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

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.