Quasi-optimal nonconforming methods for symmetric elliptic problems. II -- Overconsistency and classical nonconforming elements
For researchers in numerical analysis, this work provides a theoretical framework to improve the accuracy of classical nonconforming methods without altering the underlying discretization, though the improvements are incremental and theoretical.
The paper develops variants of classical nonconforming finite element methods (Crouzeix-Raviart and Morley) that achieve quasi-optimality by transforming test functions before applying the load functional, with the quasi-optimality constant equaling the stability constant. The methods are applied to Poisson and biharmonic problems, yielding quasi-optimal error estimates in the piecewise energy norm.
We devise variants of classical nonconforming methods for symmetric elliptic problems. These variants differ from the original ones only by transforming discrete test functions into conforming functions before applying the load functional. We derive and discuss conditions on these transformations implying that the ensuing method is quasi-optimal and that its quasi-optimality constant coincides with its stability constant. As applications, we consider the approximation of the Poisson problem with Crouzeix-Raviart elements and higher order counterparts and the approximation of the biharmonic problem with Morley elements. In each case, we construct a computationally feasible transformation and obtain a quasi-optimal method with respect to the piecewise energy norm on a shape regular mesh.