Maximum principle and uniform convergence for the finite element method

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)17-31
Journal / PublicationComputer Methods in Applied Mechanics and Engineering
Volume2
Issue number1
Publication statusPublished - Feb 1973
Externally publishedYes

Abstract

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ⊂ Rn, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution uh converges uniformly to the exact solution u if u ε{lunate} W1,p (Ω), with p > n, and that ∥u-uhL∞(Ω) = O(h) if u ε{lunate} W2,p(Ω), with 2p > n. In the case of the model problem -Δu+au = f in Ω, u = uo on δΩ, with a ≥ 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem. © 1973.