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Journal of Scientific Computing

, Volume 76, Issue 1, pp 610–629 | Cite as

The Morley-Type Virtual Element for Plate Bending Problems

  • Jikun ZhaoEmail author
  • Bei Zhang
  • Shaochun Chen
  • Shipeng Mao
Article

Abstract

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in \(H^2\)-norm. Moreover, we prove the optimal error estimates in \(H^1\)- and \(L^2\)-norm. The nonconforming virtual element is constructed for any order of accuracy, but not \(C^0\)-continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in \(H^2\)-norm for the nonconforming virtual element by some numerical tests.

Keywords

Nonconforming virtual element Plate bending problem Polygonal mesh 

Mathematics Subject Classification

65N30 65N12 

Notes

Acknowledgements

We would like to thank Jiming Wu, Zhiming Gao and Shuai Su from Institute of Applied Physics and Computational Mathematics, Beijing, China, for the provision of the polygonal mesh data and specially for the encouragement to carry on with this work. We also thank Donatella Marini and Claudia Chinosi from Italy, for the guidance to compute the errors of VEM.

References

  1. 1.
    Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antonietti, P.F., da Veiga, L.B., Scacchi, S., Verani, M.: A \(C^1\) virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54, 34–56 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antonietti, P.F., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. arXiv:1611.08736 (2016)
  4. 4.
    Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 3, 199–214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24, 1541–1573 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Guan, Q., Sung, L.-Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math.  https://doi.org/10.1515/cmam-2017-0008 (2017)
  7. 7.
    Brenner, S.C., Scott, L.R.: Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  8. 8.
    Brenner, S.C., Wang, K., Zhao, J.: Poincaré-Friedrichs inequalities for piecewise \(H^2\) functions. Numer. Funct. Anal. Optim. 25, 463–478 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chinosi, C., Marini, L.D.: Virtual element method for fourth order problems: \(L^2\)-estimates. Comput. Math. Appl. 72, 1959–1967 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)zbMATHGoogle Scholar
  12. 12.
    Grisvard, P.: Singularities in Boundary Value Problems, vol. 22. Springer, Berlin (1992)zbMATHGoogle Scholar
  13. 13.
    Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numér. 9, 9–53 (1975)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mao, S., Nicaise, S., Shi, Z.-C.: Error estimates of Morley triangular element satisfying the maximal angle condition. Int. J. Numer. Anal. Model. 7, 639–655 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, 149–169 (1968)Google Scholar
  16. 16.
    Shi, Z.C.: On the error estimates of Morley element. Math. Numer. Sin. 12, 113–118 (1990). (in Chinese)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhao, J., Chen, S., Zhang, B.: The nonconforming virtual element method for plate bending problems. Math. Models Methods Appl. Sci. 26, 1671–1687 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China
  2. 2.LSEC and Institute of Computational Mathematics, Academy of Mathematics and Systems ScienceUniversity of Chinese Academy of Sciences, Chinese Academy of SciencesBeijingPeople’s Republic of China

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