Mixed and nonconforming finite element approximations of Reissner–Mindlin plates

Abstract

We consider a mixed finite element approximation of the Reissner–Mindlin plate model, which plays an important rôle in the simulation of plates and shells of small to moderate thickness. On triangles and quadrilaterals, low-order finite elements are proposed: the rotation is approximated by conforming isoparametric (bi)linear elements and the transverse displacement by constant elements and the shear stress by the lowest-order Raviart–Thomas elements. We prove that the new method satisfies the K -ellipticity and the Inf–Sup condition in the abstract framework of the Babus̆ka–Brezzi theory for mixed problems. Consequently, the method is uniformly stable and uniformly optimally convergent (independent of the thickness of the plate). Moreover, a local postprocessing approach and the implementation by either the augmented Lagrangian algorithm or the hybridization method are discussed, including equivalence to some nonconforming methods. In particular, a new nonconforming method is deduced, which can accommodate arbitrary regular quadrilaterals.