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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 15, Issue 4


On Preservation of Positivity in Some Finite Element Methods for the Heat Equation

Panagiotis Chatzipantelidis
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  • Department of Mathematics and Applied Mathematics, University of Crete, GR-70013 Heraklion, Greece
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/ Zoltan Horváth
  • Department of Mathematics and Computational Sciences, Széchenyi István University, 1 Egyetem Square, H-9026 Györ, Hungary
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/ Vidar Thomée
  • Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-41296 Göteborg, Sweden; and Institute of Applied and Computational Mathematics, FORTH, GR-71110 Heraklion, Greece
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Published Online: 2015-08-04 | DOI: https://doi.org/10.1515/cmam-2015-0018


We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings.

Keywords: Heat Equation; Finite Element Method; Lumped Mass; Finite Volume Element Method; Spatially Semidiscrete; Fully Discrete; Positivity Preserving; Finite Element Discretization

MSC: 65M12; 65M60

About the article

Received: 2015-02-11

Revised: 2015-06-21

Accepted: 2015-06-29

Published Online: 2015-08-04

Published in Print: 2015-10-01

Funding Source: University of Crete Research Committee

Award identifier / Grant number: KA 4179

Funding Source: European Social Fund and the government of Hungary

Award identifier / Grant number: TÁMOP-4.2.2.A-11/1/KONV-2012-0012

Citation Information: Computational Methods in Applied Mathematics, Volume 15, Issue 4, Pages 417–437, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2015-0018.

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