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Volume 12, Issue 1
A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

Bei Zhang & Jikun Zhao

Adv. Appl. Math. Mech., 12 (2020), pp. 278-300.

Published online: 2019-12

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  • Abstract

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement. Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement. The mixed method is locking-free, i.e., the convergence does not deteriorate in the nearly incompressible or incompressible case. The optimal convergence order is shown in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement, respectively. Finally, some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

beizhang@haut.edu.cn (Bei Zhang)

jkzhao@zzu.edu.cn (Jikun Zhao)

  • BibTex
  • RIS
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@Article{AAMM-12-278, author = {Zhang , Bei and Zhao , Jikun}, title = {A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {278--300}, abstract = {

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement. Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement. The mixed method is locking-free, i.e., the convergence does not deteriorate in the nearly incompressible or incompressible case. The optimal convergence order is shown in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement, respectively. Finally, some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0048}, url = {http://global-sci.org/intro/article_detail/aamm/13427.html} }
TY - JOUR T1 - A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity AU - Zhang , Bei AU - Zhao , Jikun JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 278 EP - 300 PY - 2019 DA - 2019/12 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0048 UR - https://global-sci.org/intro/article_detail/aamm/13427.html KW - Mixed method, nonconforming finite element, elasticity, locking-free, stabilization. AB -

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement. Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement. The mixed method is locking-free, i.e., the convergence does not deteriorate in the nearly incompressible or incompressible case. The optimal convergence order is shown in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement, respectively. Finally, some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.

Zhang , Bei and Zhao , Jikun. (2019). A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity. Advances in Applied Mathematics and Mechanics. 12 (1). 278-300. doi:10.4208/aamm.OA-2019-0048
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