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Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem


Authors: Jun Hu and Zhong-Ci Shi
Journal: Math. Comp. 78 (2009), 673-711
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-08-02153-4
Published electronically: August 1, 2008
MathSciNet review: 2476556
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Abstract: The present paper is made up of two parts. In the first part, we study the mathematical stability and convergence of the quadrilateral MITC elements for the Reissner-Mindlin plate problem in an abstract setting. We generalize the Brezzi-Bathe-Fortin conditions to the quadrilateral MITC elements by weakening the second and fourth conditions. Under these conditions, we show the well-posedness of the discrete problem and establish an abstract error estimate in the energy norm. The conclusion of this part is sparsity in the mathematical research of the quadrilateral MITC elements in the sense that one only needs to check these five conditions.

In the second part, we extend four families of rectangular MITC elements of Stenberg and Süri to the quadrilateral meshes. We prove that these quadrilateral elements satisfy the generalized Brezzi-Bathe-Fortin conditions from the first part. We develop the h-p error estimates in both energy and $ L^2$ norm for these quadrilateral elements. For the first three families of quadrilateral elements, the error estimates indicate that their convergent rates in both energy and $ L^2$ norm depend on the mesh distortion parameter $ \alpha$. We can get optimal error estimates for them provided that $ \alpha=1$. In addition, we show the optimal convergence rates in energy norm uniformly in $ \alpha$ for the fourth family of quadrilateral elements. Like their rectangular counterparts, these quadrilateral elements are locking-free.


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Additional Information

Jun Hu
Affiliation: LMAM and School of Mathematical Sciences, Peking University, 100871 Beijing, China
Email: hujun@math.pku.edu.cn

Zhong-Ci Shi
Affiliation: No 55, Zhong-Guan-Cun Dong Lu, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, China
Email: shi@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/S0025-5718-08-02153-4
Keywords: Reissner-Mandlin plate, MITC, quadrilateral element, locking-free.
Received by editor(s): October 26, 2006
Received by editor(s) in revised form: February 16, 2008
Published electronically: August 1, 2008
Additional Notes: This research was supported by the Special Funds for Major State Basic Research Project. The first author was partially supported by the National Science Foundation of China under Grant No.10601003 and A Foundation for the Author of National Excellent Doctoral Dissertation of PR China 200718.
Article copyright: © Copyright 2008 American Mathematical Society

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