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The Morley-Type Virtual Element for Plate Bending Problems

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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.

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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.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Jikun Zhao, Bei Zhang & Shaochun Chen

  2. LSEC and Institute of Computational Mathematics, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shipeng Mao

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  1. Jikun Zhao
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  2. Bei Zhang
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  3. Shaochun Chen
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  4. Shipeng Mao
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Corresponding author

Correspondence to Jikun Zhao.

Additional information

This work is supported by National Natural Science Foundation of China (11701522, 11371331), National Key Research and Development Program of China (2016YFB0201304) and National Magnetic Confinement Fusion Science Program of China (2015GB110003).

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Zhao, J., Zhang, B., Chen, S. et al. The Morley-Type Virtual Element for Plate Bending Problems. J Sci Comput 76, 610–629 (2018). https://doi.org/10.1007/s10915-017-0632-3

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  • DOI: https://doi.org/10.1007/s10915-017-0632-3

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