A Discontinuous Galerkin Method for the Plate Equation

We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing, non matching grids and diierent types of approximation spaces. Continuity is enforced weakly through the vari-ational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element level, occur naturally in the method. We show optimal a priori error estimates in various norms and investigate locking phenomena when certain stabilization parameters tend to innnity. Finally , we relate the method to two classical elements; the nonconforming Morley element and the C 1 Argyris element. 1. Introduction In this paper we propose and analyze a discontinuous Galerkin (dG) method for the plate equation describing the transversal deeection of a thin plate under a transversal load. The method is based on the classical method rst proposed by Nitsche in the context of weak enforcement of boundary conditions 14] and later extended to a discontinuous method with weak enforcement of the continuity of the solution at interior edges by Douglas and Dupont 8], Baker 3], Wheeler 15], and Arnold 2]. In the last few years there has been a renewed interest in these methods, see for instance the proceedings 7] for a comprehensive overview of recent work. The use of discontinuous approximation spaces lead to several advantages, for instance, one can use diierent types of approximation spaces on diierent elements without enforcing continuity; non matching grids, see Becker and Hansbo 4], can also be used. Using the added richness of the spaces one can also construct locking free schemes for nearly incompressible linear elastic materials, see Hansbo and Larson, 9]. Further, discontinuous methods enjoy a local elementwise conservation property, a property often desired in applications. The obvious disadvantage of the dG method is the increased number of degrees