • Hackbush, W. (1985): Multi-grid methods and applications. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Jung, M. (1987): Konvergenzfaktoren für Mehrgitterverfahren zur Lösung von Problemen der Ebenen, Linearen Elastizitäts-theorie. ZAMM 67, 165–173

    Google Scholar 

  • Park, K. C.; Flaggs, D. L. (1984): A Fourier analysis of spurious mechanisms and locking in the finite element method. Comput. Meths. Appl. Mech. Eng. 46, 65–81

    Google Scholar 

  • Peisker, P.; Braess, D. (1987): A conjugate gradient method and a multigrid algorithm for Morley's finite element approximation of the biharmonic equation. Numer. Math. 50, 567–586

    Google Scholar 

  • Prathap, G. (1984): Field consistency and the finite element analysis of multi-field structural problems. Strukturberechnung, pp. 73–139, Rpt. Deutsche Forschungs- and Versuchsanstalt für Luft- and Raumfahrt, DFVLR-Mitt. 84-21.

  • Prathap, G. (1985): An additional stiffness parameter measure of error of the second kind in the finite element method. Int. J. Numer. Meth. Eng. 21, 1001–1012

    Google Scholar 

  • Viswanath, S.; Prathap, G. (1983): A note on locking in a shear flexible triangular plate bending element. Int. J. Numer. Meth. Eng. 19, 305–309

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität, D-4630, Bochum, Germany

    D. Braess

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  1. D. Braess
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Communicated by E. Stein, May 4, 1987

To the memory of Professor Helmut Werner

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Braess, D. A multigrid method for the membrane problem. Computational Mechanics 3, 321–329 (1988). https://doi.org/10.1007/BF00712146

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  • DOI: https://doi.org/10.1007/BF00712146

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