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Positivity preserving finite element approximation

Authors:	Ricardo H Nochetto Lars B Wahlbin

Institution:	Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 ; Department of Mathematics, Cornell University, Ithaca, New York 14853

Abstract:	We consider finite element operators defined on ``rough' functions in a bounded polyhedron $\Omega$ in $\mathbb{R} ^N$ . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of $\Omega$ 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 $\Omega$ . 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.

Keywords:	Positive operators finite elements extreme points second order accuracy

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