Geodesic
Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system
Cancès, Clément1 ; Ibrahim, Moustafa2 ; Saad, Mazen3
1 Inria Lille - Nord Europe, 40, Avenue Halley, 59650 Villeneuve d’Ascq, France
2 American College of the Middle East, Math and science division. 220 Dasman, 15453, Kuwait.
3 Ecole Centrale de Nantes, Laboratoire de Mathématiques Jean Leray, 1, rue de la Noé, BP 92101, 44321 Nantes, France
The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 1-28 Cet article a éte moissonné depuis la source Numdam

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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.

Publié le :
DOI : 10.5802/smai-jcm.18
Classification : 65N08, 65N30
Keywords: Finite Volume, Finite Element, Degenerate Problem, Godunov Scheme, Maximum Principle

Cancès, Clément 1 ; Ibrahim, Moustafa 2 ; Saad, Mazen 3

1 Inria Lille - Nord Europe, 40, Avenue Halley, 59650 Villeneuve d’Ascq, France
2 American College of the Middle East, Math and science division. 220 Dasman, 15453, Kuwait.
3 Ecole Centrale de Nantes, Laboratoire de Mathématiques Jean Leray, 1, rue de la Noé, BP 92101, 44321 Nantes, France 
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     title = {Positive nonlinear {CVFE} scheme for degenerate anisotropic {Keller-Segel} system},
     journal = {The SMAI Journal of computational mathematics},
     pages = {1--28},
     year = {2017},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {3},
     doi = {10.5802/smai-jcm.18},
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     zbl = {1416.65339},
     language = {en},
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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

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