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Finite volume element method for monotone nonlinear elliptic problems
Author(s) -
Bi Chunjia,
Lin Yanping,
Yang Min
Publication year - 2013
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21747
Subject(s) - mathematics , monotone polygon , lipschitz continuity , norm (philosophy) , finite element method , rate of convergence , nonlinear system , estimator , upper and lower bounds , mathematical analysis , finite volume method , a priori and a posteriori , geometry , channel (broadcasting) , statistics , physics , electrical engineering , philosophy , epistemology , quantum mechanics , political science , mechanics , law , thermodynamics , engineering
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 ∈ H 1 (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 H 1 ‐norm. If u ∈ H 1+ 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 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. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013