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H\(^2\)-Korn’s Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

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Abstract

We establish a new H\(^2\)-Korn’s inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle (Specht in Int J Numer Methods Eng 28:705–715, 1988) and the NZT tetrahedron (Wang et al. in Numer Math 106:335–347, 2007) are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct the regularized interpolation operators and the enriching operators for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results for the smooth solution, and the solution with boundary layer are consistent with the corresponding theoretical prediction.

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Notes

  1. The inequality below (3.15), which is exactly (7) for a vector filed satisfying periodic boundary condition over a thin domain.

  2. http://lsec.cc.ac.cn/phg.

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Acknowledgements

The work of Ming was supported by the National Natural Science Foundation of China through Grant No. 11971467 and Beijing Academy of Artificial Intelligence (BAAI). We are grateful to the anonymous referees for their valuable suggestions.

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Authors and Affiliations

  1. School of Mathematics and Science, Sichuan Normal University, Chengdu, 610066, China

    Hongliang Li

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS Chinese Academy of Sciences, No. 55, East Road Zhong-Guan-Cun, Beijing, 100190, China

    Pingbing Ming

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Pingbing Ming

  4. Beijing 101 Middle School, Beijing, China

    Huiyu Wang

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  1. Hongliang Li
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Correspondence to Pingbing Ming.

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Li, H., Ming, P. & Wang, H. H\(^2\)-Korn’s Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model. J Sci Comput 88, 78 (2021). https://doi.org/10.1007/s10915-021-01597-7

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