Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity
Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity
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DOI：
10.1137/060669681
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发表时间：
2007-12
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影响因子：
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通讯作者：
Jun-Jue Hu;Zhongci Shi
中科院分区：
文献类型：
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作者：
Jun-Jue Hu;Zhongci Shi
在本文中，我们提出了两种稳定的矩形非相容混合有限元方法，用于求解二维空间中的线弹性方程，这些方法可以直接逼近应力和位移。在第一种方法中，矩阵值应力空间的法向应力空间被视为二阶旋转Brezzi-Douglas-Fortin-Marini元素空间[F. Brezzi 和 M. Fortin，混合和混合有限元方法，Springer-Verlag，纽约，1991 年]，丰富的非相容旋转 $Q_1$ 元素 [Q. Lin、L. Tobiska 和 A. H. Zhou，IMA J. Numer。 Anal., 25 (2005), pp. 160-181] 取剪应力，并采用最低阶 Raviart-Thomas 元素空间 [P. Anal., 25 (2005), pp. 160-181]。 A. Raviart 和 J. M. Thomas，《有限元方法的数学方面》，数学讲义。 606，Springer-Verlag，纽约，1977 年，第 292-315 页]用于近似矢量位移场。第二种方法是由第一种方法通过降低每个单元上的法向应力的内度而得到的。基于丰富的非相容旋转 $Q_1$ 元素的超收敛，这些方法获得了应力和位移的一阶收敛速率。
In this paper, we present two stable rectangular nonconforming mixed finite element methods for the equations of linear elasticity in two space dimensions which produce direct approximations for the stress and displacement. In the first method, the normal stress space of the matrix-valued stress space is taken as the second order rotated Brezzi-Douglas-Fortin-Marini element space [F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991], the enriched nonconforming rotated $Q_1$ element [Q. Lin, L. Tobiska, and A. H. Zhou, IMA J. Numer. Anal., 25 (2005), pp. 160-181] is taken for the shear stress, and the lowest order Raviart-Thomas element space [P. A. Raviart and J. M. Thomas, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Math. 606, Springer-Verlag, New York, 1977, pp. 292-315] is employed to approximate the vector displacement field. The second method is obtained from the first one through dropping the interior degrees of the normal stress on each element. A first order convergence rate is obtained for both the stress and the displacement for these methods based on the superconvergence of the enriched nonconforming rotated $Q_1$ element.