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A Nonconforming Virtual Element Method for a Fourth-order Hemivariational Inequality in Kirchhoff Plate Problem

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Abstract

This paper is devoted to a fourth-order hemivariational inequality for a Kirchhoff plate problem. A solution existence and uniqueness result is proved for the hemivariational inequality through the analysis of a corresponding minimization problem. A nonconforming virtual element method is developed to solve the hemivariational inequality. An optimal order error estimate in a broken \(H^2\)-norm is derived for the virtual element solutions under appropriate solution regularity assumptions. The discrete problem can be formulated as an optimization problem for a difference of two convex (DC) functions and a convergent algorithm is used to solve it. Computer simulation results on a numerical example are reported, providing numerical convergence orders that match the theoretical prediction.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Code Availability

The codes during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank Profs. K. Joki and O. Montonen for providing their algorithms freely at the website http://napsu.karmitsa.fi/nsosoftware/, which are very helpful for solving the discrete problems given in this paper. The authors also thank the two anonymous referees for valuable comments and suggestions which helped to improve an early version of the paper.

Funding

The work of Prof. Weimin Han was partially supported by Simons Foundation Collaboration Grants, No. 850737. The work of Prof. Jianguo Huang was partially supported by NSFC (Grant No. 12071289).

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Authors and Affiliations

  1. School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Fang Feng & Jianguo Huang

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

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  1. Fang Feng
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  2. Weimin Han
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All authors contributed equally to this manuscript.

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Correspondence to Jianguo Huang.

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Feng, F., Han, W. & Huang, J. A Nonconforming Virtual Element Method for a Fourth-order Hemivariational Inequality in Kirchhoff Plate Problem. J Sci Comput 90, 89 (2022). https://doi.org/10.1007/s10915-022-01759-1

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  • DOI: https://doi.org/10.1007/s10915-022-01759-1

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