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

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.

Figures (2)

• Figure 1: Convergence history of the relative errors $\lVert F(\nabla u) - F(\nabla u_{h,n})\rVert_{L^2(\Omega)}^2 / \lVert F(\nabla u)\rVert_{L^2(\Omega)}^2$ for the lowest-order Lagrange and Crouzeix--Raviart FEM with $p=1.1$. The left-hand side displays the error of iterates $u_{h,n}$ before a mesh refinement, plotted against the degrees of freedom. The right-hand side displays the error of any iterate $u_{h,n}$ plotted against the accumulated number of degrees of freedom.
• Figure 2: Convergence history of the relative errors $\lVert F(\nabla u) - F(\nabla u_{h,n})\rVert_{L^2(\Omega)}^2 / \lVert F(\nabla u)\rVert_{L^2(\Omega)}^2$ for the lowest-order Lagrange and Crouzeix--Raviart FEM with $p=10$. The left-hand side displays the error of iterates $u_{h,n}$ before a mesh refinement, plotted against the degrees of freedom. The right-hand side displays the error of any iterate $u_{h,n}$ plotted against the accumulated number of degrees of freedom.

Theorems & Definitions (23)

• theorem 1: Main result
• theorem 2: A priori error estimate -- non-conforming
• theorem 3: A priori error estimate -- conforming
• remark 1: Comparison to existing estimates
• definition 1: N-function
• proposition 1: Concept of distance
• proof
• remark 2: Quasi-norm of Barrett and Liu
• proposition 2: Properties of N-functions
• proof