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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Positivity Preserving Gradient Approximation with Linear Finite Elements

Andreas VeeserORCID iD: http://orcid.org/0000-0002-2152-2911
Published Online: 2018-06-30 | DOI: https://doi.org/10.1515/cmam-2018-0017

Abstract

Preserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.

Keywords: Approximation of Gradients; Courant Elements; Best Error Decompositions; Positivity; Obstacles; A Priori Error Bounds

MSC 2010: 65N30; 41A29; 41A05; 41A36; 65N15

Dedicated to Amiya Kumar Pani on the occasion of his 60th birthday

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About the article

Received: 2018-12-18

Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-30


The author is a member of the INdAM research group GNCS, the support of which is gratefully acknowledged.


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0017.

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