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Mathematics of Computation

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Positivity preserving finite element approximation

Authors: Ricardo H. Nochetto and Lars B. Wahlbin
Journal: Math. Comp. 71 (2002), 1405-1419
MSC (2000): Primary 41A25, 41A36, 65D05, 65N15, 65N30
Published electronically: November 20, 2001
MathSciNet review: 1933037
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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.

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Additional Information

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742

Lars B. Wahlbin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Keywords: Positive operators, finite elements, extreme points, second order accuracy
Received by editor(s): November 5, 1999
Received by editor(s) in revised form: November 21, 2000
Published electronically: November 20, 2001
Article copyright: © Copyright 2001 American Mathematical Society