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Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

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Abstract

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion.

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References

  1. 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.

    Article  MATH  MathSciNet  Google Scholar 

  2. D.N. Arnold and R.S. Falk, Edge effects in the Reissner-Mindlin plate theory, in: Analytical and Computational Models for Shells, eds. A.K. Noor, T. Belytschko and J. Simo (American Society of Mechanical Engineers, New York, 1989) pp. 71–90.

    Google Scholar 

  3. D.N. Arnold and R.S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21 (1990) 281–312.

    Article  MATH  MathSciNet  Google Scholar 

  4. D.N. Arnold and R.S. Falk, Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 27 (1996) 486–514.

    Article  MATH  MathSciNet  Google Scholar 

  5. D.N. Arnold and X. Liu, Local error estimates for finite element discretizations of the Stokes equations, Math. Modelling Numer. Anal. 29 (1995) 367–389.

    MATH  MathSciNet  Google Scholar 

  6. K.J. Bathe, Finite Element Procedure in Engineering Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1982).

    Google Scholar 

  7. F. Brezzi, K.J. Bathe and M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates, Internat. J. Numer. Methods Engrg. 28 (1989) 1787–1802.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47 (1986) 151–158.

    Article  MATH  MathSciNet  Google Scholar 

  9. F. Brezzi and M. Fortin, Mixed Finite Element Methods (Springer, New York, 1991).

    Google Scholar 

  10. J. Douglas, Jr and F.A. Milner, Interior and superconvergence estimates for mixed methods for second order elliptic problems, RAIRO Modél. Math. Anal. Numér. 19 (1985) 397–428.

    MATH  MathSciNet  Google Scholar 

  11. L. Gastaldi, Uniform interior error estimates for the Reissner-Mindlin plate model, Math. Comp. 61 (1993) 539–567.

    Article  MATH  MathSciNet  Google Scholar 

  12. T.J.R. Hughes, The Finite Element Method, Linear Static and Dynamic Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1987).

    Google Scholar 

  13. X. Liu, Interior error estimates for nonconforming finite element methods, Numer. Math. 74 (1996) 49–67.

    Article  MATH  MathSciNet  Google Scholar 

  14. X. Liu, Interior estimates for some nonconforming and mixed finite element methods, Ph.D. Dissertation, The Pennsylvania State University, University Park (December 1993).

    Google Scholar 

  15. J.A. Nitsche and A.H. Schatz, Interior estimate for Ritz-Galerkin methods, Math. Comp. 28 (1974) 937–958.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Temam, Navier-Stokes Equations (North-Holland, Amsterdam, 1984).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Douglas N. Arnold

  2. Department of Mathematics and Computer Science, Clarkson University, Box 5815, Potsdam, NY, 13676-5815, USA

    Xiaobo Liu

Authors
  1. Douglas N. Arnold
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  2. Xiaobo Liu
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Arnold, D.N., Liu, X. Interior estimates for a low order finite element method for the Reissner–Mindlin plate model. Advances in Computational Mathematics 7, 337–360 (1997). https://doi.org/10.1023/A:1018907205385

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