Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. III---Discontinuous Galerkin and Other Interior Penalty Methods

Abstract

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix--Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases.