Quasi-optimality of the Crouzeix-Raviart FEM for p-Laplace-type problems

TL;DR

This paper proves the quasi-optimality of the Crouzeix-Raviart finite element method for p-Laplace problems, establishing error bounds related to best approximation and data oscillation.

Contribution

It demonstrates the quasi-optimality of the Crouzeix-Raviart FEM for nonlinear p-Laplace problems and introduces a localized a priori error estimate for conforming FEM.

Findings

Error of Crouzeix-Raviart FEM bounded by best-approximation plus data oscillation

Quasi-optimality constant is uniform and independent of mesh size

New localized a priori error estimate for conforming Lagrange FEM

Abstract

We verify quasi-optimality of the Crouzeix-Raviart FEM for nonlinear problems of $p$-Laplace type. More precisely, we show that the error of the Crouzeix-Raviart FEM with respect to a quasi-norm is bounded from above by a uniformly bounded constant times the best-approximation error plus a data oscillation term. As a byproduct, we verify a novel more localized a priori error estimate for the conforming lowest-order Lagrange FEM.