Quasi-norm local error estimators for p-Laplacian (Q2706405)

The authors consider the piecewise linear finite element approximation of the \(p\)-Laplacian NEWLINE\[NEWLINE -\text{div} (|\nabla u|^{p-2} \nabla u) =f NEWLINE\]NEWLINE subject to Dirichlet boundary conditions in a domain \(\Omega \subset \mathbb{R}^n\), \(n=1, 2,\) or \(3\). The variational formulation of the problem leads to a monotone operator in \(L^p(\Omega)\) for \(p \in (1,\infty)\). In order to obtain sharp error bounds as a base of an adaptive grid refinement the authors use the quasi-norm defined by NEWLINE\[NEWLINE \|v \|^\rho_{(w,p)}:= \int_{\Omega} |\nabla v|^{2}(|\nabla u|+|\nabla v|)^{p-2}, NEWLINE\]NEWLINE where \(\rho:= \max\{2,p \}\). The theoretical analysis is complemented by numerical results.