Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity

• Authors: Jun Hu;Zhong-Ci Shi
• Venue: SIAM Journal on Numerical Analysis
• Year: 2007

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.