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Virtual Element Methods for Elliptic Variational Inequalities of the Second Kind

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Abstract

This paper is devoted to virtual element methods for solving elliptic variational inequalities (EVIs) of the second kind. First, a general framework is provided for the numerical solution of the EVIs and for its error analysis. Then virtual element methods are applied to solve two representative EVIs: a simplified friction problem and a frictional contact problem. Optimal order error estimates are derived for the virtual element solutions of the two representative EVIs, including the effects of numerical integration for the non-smooth term in the EVIs. A fast solver is introduced to solve the discrete problems. Several numerical examples are included to show the numerical performance of the proposed methods.

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Acknowledgements

The authors are grateful to the referees for their valuable suggestions and comments which improved an early version of the paper.

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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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  3. Jianguo Huang
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Correspondence to Jianguo Huang.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of W. Han was partially supported by NSF under the Grant DMS-1521684.

The work of J. Huang was partially supported by NSFC (Grant No. 11571237).

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Feng, F., Han, W. & Huang, J. Virtual Element Methods for Elliptic Variational Inequalities of the Second Kind. J Sci Comput 80, 60–80 (2019). https://doi.org/10.1007/s10915-019-00929-y

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  • DOI: https://doi.org/10.1007/s10915-019-00929-y

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