Quasi-optimal nonconforming methods for symmetric elliptic problems. III -- DG and other interior penalty methods
This work improves the accuracy of existing nonconforming finite element methods for elliptic problems, offering a simple modification that yields near-optimal error bounds.
The paper introduces new variants of DG and interior penalty methods for symmetric elliptic problems that achieve quasi-optimality by modifying the right-hand side discretization, with quasi-optimality constants uniformly bounded and approaching 1 as the penalty parameter increases.
We devise new variants of the following nonconforming finite element methods: DG 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.