Finite volume element method for monotone nonlinear elliptic problems

  1. Chunjia Bi1,*,
  2. Yanping Lin2,
  3. Min Yang1

Article first published online: 27 OCT 2012

DOI: 10.1002/num.21747

Issue

Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Volume 29, Issue 4, pages 1097–1120, July 2013

Additional Information

How to Cite

Bi, C., Lin, Y. and Yang, M. (2013), Finite volume element method for monotone nonlinear elliptic problems. Numer. Methods Partial Differential Eq., 29: 1097–1120. doi: 10.1002/num.21747

Author Information

  1. 1

    Department of Mathematics, Yantai University, Shandong 264005, China

  2. 2

    Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

*Department of Mathematics, Yantai University, Shandong 264005, China

Publication History

  1. Issue published online: 23 APR 2013
  2. Article first published online: 27 OCT 2012
  3. Manuscript Accepted: 25 SEP 2012
  4. Manuscript Received: 14 NOV 2011

Funded by

  • Shandong Rovince Natural Science Foundation. Grant Numbers: ZR2010AM004, ZR2010AQ020
  • National Natural Science Foundation of China. Grant Number: 11201405
  • Projects of Shandong Province Higher Educational Science and Technology. Grant Numbers: J10LA01, J11LA09
  • RGF of SAR Hong Kong, China (PolyU 5017/09P), PolyU G-U946 and NSERC (Canada)

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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, 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 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

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