Convergence estimates of finite elements for a class of quasilinear elliptic problems

Abstract

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p-Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W1,p-seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W1,p-seminorm. We also give a simplified proof of known near-best approximation results in W1,p-seminorm starting from the corresponding result in the mentioned quasinorm.