Finite volume element method for monotone nonlinear elliptic problems

In 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 O(hε) 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.