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Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations


Authors: Clément Cancès and Cindy Guichard
Journal: Math. Comp. 85 (2016), 549-580
MSC (2010): Primary 65M12, 65M08
DOI: https://doi.org/10.1090/mcom/2997
Published electronically: July 6, 2015
MathSciNet review: 3434871
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Abstract: In this paper, we propose and analyze a Control Volume Finite Elements (CVFE) scheme for solving possibly degenerated parabolic equations. This scheme does not require the introduction of the so-called Kirchhoff transform in its definition. We prove that the discrete solution obtained via the scheme remains in the physical range, and that the natural entropy of the problem decreases with time. The convergence of the method is proved as the discretization steps tend to 0. Finally, numerical examples illustrate the efficiency of the method.


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Clément Cancès
Affiliation: Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France; CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
Email: cances@ljll.math.upmc.fr

Cindy Guichard
Affiliation: Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France; CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France; Inria, ANGE project-team, Rocquencourt - B.P. 105, F78153 Le Chesnay cedex, France; CEREMA, ANGE project-team, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France
Email: guichard@ljll.math.upmc.fr

DOI: https://doi.org/10.1090/mcom/2997
Received by editor(s): March 14, 2014
Received by editor(s) in revised form: August 29, 2014
Published electronically: July 6, 2015
Additional Notes: This work was supported by the French National Research Agency ANR (project GeoPor, grant ANR-13-JS01-0007-01).
Article copyright: © Copyright 2015 American Mathematical Society