Numerical methods and HPC
Open Access
Issue
Oil & Gas Science and Technology - Rev. IFP Energies nouvelles
Volume 73, 2018
Numerical methods and HPC
Article Number 78
Number of page(s) 18
DOI https://doi.org/10.2516/ogst/2018067
Published online 12 December 2018
  • van Duijn C.J., Molenaar J., de Neef M.J. (1995) The effect of capillary forces on immiscible two-phase flows in heterogeneous porous media, Transp. Porous Media 21, 71–93. [Google Scholar]
  • Bertsch M., Dal Passo R., van Duijn C.J. (2003) Analysis of oil trapping in porous media flow, SIAM J. Math. Anal. 35, 1, 245–267. ISSN 0036-1410. [CrossRef] [MathSciNet] [Google Scholar]
  • Buzzi F., Lenzinger M., Schweizer B. (2009) Interface conditions for degenerate two-phase flow equations in one space dimension, Analysis 29, 299–316. [Google Scholar]
  • Cancès C., Gallouët T., Porretta A. (2009) Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound. 11, 2, 239–258. [CrossRef] [Google Scholar]
  • Cancès C., Pierre M. (2012) An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field, SIAM J. Math. Anal. 44, 2, 966–992. doi: 10.1137/11082943X. URL http://hal.archives-ouvertes.fr/hal-00518219. [CrossRef] [MathSciNet] [Google Scholar]
  • Cancès C., Gallouët T.O., Monsaingeon L. (2015) The gradient flow structure of immiscible incompressible two-phase flows in porous media, C. R. Acad. Sci. Paris Ser. I Math. 353, 985–989. [CrossRef] [Google Scholar]
  • Cancès C., Gallouët T.O., Monsaingeon L. (2017) Incompressible immiscible multiphase flows in porous media: a variational approach, Anal. PDE 10, 8, 1845–1876. [CrossRef] [Google Scholar]
  • Cancès C., Gallouët T.O., Laborde M., Monsain-Geon L. (2018) Simulation of multiphase porous media flows with minimizing movement and finite volume schemes, HAL, hal-01700952. URL https://hal.archives-ouvertes.fr/hal-01700952/document . [Google Scholar]
  • Murphy T.J., Walkington N.J. Control volume approximation of degenerate two-phase porous media flows, submitted for publication. [Google Scholar]
  • Mielke A. (2011) A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity 24, 4, 1329–1346, ISSN 0951-7715. doi: 10.1088/0951-7715/24/4/016. URL http://dx.doi.org/10.1088/0951-7715/24/4/016. [Google Scholar]
  • Otto F. (2001) The geometry of dissipative evolution equations: the porous medium equation, Comm. PDE 26, 1–2, 101–174, ISSN 0360-5302. [Google Scholar]
  • Ambrosio L., Gigli N., Savaré G. (2008) Gradient flows in metric spaces and in the space of probability measures, 2nd edn, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, ISBN 978-3-7643-8721-1. [Google Scholar]
  • Bessemoulin-Chatard M. (2012) Développement et analyse de schémas volumes finis motivés par la préservation de comportements asymptotiques. Application à des modèles issus de la physique et de la biologie, PhD Thesis, Université Blaise Pascal – Clermont-Ferrand II, 2012. URL http://tel.archives-ouvertes.fr/tel-00836514 [Google Scholar]
  • Bear J., Bachmat Y. (1990) Introduction to modeling of transport phenomena in porous media, Kluwer Academic Publishers, Dordrecht, The Netherlands. [Google Scholar]
  • Maury B., Roudneff-Chupin A., Santambrogio F. (2010) A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci. 20, 10, 1787–1821. ISSN 0218-2025. doi: 10.1142/S0218202510004799. URL http://dx.doi.org/10.1142/S0218202510004799. [Google Scholar]
  • Kumar K., Pop I.S., Radu F.A. (2013) Convergence analysis of mixed numerical schemes for reactive flow in a porous medium, SIAM J. Numer. Anal. 51, 4, 2283–2308. [Google Scholar]
  • Zarba R.L., Bouloutas E.T., Celia M. (1990) General massconservative numerical solution for the unsaturated flow equation, Water Resour. Res. 26, 7, 1483–1496. [Google Scholar]
  • Jäger W., Kacur J. (1991) Solution of porous medium type systems by linear approximation schemes, Numer. Math. 60, 3, 407–427. [Google Scholar]
  • Jäger W., Kacur J. (1995) Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, RAIRO Modél. Math. Anal. Numér 29, 5, 605–627. [Google Scholar]
  • Pop I.S., Radu F.A., Knabner P. (2004) Mixed finite elements for the Richards equation: linearization procedure, J. Comput. Appl. Math. 168, 1, 365–373. [Google Scholar]
  • Radu F.A., Nordbotten J.M., Pop I.S., Kumar K. (2015) A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media, J. Comput. Appl. Math. 289, 134–141, ISSN 0377-0427. URL https://doi.org/10.1016/j.cam.2015.02.051. [Google Scholar]
  • Radu F.A., Kumar K., Nordbotten J.M., Pop I.S. (2018) A robust, mass conservative scheme for two-phase flow in porous media including hlder continuous nonlinearities, IMA J. Numer. Anal. 38, 2, 88420. doi: 10.1093/imanum/drx032. URL http://dx.doi.org/10.1093/imanum/drx032 [Google Scholar]
  • Casulli V., Zanolli P. (2010) A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form, SIAM J. Sci. Comp. 32, 4, 2255–2273. doi: 10.1137/100786320. URL https://doi.org/10.1137/100786320 . [CrossRef] [Google Scholar]
  • Younis R., Tchelepi H.A., Aziz K. (2010) Adaptively localized continuation-Newton method-nonlinear solvers that converge all the time, SPE J. 15, 2, 526–544. [CrossRef] [Google Scholar]
  • Wang X., Tchelepi H.A. (2013) Trust-region based solver for nonlinear transport in heterogeneous porous media, J. Comput. Phys. 253, 114–137. [Google Scholar]
  • Lehmann F., Ackerer P.H. (1998) Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media, Transp. Porous Media. 31, 3, 275–292. [Google Scholar]
  • Bergamaschi L., Putti M. (1999) Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation, Int. J. Numer. Meth. Eng. 45, 8, 1025–1046. [CrossRef] [Google Scholar]
  • Radu F.A., Pop I.S., Knabner P. (2006) Newton-type methods for the mixed finite element discretization of some degenerate parabolic equations. Numerical mathematics and advanced applications, Springer. [Google Scholar]
  • List F., Radu F.A. (2016) A study on iterative methods for solving Richards’ equation, Comput. Geosci. 1–13. [Google Scholar]
  • Marchand E., Müller T., Knabner P. (2012) Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results, Comput. Geosci. 16, 3, 691–708. doi: 10.1007/s10596-012-9279-1. URL https://doi.org/10.1007/s10596-012-9279-1. [Google Scholar]
  • Marchand E., Müller T., Knabner P. (2013) Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part I: Formulation and properties of the mathematical model, Comput. Geosci. 17, 2, 431–442, ISSN 1573-1499. doi: 10.1007/s10596-013-9341-7. URL https://doi.org/10.1007/s10596-013-9341-7. [Google Scholar]
  • Ben Gharbia I. (2012) Résolution de problèmes de complémentarité : application à un écoulement diphasique dans un milieu poreux, Thesis, Université Paris Dauphine - Paris IX, December 2012. URL https://tel.archives-ouvertes.fr/tel-00776617 [Google Scholar]
  • Diersch H.-J.G., Perrochet P. (1999) On the primary variable switching technique for simulating unsaturated-saturated flows, Adv. Water Resour. 23, 3, 271–301. [Google Scholar]
  • Brenner K., Cancès C. (2017) Improving Newton’s method performance by parametrization: The case of the Richards equation, SIAM J. Numer. Anal. 55, 4, 1760–1785. doi: 10.1137/16M1083414. URL https://doi.org/10.1137/16M1083414 [Google Scholar]
  • Brenner K., Groza M., Jeannin L., Masson R., Pellerin J. (2017) Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media, Comput. Geosci. 21, 5–6, 1075–1094. [Google Scholar]
  • Ciarlet P.G. (1978) The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, ISBN 0-444-85028-7. Studies in Mathematics and its Applications, Vol. 4. [Google Scholar]
  • Ern A., Guermond J.L. (2004) Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Series, Springer, New York. [CrossRef] [Google Scholar]
  • Franco Brezzi and Michel Fortin (1991) Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York. ISBN 0-387-97582-9 [Google Scholar]
  • Arbogast T., Wheeler M.F., Yotov I. (1997) Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34, 2, 828–852. doi: 10.1137/S0036142994262585. URL https://doi.org/10.1137/S0036142994262585. [Google Scholar]
  • Aavatsmark I., Barkve T., Bøe Ø., Mannseth T. (1998) Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods, SIAM J. Sci. Comput. 19, 5, 1700–1716. doi: 10.1137/S1064827595293582. URL http://dx.doi.org/10.1137/S1064827595293582 . [Google Scholar]
  • Edwards M.G., Rogers C.F. (1998) Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geosci. 2, 4, 259–290. doi: 10.1023/A:1011510505406. URL http://dx.doi.org/10.1023/A:1011510505406. [Google Scholar]
  • Edwards M.G. (2002) Unstructured, control-volume distributed, full- tensor finite-volume schemes with flow based grids, Comput. Geosci. 6, 3–4, 433–452, ISSN 1420-0597. doi: 10.1023/A:1021243231313. URL https://doi.org/10.1023/A:1021243231313 . [Google Scholar]
  • Agelas L., Guichard C., Masson R. (2010) Convergence of finite volume MPFA O type schemes for heterogeneous anisotropic diffusion problems on general meshes, Int. J. Finite 7, 2, 33. [Google Scholar]
  • Arnold D., Brezzi F., Cockburn B., Marini L. (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39, 5, 1749–1779. doi: 10.1137/S0036142901384162. URL https://doi.org/10.1137/S0036142901384162. [Google Scholar]
  • Rivière B. (2008) Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM. doi: 10.1137/1.9780898717440. URL https://epubs.siam.org/doi/abs/10.1137/1.9780898717440 . [Google Scholar]
  • Di Pietro D.A., Ern A. (2012) Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, Heidelberg, ISBN 978-3-642-22979-4. doi: 10.1007/978-3-642-22980-0. URL http://dx.doi.org/10.1007/978-3-642-22980-0. [Google Scholar]
  • Herbin R. (1995) An error estimate for a finite volume scheme for a diffusiononvection problem on a triangular mesh, Numer. Methods Partial Differ. Equ. 11, 2, 165–173. doi: 10.1002/num.1690110205. URL https://doi.org/10.1002/num.1690110205. [Google Scholar]
  • Eymard R., Gallouët T., Herbin R. (2000) Finite volume methods, in: Ciarlet P.G., et al. (eds), Handbook of numerical analysis, North-Holland: Amsterdam, p. 713 1020. [Google Scholar]
  • Eymard R., Gallouët T., Guichard C., Herbin R., Masson R. (2014) TP or not TP, that is the question, Comput. Geosci. 18, 285–296. [Google Scholar]
  • Hackbusch W. (1989) On first and second order box schemes, Computing 41, 4, 277–296, ISSN 0010-485X. doi: 10.1007/BF02241218. URL https://doi.org/10.1007/BF02241218 . [CrossRef] [MathSciNet] [Google Scholar]
  • Droniou J., Eymard R., Gallouët T., Herbin R. (2010) A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci. 20, 2, 265–295. [Google Scholar]
  • Eymard R., Gallouët T., Herbin R. (2010) Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30, 4, 1009–1043. [CrossRef] [MathSciNet] [Google Scholar]
  • Droniou J., Eymard R. (2006) A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math. 105, 35–71. [Google Scholar]
  • Brezzi F., Lipnikov K., Simoncini V. (2005) A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci. 15, 10, 1533–1551. [Google Scholar]
  • Brezzi F., Lipnikov K., Shashkov M. (2005) Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43, 5, 1872–1896. [Google Scholar]
  • Domelevo K., Omnes P. (2005) A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, M2AN: Math. Model. Numer. Anal. 39, 6, 1203–1249. [Google Scholar]
  • Droniou J. (2014) Finite volume schemes for diffusion equations: introduction to and review of modern methods, Math. Models Methods Appl. Sci. 24, 8, 1575–1620. [Google Scholar]
  • Droniou J., Eymard R., Gallouët T., Guichard C., Herbin R. (2018) The gradient discretisation method, Vol. 42, Mathématiques et Applications, Springer International Publishing, https://doi.org/10.1007/978-3-319-79042-8. [Google Scholar]
  • Eymard R., Guichard C., Herbin R. (2012) Small-stencil 3D schemes for diffusive flows in porous media, ESAIM: Math. Model. Numer. Anal. 46, 2, 265–290. doi: 10.1051/m2an/2011040. URL http://dx.doi.org/10.1051/m2an/2011040 . [Google Scholar]
  • Eymard R., Guichard C., Herbin R. (2011) Benchmark 3D: the VAG scheme, in: Fort J., Fürst J., Halama J., Herbin R., Hubert F. (eds), Finite Volumes for Complex Applications VI Problems & Perspectives, volume 4 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, pp. 1013–1022. ISBN 978-3-642-20670-2. doi: 10.1007/978-3-642-20671-9_99. URL http://dx.doi.org/10.1007/978-3-642-20671-9_99 . [Google Scholar]
  • Cancès C., Guichard C. (2017) Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure, Found. Comput. Math. 17, 6, 1525–1584. doi: 10.1007/s10208-016-9328-6. [CrossRef] [Google Scholar]
  • Cancès C., Chainais-Hillairet C., Krell S. (2017) A nonlinear Discrete Duality Finite Volume Scheme for convection- diffusion equations, in: Cancès C., Omnes P. (eds), FVCA8 2017 – International Conference on Finite Volumes for Complex Applications VIII, volume 199 of Springer Proceedings in Mathematics & Statistics, Lille, France, Springer International Publishing, pp. 439–447. URL https://hal.archives-ouvertes.fr/hal-01468811. [Google Scholar]
  • Cancès C., Chainais-Hillairet C., Krell S. (2017) Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations, Comput Methods Appl. Math. doi: 10.1515/cmam-2017-0043. URL https://hal.archives-ouvertes.fr/hal-01529143. Special issue on “Advanced numerical methods: recent developments, analysis and application”. [Google Scholar]
  • Cancès C., Guichard C. (2016) Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations, Math. Comp. 85, 298, 549–580. [CrossRef] [Google Scholar]
  • Chavent G., Jaffré J. (1986), Mathematical Models and Finite Elements for Reservoir Simulation, Vol. 17, Stud. Math. Appl. edition, North-Holland, Amsterdam. [Google Scholar]
  • Antontsev S.N., Kazhikhov A.V., Monakhov V.N. (1990) Boundary value problems in mechanics of nonhomogeneous fluids, vol. 22 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, ISBN 0-444-88382-7. Translated from the Russian. [Google Scholar]
  • Gagneux G., Madaune-Tort M. (1996) Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, vol. 22 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, ISBN 3-540-60588-6. [Google Scholar]
  • Chen Z. (2001) Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution, J. Diff. Equ. 171, 2, 203–232. [CrossRef] [MathSciNet] [Google Scholar]
  • Nochetto R.H., Verdi C. (1988) Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25, 4, 784–814. doi: 10.1137/0725046. URL https://doi.org/10.1137/0725046. [Google Scholar]
  • Arbogast T., Wheeler M.F., Zhang N.-Y. (1996) A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal. 33, 4, 1669–1687. doi: 10.1137/S0036142994266728. URL http://dx.doi.org/10.1137/S0036142994266728. [Google Scholar]
  • Eymard R., Gallouët T., Hilhorst D., Naït Slimane Y. (1998) Finite volumes and nonlinear diffusion equations, RAIRO Modél. Math. Anal. Numér. 32, 6, 747–761. [Google Scholar]
  • Eymard R., Gutnic M., Hilhorst D. (1999) The finite volume method for Richards equation, Comput. Geosci. 3, 3–4, 259–294. doi: 10.1023/A:1011547513583. URL http://dx.doi.org/10.1023/A%3A1011547513583. [Google Scholar]
  • Woodward C.S., Dawson C.N. (2000) Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J. Numer. Anal. 37, 3, 701–724. doi: 10.1137/S0036142996311040. URL https://doi.org/10.1137/S0036142996311040. [Google Scholar]
  • Eymard R., Gallouët T., Herbin R., Michel A. (2002) Convergence of finite volume schemes for parabolic degenerate equations, Numer. Math. 92, 41–82. [Google Scholar]
  • Pop I.S. (2002) Error estimates for a time discretization method for the Richards’ equation, Comput. Geosci. 6, 141–160. [Google Scholar]
  • Radu F.A., Pop I.S., Knabner P. (2004) Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal. 42, 4, 1452–1478. [Google Scholar]
  • Eymard R., Hilhorst D., Vohralík M. (2006) A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numer. Math. 105, 1, 73–131. doi: 10.1007/s00211-006-0036-z. URL http://dx.doi.org/10.1007/s00211-006-0036-z [Google Scholar]
  • Radu F.A., Pop I.S., Knabner P. (2008) Error estimates for a mixed finite element discretization of some degenerate parabolic equations, Numer. Math. 109, 2, 285–311. doi: 10.1007/s00211-008-0139-9. URL http://dx.doi.org/10.1007/s00211-008-0139-9. [Google Scholar]
  • Angelini O., Brenner K., Hilhorst D. (2013) A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation, Numer. Math. 123, 219–257, ISSN 0029-599X. doi: 10.1007/s00211-012-0485-5. URL http://dx.doi.org/10.1007/s00211-012-0485-5. [Google Scholar]
  • Chen Z., Ewing R.E. (1997) Fully discrete finite element analysis of multiphase flow in groundwater hydrology, SIAM J. Numer. Anal. 34, 6, 2228–2253. [Google Scholar]
  • Chen Z., Ewing R.E. (2001) Degenerate two-phase incompressible flow. III. Sharp error estimates, Numer. Math. 90, 2, 215–240, ISSN 0029-599X. doi: 10.1007/s002110100291. URL http://dx.doi.org/10.1007/s002110100291. [Google Scholar]
  • Michel A. (2003) A finite volume scheme for two-phase immiscible flow in porous media, SIAM J. Numer. Anal. 41, 4, 1301–1317. [Google Scholar]
  • Epshteyn Y., Rivière B. (2009) Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow, J. Comput. Appl. Math. 225, 2, 487–509. doi: 10.1016/j.cam.2008.08.026. URL https://doi.org/10.1016/j.cam.2008.08.026. [Google Scholar]
  • Brenner K., Masson R. (2013) Convergence of a vertex centered discretization of two-phase Darcy flows on general meshes, Int. J. Finite 10, 1–37. [Google Scholar]
  • Cancès C., Pop I.S., Vohralík M. (2014) An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow, Math. Comp. 83, 285, 153–188. doi: 10.1090/S0025-5718-2013-02723-8. URL http://dx.doi.org/10.1090/S0025-5718-2013-02723-8. [CrossRef] [MathSciNet] [Google Scholar]
  • Cancès C., Nabet F., Vohralik M. (2018) Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations, in preparation. [Google Scholar]
  • Forsyth P.A. (1991) A control volume finite element approach to NAPL groundwater contamination, SIAM J. Sci. Statist. Comput. 12, 5, 1029–1057. [CrossRef] [MathSciNet] [Google Scholar]
  • Ait Hammou Oulhaj A., Cancès C., Chainais-Hillairet C. (2018) Numerical analysis of a nonlinearly stable and positive Control Volume Finite Element scheme for Richards equation with anisotropy, ESAIM Math. Model. Numer. Anal. 52, 1532—1567. [Google Scholar]
  • Ait Hammou Oulhaj A. (2018) Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer, Numer. Methods Partial Differ. Equ. 34, 3, 857–880. doi: 10.1002/num.22234. URL https://doi.org/10.1002/num.22234. [Google Scholar]
  • Cancès C., Nabet F. (2017) Finite volume approximation of a degenerate immiscible two-phase flow model of Cahn-Hilliard type, in: Cancès C., Omnes P. (eds), Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects : FVCA 8, Lille, France, June 2017, number 199 in Proceedings in Mathematics and Statistics, Cham, Springer International Publishing, pp. 431–438, ISBN 978-3-319-57397-7. doi: 10.1007/978-3-319-57397-7_36. http://dx.doi.org/10.1007/978-3-319-57397-7_36. [CrossRef] [Google Scholar]
  • Otto F., Weinan E. (1997) Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys. 107, 23, 10177–10184. [Google Scholar]
  • Cancès C., Matthes D., Nabet F. (2017) A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow, HAL, hal-01665338, December 2017. URL https://hal.archives-ouvertes.fr/hal-01665338. [Google Scholar]
  • Herbin R., Hubert F. (2008) Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in: Eymard R., Herard J.-M. (eds), Finite Volumes for Complex Applications V, Wiley, pp. 659–692. https://www.latp.univ-mrs.fr/fvca5/benchmark/ [Google Scholar]
  • Eymard R., Herbin R., Michel A. (2003) Mathematical study of a petroleum-engineering scheme, M2AN: Math. Model. Numer. Anal. 37, 6, 937–972. [CrossRef] [Google Scholar]
  • Chainais-Hillairet C., Filbet F. (2007) Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model, IMA J. Numer. Anal. 27, 4, 689–716. doi: 10.1093/imanum/drl045. URL http://dx.doi.org/10.1093/imanum/drl045. [CrossRef] [Google Scholar]
  • Bessemoulin-Chatard M., Chainais-Hillairet C. (2017) Exponential decay of a finite volume scheme to the thermal equilibrium for drift-diffusion systems, J. Numer. Math. 25, 3, 147—168. [CrossRef] [Google Scholar]
  • Filbet F., Herda M. (2017) A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure, Numer. Math. URL https://hal.archives-ouvertes.fr/hal-01326029. [Google Scholar]
  • Ganis B., Kumar K., Pencheva G., Wheeler M., Yotov I. (2014) A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model, Multiscale Model. Simul. 12, 4, 1401–1423. doi: 10.1137/140952922. URL https://doi.org/10.1137/140952922. [Google Scholar]

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