Journal of Scientific Computing

, Volume 75, Issue 2, pp 782–802 | Cite as

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

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Abstract

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.

Keywords

Weak Galerkin Finite element methods Weak gradient The Reissner–Mindlin plate Polygonal partitions 

Mathematics Subject Classification

Primary 65N15 65N30 Secondary 35J50 

References

  1. 1.
    Arnold, D., Brezzi, F., Marini, D.: A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate. J. Sci. Comput. 22, 25–45 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold, D., Brezzi, F., Falk, R., Marini, D.: Locking-free Reissner–Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Eng. 96, 3660–3671 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Falk, R.S.: A Uniformly accurate finite element method for the Reissner–Mindlin plate. SIAM J. Numer. Anal. 26, 1276–1290 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Liu, X.: Interior estimates for a low order finite element method for the Reissner–Mindlin plate model. Adv. Comput. Math. 7, 337–360 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brenner, S.: Korns inequalities for piecewise \(H^1\) vector fields. Math. Comput. 73, 1067–1087 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Numerical approximation of Mindlin–Reissner plates. Math. Comput. 47, 151–158 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brezzi, F., Bathe, K., Fortin, M.: Mixed interpolated elements for Reissner–Mindlin plates. Int. J. Numer. Methods Eng. 28, 1787–1801 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brezzi, G., Fortin, M., Stenberg, R.: Error analysis of mixed-interpolated elements for Reissner–Mindlin plates. Math. Models Methods Appl. Sci. 1, 125–151 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chapelle, D., Stenberg, R.: An optimal low-order locking-free finite element method for Reissner–Mindlin plates. Math. Models Methods Appl. Sci. 8, 407–430 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Duran, R., Liberman, E.: On mixed nite element methods for the Reissner–Mindlin plate model. Math. Comput. 58, 561–573 (1992)CrossRefMATHGoogle Scholar
  11. 11.
    Falk, R., Tu, T.: Locking-free nite elements for the Reissner–Mindlin plate. Math. Comput. 69, 911–928 (2000)CrossRefMATHGoogle Scholar
  12. 12.
    Hansbo, P., Heintz, D., Larson, M.: A finite element method with discontinuous rotations for the Mindlin–Reissner plate model. Comput. Methods Appl. Mech. Eng. 200, 638–648 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lovadina, C., Marini, D.: Nonconforming locking-free nite elements for Reissner–Mindlin plates. Comput. Methods Appl. Mech. Eng. 195, 3448–3460 (2006)CrossRefMATHGoogle Scholar
  14. 14.
    Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes. Int. J. Numer. Anal. Model. 12, 31–53 (2015)MathSciNetMATHGoogle Scholar
  15. 15.
    Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285, 45–58 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mu, L., Wang, J., Ye, X.: A hybridized Weak Galerkin mixed finite element method. J. Comput. Appl. Math. 307, 335–345 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pierre, R.: Convergence properties and numerical approximation of the solution of the Mindlin plate bending problem. Math. Comput. 51, 15–25 (1988)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wang, J., Ye, X.: A superconvergent finite element scheme for the Reissner–Mindlin plate by projection methods. Int. J. Numer. Anal. Model. 1, 99–110 (2004)MathSciNetMATHGoogle Scholar
  19. 19.
    Wang, C., Wang, J.: A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form. Math. Comput. (2017). doi: 10.1090/mcom/3220
  20. 20.
    Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013). arXiv:1104.2897v1
  21. 21.
    Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83, 2101–2126 (2014). arXiv:1202.3655v1
  22. 22.
    Ye, X.: Stabilized finite element approximations for the Reissner–Mindlin plate. Adv. Comput. Math. 13, 375–386 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ye, X.: A rectangular element for the Reissner–Mindlin plate. Numer. Method PDE 16, 184–193 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  2. 2.Division of Mathematical SciencesNational Science FoundationArlingtonUSA
  3. 3.Department of MathematicsUniversity of Arkansas at Little RockLittle RockUSA

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