Volume 40, Issue 6
Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

J. Comp. Math., 40 (2022), pp. 865-881.

Published online: 2022-08

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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 on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.

65N15, 65N30

beizhang@haut.edu.cn (Bei Zhang)

jkzhao@zzu.edu.cn (Jikun Zhao)

lyminghao@126.com (Minghao Li)

chenhongru5@126.com (Hongru Chen)

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@Article{JCM-40-865, author = {Zhang , BeiZhao , JikunLi , Minghao and Chen , Hongru}, title = {Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {6}, pages = {865--881}, abstract = {

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2103-m2020-0143}, url = {http://global-sci.org/intro/article_detail/jcm/20839.html} }
TY - JOUR T1 - Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes AU - Zhang , Bei AU - Zhao , Jikun AU - Li , Minghao AU - Chen , Hongru JO - Journal of Computational Mathematics VL - 6 SP - 865 EP - 881 PY - 2022 DA - 2022/08 SN - 40 DO - http://doi.org/10.4208/jcm.2103-m2020-0143 UR - https://global-sci.org/intro/article_detail/jcm/20839.html KW - Mixed finite element method, Nonconforming rectangular or cubic elements, 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 on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.

Bei Zhang, Jikun Zhao, Minghao Li & Hongru Chen. (2022). Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes. Journal of Computational Mathematics. 40 (6). 865-881. doi:10.4208/jcm.2103-m2020-0143
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