Sharply localized pointwise and $W_\infty ^{-1}$ estimates for finite element methods for quasilinear problems

Abstract:

We establish pointwise and $W_\infty ^{-1}$ estimates for finite element methods for a class of second-order quasilinear elliptic problems defined on domains $\Omega$ in $\mathbb {R}^n$. These estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order. Our pointwise estimates are similar to and rely on results and analysis techniques of Schatz for linear problems. We also extend estimates of Schatz and Wahlbin for pointwise differences $e(x_1)-e(x_2)$ in pointwise errors to quasilinear problems. Finally, we establish estimates for the error in $W_\infty ^{-1}(D)$, where $D \subset \Omega$ is a subdomain. These negative norm estimates are novel for linear as well as for nonlinear problems. Our analysis heavily exploits the fact that Galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships, thus allowing easy application of properly formulated results for linear problems.