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The Stabilized Nonconforming Virtual Element Method for Linear Elasticity Problem

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Abstract

We present the stabilized nonconforming virtual element method for linear elasticity problem in two dimensions. The jump penalty term is introduced to guarantee the stability of the discrete formulation as the stabilization term, which is obtained based on the discrete Korn’s inequality. In order to obtain the computability of jump penalty term, we reconstruct the lowest-order nonconforming virtual element by imposing some restrictions on the conforming virtual element space of order 2. We prove the interpolation error estimate for the virtual element and the ellipticity of the discrete bilinear form, so the resulting stabilized method is well-posed. Then we show the optimal convergence in the \(L^2\) and \(H^1\) norms. Moreover, this method is locking-free, i.e. the convergence is uniform with respect to the Lamé constant. Numerical results are provided to confirm the theoretical results.

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Data Availibility

The datasets generated during and/or analyzed 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 first author was partially supported by National Natural Science Foundation of China (Nos. 11701522, 11971379) and Natural Science Foundation of Henan Province (No. 222300420553). The third author was partially supported by National Natural Science Foundation of China (No. 12001170) and Research Foundation for Advanced Talents of Henan University of Technology (No. 2018BS013).

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Authors and Affiliations

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

    Jikun Zhao & Tianle Wang

  2. School of Sciences, Henan University of Technology, Zhengzhou, 450001, People’s Republic of China

    Bei Zhang

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  1. Jikun Zhao
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  2. Tianle Wang
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  3. Bei Zhang
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Correspondence to Jikun Zhao.

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Zhao, J., Wang, T. & Zhang, B. The Stabilized Nonconforming Virtual Element Method for Linear Elasticity Problem. J Sci Comput 92, 68 (2022). https://doi.org/10.1007/s10915-022-01927-3

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