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Volume 41, Issue 6
The Nonconforming Crouzeix-Raviart Element Approximation and Two-Grid Discretizations for the Elastic Eigenvalue Problem

Hai Bi, Xuqing Zhang & Yidu Yang

DOI: 10.4208/jcm.2201-m2020-0128

J. Comp. Math., 41 (2023), pp. 1041-1063.

Published online: 2023-11

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  • Abstract

In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321–338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lamé constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem, and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve the optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

  • Keywords

Elastic eigenvalue problem, Nonconforming Crouzeix-Raviart element, Two-grid discretizations, Error estimates, Locking-free.

  • AMS Subject Headings

65N25, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-41-1041, author = {Bi , HaiZhang , Xuqing and Yang , Yidu}, title = {The Nonconforming Crouzeix-Raviart Element Approximation and Two-Grid Discretizations for the Elastic Eigenvalue Problem}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {6}, pages = {1041--1063}, abstract = {

In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321–338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lamé constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem, and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve the optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2201-m2020-0128}, url = {http://global-sci.org/intro/article_detail/jcm/22103.html} }
TY - JOUR T1 - The Nonconforming Crouzeix-Raviart Element Approximation and Two-Grid Discretizations for the Elastic Eigenvalue Problem AU - Bi , Hai AU - Zhang , Xuqing AU - Yang , Yidu JO - Journal of Computational Mathematics VL - 6 SP - 1041 EP - 1063 PY - 2023 DA - 2023/11 SN - 41 DO - http://doi.org/10.4208/jcm.2201-m2020-0128 UR - https://global-sci.org/intro/article_detail/jcm/22103.html KW - Elastic eigenvalue problem, Nonconforming Crouzeix-Raviart element, Two-grid discretizations, Error estimates, Locking-free. AB -

In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321–338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lamé constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem, and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve the optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.

Hai Bi, Xuqing Zhang & Yidu Yang. (2023). The Nonconforming Crouzeix-Raviart Element Approximation and Two-Grid Discretizations for the Elastic Eigenvalue Problem. Journal of Computational Mathematics. 41 (6). 1041-1063. doi:10.4208/jcm.2201-m2020-0128
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