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
Keywords: Finite Volume, Finite Element, Degenerate Problem, Godunov Scheme, Maximum Principle
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2017__3__1_0,
author = {Cl\'ement Canc\`es and Moustafa Ibrahim and Mazen Saad},
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},
zbl = {1416.65339},
mrnumber = {3638625},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.18/}
}
TY - JOUR AU - Clément Cancès AU - Moustafa Ibrahim AU - Mazen Saad 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 - https://smai-jcm.centre-mersenne.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 Clément Cancès %A Moustafa Ibrahim %A Mazen Saad %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 https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.18/ %R 10.5802/smai-jcm.18 %G en %F SMAI-JCM_2017__3__1_0
Clément Cancès; Moustafa Ibrahim; Mazen Saad. Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system. The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 1-28. doi : 10.5802/smai-jcm.18. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.18/
[1] Convergence of finite volume schemes for a degenerate convection–diffusion equation arising in flow in porous media, Computer methods in applied mechanics and engineering, Volume 191 (2002) no. 46, pp. 5265-5286 | DOI | MR | Zbl
[2] Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients. Application to a convection diffusion problem, EAST WEST J NUMER MATH, Volume 3 (1995) no. 4, pp. 237-254 | MR | Zbl
[3] Quasilinear elliptic-parabolic differential equations, Math. Z., Volume 183 (1983) no. 3, pp. 311-341 | DOI | MR | Zbl
[4] Finite volume methods for degenerate chemotaxis model, Journal of computational and applied mathematics, Volume 235 (2011) no. 14, pp. 4015-4031 | DOI | MR | Zbl
[5] A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs (2015) (https://hal.archives-ouvertes.fr/hal-01142499/document) | Zbl
[6] Convergence of a vertex centered discretization of two-phase Darcy flows on general meshes, International Journal of Finite Volume, Volume 10 (2013), pp. 1-37 | MR
[7] On the finite volume element method, Numerische Mathematik, Volume 58 (1990) no. 1, pp. 713-735 | DOI | MR
[8] Entropy-diminishing CVFE scheme for solving anisotropic degenerate diffusion equations, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects (2014), pp. 187-196 | DOI | Zbl
[9] Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations, Math. Comp., Volume 85 (2016) no. 298, pp. 549-580 | DOI | MR | Zbl
[10] Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure, Found. Comput. Math. (2016), pp. 1-60 | DOI
[11] Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations, Numerische Mathematik, Volume 3 (2013), pp. 387-417 | DOI | MR | Zbl
[12] Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model, Numerical Methods for Partial Differential Equations, Volume 30 (2014) no. 3, pp. 1030-1065 | DOI | MR | Zbl
[13] Mixed-hybrid finite elements and cell-centred finite volumes for two-phase flow in porous media, Mathematical Modelling of Flow Through Porous Media (1995), pp. 100-114
[14] Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 33 (1999) no. 03, pp. 493-516 | DOI | Numdam | MR | Zbl
[15] Finite volume schemes for diffusion equations: introduction to and review of modern methods, Mathematical Models and Methods in Applied Sciences, Volume 24 (2014) no. 08, pp. 1575-1619 | DOI | MR | Zbl
[16] Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations, Numer. Math., Volume 132 (2016) no. 4, pp. 721-766 | DOI | MR | Zbl
[17] The gradient discretisation method (2016) (https://hal.archives-ouvertes.fr/hal-01382358) | Zbl
[18] Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations, Mathematical Models and Methods in Applied Sciences, Volume 23 (2013) no. 13, pp. 2395-2432 | DOI | MR | Zbl
[19] Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal., Volume 18 (1998) no. 4, pp. 563-594 | DOI | MR | Zbl
[20] Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numerische Mathematik, Volume 92 (2002) no. 1, pp. 41-82 | DOI | MR
[21] Finite volume methods, Handbook of numerical analysis, Volume 7 (2000), pp. 713-1018
[22] Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, Volume 30 (2010) no. 4, pp. 1009-1043 | DOI | MR | Zbl
[23] , Finite Volumes for Complex Applications VIII (Proceedings in Mathematics and Statistics) (2017) | Zbl
[24] Hyperbolic systems of conservation laws, Mathematics and Applications, 3–4, Ellipses, Paris, 1991, 252 pages
[25] On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model, Computers & Mathematics with Applications, Volume 68 (2014) no. 9, pp. 1032 -1051 | DOI | MR | Zbl
[26] Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, Volume 26 (1970) no. 3, pp. 399-415 | DOI | MR | Zbl
[27] Model for chemotaxis, Journal of Theoretical Biology, Volume 30 (1971) no. 2, pp. 225-234 | DOI | Zbl
[28] Nonlinear conservation laws and finite volume methods, Computational methods for astrophysical fluid flow, Springer, 1998, pp. 1-159
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