Quasi-optimal nonconforming methods for symmetric elliptic problems. II -- Overconsistency and classical nonconforming elements

TL;DR

This paper introduces variants of classical nonconforming methods for symmetric elliptic problems, transforming test functions into conforming functions to achieve quasi-optimality with practical transformations for key problems.

Contribution

It develops a framework for transforming nonconforming methods into quasi-optimal ones by modifying test functions, applicable to Poisson and biharmonic problems.

Findings

Achieves quasi-optimality with shape regular meshes

Constructs feasible transformations for Crouzeix-Raviart and Morley elements

Ensures the quasi-optimality constant equals the stability constant

Abstract

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.