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Keywords:

  • finite volume element method;
  • monotone nonlinear elliptic problems;
  • a priori;
  • a posteriori;
  • error estimates

Abstract

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, uH1(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 uH1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate equation image 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