# Positivity preserving finite element approximation

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Math. Comp., 71 (2001), 1405-1419.
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Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Lars B. Wahlbin

Department of Mathematics

Cornell University

Ithaca, NY 14853, USA

wahlbin@math.cornell.edu
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Abstract

We consider finite element operators defined on ``rough'' functions
in a bounded polyhedron $\Om$ in $\R^N$.
Insisting on preserving positivity in the approximations, we
discover an intriguing and basic
difference between approximating functions which vanish on the
boundary of $\Om$ and approximating general functions which do not.
We give impossibility results for approximation
of general functions to more than first order accuracy
at extreme points of $\Om$. We also give impossibility results
about invariance of positive operators on finite element functions.
This is in striking contrast to the
well-studied case without positivity.