Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations
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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.References
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Additional Information
- 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
- 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).
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 549-580
- MSC (2010): Primary 65M12, 65M08
- DOI: https://doi.org/10.1090/mcom/2997
- MathSciNet review: 3434871