A Posteriori Finite Element Error Control for the P-Laplace Problem

Abstract

Three a posteriori error estimates are discussed for the p-Laplace problem as a model equation for nonquadratic growth of order p>2. The standard residual-based error estimate is addressed with emphasis on involved constants determined as local eigenvalues. Up to errors in their numerical computation, they provide a guaranteed upper bound for the W1,p seminorm and a weighted W1,2 seminorm of u-uh under the major assumption that $|\nabla u_h|$ is positive. A second, sharper, a posteriori estimate is based on the numerical solution of local interface problems. Averaging techniques for a posteriori error control on the flux error motivate a third estimator. The results are presented for any shape regular but unstructured triangulation into triangles. They partly rely on a positive weight function $\rho:=(|\nabla u|^{p-2} + |\nabla u_h|^{p-2})/2$ similar to the notion of quasi norms due to Barrett and Liu. Numerical experiments indicate the surprising accuracy of our averaging error estimators, confirm our estimates, and allow for a comparison.