In this paper we propose a finite element method for the approximation ofsecond order elliptic problems on composite grids. The method isbased on continuous piecewise polynomial approximation on eachgrid and weak enforcement of the proper continuity at anartificial interface defined by edges (or faces) of one the grids.We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions,and present some numerical examples.

In this note, we propose and analyse a method for handlinginterfaces between non-matching grids based on an approachsuggested by Nitsche (1971) for the approximation ofDirichlet boundary conditions. The exposition is limited toself-adjoint elliptic problems, using Poisson's equation as amodel. A priori and a posteriori error estimates are given. Somenumerical results are included.

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.