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 epsilon H-1(Omega), 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 H-1 -norm. If u epsilon H1+epsilon(Omega),0 < epsilon <= 1, we develop the optimal convergence rate O (h(epsilon)) in the H-1 -norm. Moreover, we propose a natural and computationally easy residual-based H-1 -norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. (c) 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013