The stabilized mixed finite element scheme of elasticity problem
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The stabilized mixed finite element scheme of elasticity problem Ming-hao Li1 · Dong-yang Shi2 · Zhen-zhen Li3
Received: 26 June 2016 / Revised: 9 June 2017 / Accepted: 21 June 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, we consider a new mixed finite element scheme of the elasticity problem in two and three dimensions, and the new scheme can impose Neumann boundary condition directly. A new stabilized method is proposed for this scheme, in which the equal order linear element pair is employed to approximate the stress and displacement, and an abstract operator is used to characterize the lack of the inf-sup condition. The new stabilized method is locking free. Numerical results show the excellent stability and accuracy of the new method. Keywords Elasticity problem · Weaker inf-sup condition · Stabilized method · Locking free Mathematics Subject Classification 65N12 · 65N30
Communicated by Abimael Loula.
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Dong-yang Shi [email protected] Ming-hao Li [email protected] Zhen-zhen Li [email protected]
1
College of Science, Henan University of Technology, Zhengzhou 450001, People’s Republic of China
2
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
3
College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, People’s Republic of China
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M. Li et al
1 Introduction In this paper, we consider a mixed finite element (MFE) method of the elasticity problem. As is known to all, this approach requires the pairs of the finites element space satisfy the so-called inf-sup condition (Boffi et al. 2013). Since the stress tensor requires symmetry, it is difficult to construct the stable MFEs. There are two ways to impose the symmetry of the stress tensor. The first way is imposing the symmetry of stress tensor exactly. Some early works employed composite elements (Johnson and Mercier 1978; Arnold et al. 1984). Until 2002, Arnold and Winther proposed the first family of stable MFEs which used polynomial shape functions in two dimensions in Arnold and Winther (2002), which was extended to tetrahedral grids in three dimensions in Arnold et al. (2008). Based on the ideas of Arnold and Winther (2002) and Arnold et al. (2008), some conforming and nonconforming MFEs were also developed in Arnold and Awanou (2005), Chen and Wang (2011), Shi and Li (2014), Chen et al. (2015), Arnold and Winther (2003), Yi (2005), Yi (2006), Hu and Shi (2007), Man et al. (2009), Gopalakrishnan and Guzmán (2011), Awanou (2009) and Arnold et al. (2012). But these elements have too many degrees of freedom (DOF), and the implementations are expensive (Carstensen et al. 2008, 2011). Recently, Hu et al. (2014, 2016), Hu and Zhang (2015, 2016) and Hu (2015a, b) proposed some new methods to construct stable elements. These mixed elements are more compact, their basis functions can be easily obtained, and have less DOF than Arnold–Winther elements, so the numerical implementations ar
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