Finite volume element method for monotone nonlinear elliptic problems

Abstract

AbstractIn this article, we consider the finite volume element method for the monotone nonlinear second‐order elliptic boundary value problems. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz‐continuous, and with the minimal regularity assumption on the exact solution, that is, u∈H1(Ω), we show that the finite volume element method has a unique solution, and the finite volume element approximation is uniformly convergent with respect to the H1 ‐norm. If u∈H1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{O}(h^{\epsilon})\end{align*}\end{document} in the H1 ‐norm. Moreover, we propose a natural and computationally easy residual‐based H1 ‐norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013