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A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate

Author: Elsa Liberman
Journal: Math. Comp. 70 (2001), 1383-1396
MSC (2000): Primary 65N30, 65N15, 74K20
Published electronically: November 17, 2000
MathSciNet review: 1836909
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Abstract: We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Durán and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the $H^1$ norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.

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  • 1. D. N. ARNOLD AND F. BREZZI, Mixed and non conforming finite element methods implementation, postprocessing and error estimates, R.A.I.R.O., Modél. Math. Anal. Numer. 19, 1985, pp. 7-32. MR 87g:65126
  • 2. D. N. ARNOLD AND R. S. FALK, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), 1276-1290. MR 91c:65068
  • 3. I. BABUSSKA AND A. MILLER, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comp. Meth. Appl. Mech. Eng. 61 (1987), 1-40. MR 88d:73036
  • 4. I. BABUSSKA AND W. C. RHEINBOLDT, A posteriori error estimators in the finite element method, Int. J. Numer. Methods Eng. 12 (1978), 1587-1615.
  • 5. R. E. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44, (1985), 283-301. MR 86g:65207
  • 6. K. J. BATHE AND F. BREZZI, On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, MAFELAP V (J. R. Witheman, ed.), Academic Press, London, 1985, pp. 491-503. MR 87f:65125
  • 7. K. J. BATHE AND F. BREZZI, A simplified analysis of two plate bending elements--the MITC4 and MITC9 elements, NUMETA 87 (G. N. Pande and J. Middleton, eds.), Numerical Techniques for Engineering Analysis and Design, vol. 1, Martinus Nijhoff, Dordrecht, 1987.
  • 8. K.J. BATHE, F. BREZZI AND M. FORTIN, Mixed-interpolated elements for Reissner-Mindlin plates, Internat. J. Numer. Methods Eng. 28 (1989), 1787-1801. MR 90g:73090
  • 9. K. J. BATHE AND E. N. DVORKIN, A four-node plate bending element based on Mindlin Reissner plate theory and a mixed interpolation, J. Numer. Methods Engrg. 21 (1985),367-383.
  • 10. F. BREZZI AND M. FORTIN, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47 (1986), 151-158. MR 87g:75057
  • 11. F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, Springer-Verlag, New York (1991). MR 92d:65187
  • 12. F. BREZZI, M. FORTIN AND R. STENBERG, Error analysis of mixed-interpolated elements for Mindlin-Reissner plates, Math. Models Methods Appl. Sci. 1 (1991), 125-151. MR 92e:73030
  • 13. P. CLEMENT, Approximation by finite element functions using local regularization. RAIRO Anal. Numér., 9 (1975) 77-84. MR 53:4569
  • 14. E. DARI, R. DURÁN, C. PADRA, V. VAMPA A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (5) (1996), 385-400. MR 97f:65066
  • 15. R. DURÁN AND E. LIBERMAN, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58, Num. 198 (1992), 561-573. MR 92f:65135
  • 16. R. DURÁN, L. HERVELLA-NIETO, E. LIBERMAN, R. RODRIGUEZ AND J. SOLOMIN, Approximation of the vibration modes of a plate by Reissner-Mindlin equations, Math. Comp. 68 (1999), 1447-1463. MR 99m:73045
  • 17. P. PEISKER AND D. BRAESS, Uniform Convergence of Mixed Interpolated Elements for Reissner-Mindlin Plates, RAIRO Modél Math. Anal. Numér. 26 (5) (1992), 557-574. MR 93j:73070
  • 18. L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990) 483-493. MR 90j:65021
  • 19. R. VERF¨URTH, A posteriori error estimators for the Stokes Equations, Numer. Math. 55 (1989), 309-325. MR 90d:65187
  • 20. R. VERF¨URTH, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62, Num. 206 (1994), 445-475. MR 94j:65136

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Additional Information

Elsa Liberman
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C.172, (1900) La Plata, Argentina

Received by editor(s): April 27, 1999
Received by editor(s) in revised form: January 6, 2000
Published electronically: November 17, 2000
Additional Notes: Member of C.I.C, Provincia de Buenos Aires, Argentina.
Article copyright: © Copyright 2000 American Mathematical Society

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