In this paper, a nonlinear control volume finite element (CVFE) scheme for a degenerate Keller–Segel model with anisotropic and heterogeneous diffusion tensors is proposed and analyzed. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in finite element methods. The diffusion term which involves an anisotropic and heterogeneous tensor is discretized on a dual mesh (Donald mesh) using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. The convergence of the scheme is proved under very general assumptions. Finally, some numerical experiments are carried out to prove the ability of the scheme to tackle degenerate anisotropic and heterogeneous diffusion problems over general meshes.
DOI : 10.5802/smai-jcm.18
Mots clés : Finite Volume, Finite Element, Degenerate Problem, Godunov Scheme, Maximum Principle
@article{SMAI-JCM_2017__3__1_0,
author = {Canc\`es, Cl\'ement and Ibrahim, Moustafa and Saad, Mazen},
title = {Positive nonlinear {CVFE} scheme for degenerate anisotropic {Keller-Segel} system},
journal = {The SMAI Journal of computational mathematics},
pages = {1--28},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {3},
year = {2017},
doi = {10.5802/smai-jcm.18},
mrnumber = {3638625},
zbl = {1416.65339},
language = {en},
url = {http://www.numdam.org/articles/10.5802/smai-jcm.18/}
}
TY - JOUR AU - Cancès, Clément AU - Ibrahim, Moustafa AU - Saad, Mazen TI - Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system JO - The SMAI Journal of computational mathematics PY - 2017 SP - 1 EP - 28 VL - 3 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/smai-jcm.18/ DO - 10.5802/smai-jcm.18 LA - en ID - SMAI-JCM_2017__3__1_0 ER -
%0 Journal Article %A Cancès, Clément %A Ibrahim, Moustafa %A Saad, Mazen %T Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system %J The SMAI Journal of computational mathematics %D 2017 %P 1-28 %V 3 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/smai-jcm.18/ %R 10.5802/smai-jcm.18 %G en %F SMAI-JCM_2017__3__1_0
Cancès, Clément; Ibrahim, Moustafa; Saad, Mazen. Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system. The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 1-28. doi : 10.5802/smai-jcm.18. http://www.numdam.org/articles/10.5802/smai-jcm.18/
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