Enhanced positive vertex-centered finite volume scheme for anisotropic convection-diffusion equations

Quenjel, El Houssaine

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 591-618.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

This article is about the development and the analysis of an enhanced positive control volume finite element scheme for degenerate convection-diffusion type problems. The proposed scheme involves only vertex unknowns and features anisotropic fields. The novelty of the approach is to devise a reliable upwind approximation with respect to flux-like functions for the elliptic term. Then, it is shown that the discrete solution remains nonnegative. Under general assumptions on the data and the mesh, the convergence of the numerical scheme is established owing to a recent compactness argument. The efficiency and stability of the methodology are numerically illustrated for different anisotropic ratios and nonlinearities.

[1] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2–14. | DOI | Zbl

[2] I. Aavatsmark, G. Eigestad, R. Klausen, M. Wheeler and I. Yotov, Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333–345. | DOI | MR | Zbl

[3] M. Afif and B. Amaziane, Convergence of finite volume schemes for a degenerate convection–diffusion equation arising in flow in porous media. Comput. Methods Appl. Mech. Eng. 191 (2002) 5265–5286. | DOI | MR | Zbl

[4] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray- Lions- type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23 (2007) 145–195. | DOI | MR | Zbl

[5] B. Andreianov, C. Cancès and A. Moussa, A nonlinear time compactness result and applications to discretization of degenerate parabolic–elliptic PDEs. J. Funct. Anal. 273 (2017) 3633–3670. | DOI | MR

[6] T. Arbogast and M.F. Wheeler, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33 (1996) 1669–1687. | DOI | MR | Zbl

[7] V.I. Bogachev, Measure Theory. Springer Science & Business Media 1 (2007). | DOI | MR | Zbl

[8] K. Brenner and R. Masson, Convergence of a vertex centred discretization of two-phase darcy flows on general meshes. Int. J. Finite 10 (2013) 1–37. | MR

[9] K. Brenner, C. Cancès and D. Hilhorst, Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 17 (2013) 573–597. | DOI | MR

[10] Z. Cai, On the finite volume element method. Numer. Math. 58 (1990) 713–735. | DOI | MR | Zbl

[11] C. Cancès, M. Cathala and C. Le Potier, Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125 (2013) 387–417. | DOI | MR | Zbl

[12] C. Cancès and C. Guichard, Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comput. 85 (2016) 549–580. | DOI | MR

[13] C. Cancès and C. Guichard, Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure. Found. Comput. Math. 17 (2017) 1525–1584. | DOI | MR

[14] G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation: single phase, multiphase and multicomponent flows through porous media. In: Vol. 17 of Stud. Math. Appl. North-Holland, Amsterdam (1986). | Zbl

[15] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media. SIAM 2 (2006). | MR | Zbl

[16] P. Ciarlet, The Finite Element Method for Elhptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[17] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. | DOI | Numdam | MR | Zbl

[18] J. Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. | DOI | MR | Zbl

[19] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35–71. | DOI | MR | Zbl

[20] J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The Gradient Discretisation Method. Springer 82 (2018). | DOI | MR

[21] A. Ern and J.-L. Guermond, . Theory and Practice of Finite Elements. Springer Science & Business Media 159 (2013). | MR | Zbl

[22] R.E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865–1888. | DOI | MR | Zbl

[23] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. In: Vol. 7 of Handbook of Numerical Analysis. Elsevier (2000) 713–1018. | MR | Zbl

[24] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. | DOI | MR | Zbl

[25] R. Eymard, D. Hilhorst and M. Vohralk, A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105 (2006) 73–131. | DOI | MR | Zbl

[26] R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2009) 1009–1043. | DOI | MR | Zbl

[27] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2012) 265–290. | DOI | Numdam | MR | Zbl

[28] G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière. Springer Science & Business Media 22 (1995). | MR | Zbl

[29] M. Ghilani, E.H. Quenjel and M. Saad, Convergence of a positivity-preserving finite volume scheme for compressible two-phase flows in anisotropic porous media: the densities are depending on the physical pressures. submitted (2019).

[30] M. Ghilani, E.H. Quenjel and M. Saad, Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media. Comput. Geosci. 23 (2019) 55–79. | DOI | MR

[31] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer-Verlag (1997). | DOI

[32] R. Herbin, F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, edited by R. Eymard and J.-M. Herard. In: Finite Volumes for Complex Applications V. Wiley (2008) 659–692. | MR

[33] W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-dependent Advection-diffusion-reaction Equations. Springer Science & Business Media 33 (2013). | MR | Zbl

[34] M. Ibrahim and M. Saad, On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model. Comput. Math. App. 68 (2014) 1032–1051. | MR

[35] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press 31 (2002). | DOI | MR | Zbl

[36] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR | Zbl

[37] A.A.H. Oulhaj, C. Cancès and C. Chainais-Hillairet, Numerical analysis of a nonlinearly stable and positive Control Volume Finite Element scheme for Richards equation with anisotropy. ESAIM: M2AN 52 (2018) 1533–1567. | DOI | Numdam | MR

[38] E.H. Quenjel, M. Saad, M. Ghilani and M. Bessemoulin-Chatard, On the positivity of a discrete duality finite volume scheme for degenerate nonlinear diffusion equations. submitted (2018).

[39] B. Saad and M. Saad, Study of full implicit petroleum engineering finite-volume scheme for compressible two-phase flow in porous media. SIAM J. Numer. Anal. 51 (2013) 716–741. | DOI | MR | Zbl

[40] M. Schneider, L. Agélas, G. Enchéry and B. Flemisch, Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351 (2017) 80–107. | DOI | MR

[41] R.S. Varga, Matrix Iterative Analysis. Springer Science & Business Media 27 (2009). | Zbl