Y. Achdou, F. Hecht, and D. Pommier, A posteriori error estimates for parabolic variational inequalities, J. Sci. Comput, vol.37, pp.336-366, 2008.
DOI : 10.1007/s10915-008-9215-7

M. Aganagi?, Newton's method for linear complementarity problems, Math. Programming, vol.28, pp.349-362, 1984.

M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics, 2000.
DOI : 10.1002/9781118032824

M. Ainsworth, J. T. Oden, and C. Lee, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differential Equations, vol.9, pp.23-33, 1993.

B. Amaziane, M. Jurak, and A. ?galji?-keko, Modeling compositional compressible two-phase flow in porous media by the concept of the global pressure, Comput. Geosci, vol.18, pp.297-309, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00868281

P. F. Antonietti, L. Beirão-da-veiga, C. Lovadina, and M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems, SIAM J. Numer. Anal, vol.51, pp.654-675, 2013.

M. Arioli, E. H. Georgoulis, and D. Loghin, Stopping criteria for adaptive finite element solvers, SIAM, J. Sci. Comput, vol.35, pp.1537-1559, 2013.

S. Auliac, Z. Belhachmi, F. B. Belgacem, and F. Hecht, Quadratic finite elements with non-matching grids for the unilateral boundary contact, ESAIM Math. Model. Numer. Anal, vol.47, pp.1185-1203, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00933248

I. Babu?ka and W. C. Rheinboldt, Reliable error estimation and mesh adaptation for the finite element method, in Computational methods in nonlinear mechanics, Proc. Second Internat. Conf., Univ. Texas, pp.67-108, 1979.

S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math, vol.99, pp.225-249, 2004.

P. Bastian, Numerical computation of multiphase flow in porous media. Habilitationsschrift, 1999.

M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, vol.22, pp.751-756, 2003.

R. Becker, C. Johnson, and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing, vol.55, pp.271-288, 1995.
DOI : 10.1007/bf02238483

Z. Belhachmi and F. B. Belgacem, Quadratic finite element approximation of the Signorini problem, Math. Comp, vol.72, pp.83-104, 2003.

F. B. Belgacem, C. Bernardi, A. Blouza, and M. Vohralík, A finite element discretization of the contact between two membranes, M2AN Math. Model. Numer. Anal, vol.43, pp.33-52, 2008.

F. B. Belgacem, C. Bernardi, A. Blouza, and M. Vohralík, On the unilateral contact between membranes. Part 1: Finite element discretization and mixed reformulation, Math. Model. Nat. Phenom, vol.4, pp.21-43, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00461144

F. B. Belgacem, C. Bernardi, A. Blouza, and M. Vohralík, On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments, IMA J. Numer. Anal, vol.32, pp.1147-1172, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00461144

I. Ben-gharbia, J. Dabaghi, V. Martin, and M. Vohralík, A posteriori error estimates and adaptive stopping criteria for a compositional two-phase flow with nonlinear complementarity constraints. HAL Preprint 01919067, submitted for publication, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01919067

I. Ben-gharbia and J. C. Gilbert, Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix, Math. Program, vol.134, pp.349-364, 2012.
URL : https://hal.archives-ouvertes.fr/inria-00442293

I. Ben-gharbia and J. C. Gilbert, An algorithmic characterization of P-matricity, SIAM J. Matrix Anal. Appl, vol.34, pp.904-916, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00713330

I. Ben-gharbia and J. C. Gilbert, An algorithmic characterization of P-matricity II: adjustments, refinements, and validation. HAL Preprint 01672197, submitted for publication, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01672197

I. Ben-gharbia and J. Jaffré, Gas phase appearance and disappearance as a problem with complementarity constraints, Math. Comput. Simulation, pp.28-36, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00641621

A. Bensoussan and J. Lions, Applications of variational inequalities in stochastic control, vol.12, 1982.

A. Bergam, C. Bernardi, and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp, vol.74, pp.1117-1138, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00020615

C. Bernardi, Y. Maday, and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, vol.45, 2004.

D. Boffi, F. Brezzi, and M. Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol.44, 2013.

J. F. Bonnans, J. C. Gilbert, C. Lemaréchal, and C. A. , Sagastizábal, Numerical optimization

A. Bourgeat, J. Mladen, and F. Smaï, Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository, Comput. Geosci, vol.13, pp.29-42, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00965384

D. Braess, A posteriori error estimators for obstacle problems-another look, Numer. Math, vol.101, pp.415-421, 2005.

D. Braess, V. Pillwein, and J. Schöberl, Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg, vol.198, pp.1189-1197, 2009.

D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp, vol.77, pp.651-672, 2008.

S. C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comp, vol.65, pp.897-921, 1996.

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, vol.15, 1994.

H. Brezis, Inéquations variationnelles paraboliques, vol.7, pp.1-10, 1971.

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, 2011.

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol.15, 1991.

F. Brezzi, W. W. Hager, and P. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math, vol.28, pp.431-443, 1977.

F. Brezzi, W. W. Hager, and P. Raviart, Error estimates for the finite element solution of variational inequalities. II. Mixed methods, Numer. Math, vol.31, issue.79, pp.1-16, 1978.

W. L. Briggs, A multigrid tutorial, Society for Industrial and Applied Mathematics (SIAM), 1987.
DOI : 10.1137/1.9780898719505

P. N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput, vol.11, pp.450-481, 1990.
DOI : 10.1137/0911026

M. Bürg and A. Schröder, A posteriori error control of hp-finite elements for variational inequalities of the first and second kind, Comput. Math. Appl, vol.70, pp.2783-2802, 2015.

C. Cancès, I. S. Pop, and M. Vohralík, An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow, Math. Comp, vol.83, pp.153-188, 2014.

C. Carstensen and S. A. Funken, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput, vol.21, pp.1465-1484, 1999.

C. Carstensen, D. Gallistl, and Y. Huang, Saturation and reliable hierarchical a posteriori Morley finite element error control, J. Comput. Math, vol.36, pp.833-844, 2018.
DOI : 10.4208/jcm.1705-m2016-0549

C. Carstensen, R. Lazarov, and S. Tomov, Explicit and averaging a posteriori error estimates for adaptive finite volume methods, SIAM J. Numer. Anal, vol.42, pp.2496-2521, 2005.
DOI : 10.1137/s0036142903425422

URL : http://www.cs.utk.edu/~tomov/apost_SINUM.pdf

J. Céa, Approximation variationnelle des problèmes aux limites, Ann. Inst. Fourier (Grenoble), vol.14, pp.345-444, 1964.

G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation, 1986.

Z. Chen, Finite element methods and their applications, Scientific Computation, 2005.

Z. Chen, Reservoir simulation, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), vol.77, 2007.

Z. Chen, G. Huan, and Y. Ma, Computational methods for multiphase flows in porous media, Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), vol.2, 2006.
DOI : 10.1137/1.9780898718942

Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math, vol.84, pp.527-548, 2000.
DOI : 10.1007/s002110050009

S. Chippada, C. N. Dawson, M. L. Martinez, and M. F. Wheeler, A Godunov-type finite volume method for the system of shallow water equations, Symposium on Advances in Computational Mechanics, vol.151, pp.105-129, 1997.

F. Chouly, M. Fabre, P. Hild, J. Pousin, and Y. Renard, Residualbased a posteriori error estimation for contact problems approximated by Nitsche's method, IMA J. Numer. Anal, vol.38, pp.921-954, 2018.
DOI : 10.1093/imanum/drx024

F. Chouly and P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal, vol.51, pp.1295-1307, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00717711

P. G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, vol.4, 1978.

F. H. Clarke, Optimization and nonsmooth analysis, vol.5, 1990.
DOI : 10.1137/1.9781611971309

H. Class, R. Helmig, and P. Bastian, Numerical simulation of nonisothermal multiphase multi-component processes in porous media. 1. An efficient solution technique, Adv. Water. Resour, vol.25, pp.533-550, 2002.

P. Coorevits, P. Hild, and J. Pelle, A posteriori error estimation for unilateral contact with matching and non-matching meshes, Comput. Methods Appl. Mech. Engrg, vol.186, p.105, 2000.

E. Creusé and S. Nicaise, A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods, J. Comput. Appl. Math, vol.234, pp.2903-2915, 2010.

J. Dabaghi, V. Martin, and M. Vohralík, Adaptive inexact semismooth Newton methods for the contact problem between two membranes. HAL Preprint 01666845, submitted for publication, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01666845

R. Dautray and J. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, vol.8, 1988.

T. De-luca, F. Facchinei, and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming, vol.75, pp.407-439, 1996.

R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal, vol.19, pp.400-408, 1982.
DOI : 10.1137/0719025

URL : http://dml.cz/bitstream/handle/10338.dmlcz/104564/AplMat_38-1993-6_2.pdf

P. Destuynder and B. Métivet, Explicit error bounds in a conforming finite element method, Math. Comp, vol.68, pp.1379-1396, 1999.
DOI : 10.1090/s0025-5718-99-01093-5

URL : https://www.ams.org/mcom/1999-68-228/S0025-5718-99-01093-5/S0025-5718-99-01093-5.pdf

P. Deuflhard, Newton methods for nonlinear problems, vol.35, 2011.
DOI : 10.1007/978-3-642-23899-4

D. A. Di-pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, vol.69, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01820185

D. A. Di-pietro, E. Flauraud, M. Vohralík, and S. Yousef, A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media, J. Comput. Phys, vol.276, pp.163-187, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00839487

D. A. Di-pietro, E. Flauraud, M. Vohralík, and S. Yousef, A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media, J. Comput. Phys, vol.276, pp.163-187, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00839487

D. A. Di-pietro, M. Vohralík, and S. Yousef, An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media, Comput. Math. Appl, vol.68, pp.2331-2347, 2014.

S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods, SIAM J. Optim, vol.4, pp.393-422, 1994.
DOI : 10.1137/0804022

URL : http://users.wpi.edu/~walker/Papers/global_inexact_newton,SIOPT_4,1994,393-422.pdf

S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, Special issue on iterative methods in numerical linear algebra, vol.17, pp.16-32, 1994.

L. E. Alaoui and A. Ern, Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods, M2AN Math. Model. Numer. Anal, vol.38, pp.903-929, 2004.

Y. Epshteyn and B. Rivière, Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow, J. Comput. Appl. Math, vol.225, pp.487-509, 2009.

B. Erdmann, M. Frei, R. H. Hoppe, R. Kornhuber, and U. Wiest, Adaptive finite element methods for variational inequalities, East-West J. Numer. Math, vol.1, pp.165-197, 1993.

A. Ern and J. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol.159, 2004.

A. Ern, I. Smears, and M. Vohralík, Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for highorder discretizations of parabolic problems, SIAM J. Numer. Anal, vol.55, pp.2811-2834, 2017.
DOI : 10.1137/16m1097626

URL : https://hal.archives-ouvertes.fr/hal-01377086

A. Ern, I. Smears, and M. Vohralík, Equilibrated flux a posteriori error estimates in L 2 (H 1 )-norms for high-order discretizations of parabolic problems, IMA Journal of Numerical Analysis, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01489721

A. Ern and M. Vohralík, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal, vol.48, pp.198-223, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00383692

A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput, vol.35, pp.1761-1791, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00681422

A. Ern and M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal, vol.53, pp.1058-1081, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00921583

L. C. Evans, Partial differential equations, vol.19, 1997.

R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Handbook of numerical analysis, vol.VII, pp.713-1020, 2000.
URL : https://hal.archives-ouvertes.fr/hal-02100732

R. Eymard, R. Herbin, and A. Michel, Mathematical study of a petroleum-engineering scheme, M2AN Math. Model. Numer. Anal, vol.37, pp.937-972, 2003.

F. Facchinei and C. Kanzow, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Math. Programming, vol.76, pp.493-512, 1997.

F. Facchinei, C. Kanzow, and S. Sagratella, Solving quasi-variational inequalities via their KKT conditions, Math. Program, vol.144, pp.369-412, 2014.

F. Facchinei and J. Pang, Finite-dimensional variational inequalities and complementarity problems, vol.I, 2003.

F. Facchinei and J. Pang, Finite-dimensional variational inequalities and complementarity problems, vol.II, 2003.

R. W. Falta, K. Pruess, I. Javandel, and P. Witherspoon, Numerical modeling of steam injection for the removal of nonaqeous phase liquids from the subsurface, Water. Resour. Res, vol.28, pp.433-449, 1992.

F. Fierro and A. Veeser, A posteriori error estimators, gradient recovery by averaging, and superconvergence, Numer. Math, vol.103, pp.267-298, 2006.

Z. Ge, Q. Ni, and X. Zhang, A smoothing inexact Newton method for variational inequalities with nonlinear constraints, J. Inequal. Appl, 2017.

R. Glowinski, Numerical methods for nonlinear variational problems, Scientific Computation, 2008.

R. Glowinski, J. Lions, and R. Trémolières, Théorie générale premiéres applications, Méthodes Mathématiques de l'Informatique, p.5, 1976.

R. Glowinski, J. Lions, and R. Trémolières, Applications aux phénomènes stationnaires et d'évolution, Méthodes Mathématiques de l'Informatique, p.5, 1976.

R. Glowinski, J. Lions, and R. Trémolières, Numerical analysis of variational inequalities, vol.8, 1981.

E. Godlewski and P. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol.118, 1996.

S. Gross and A. Reusken, Numerical methods for two-phase incompressible flows, Springer Series in Computational Mathematics, vol.40, 2011.

T. Gudi and K. , A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems, Math. Comp, vol.83, pp.579-602, 2014.

T. Gudi and K. , A remark on the a posteriori error analysis of discontinuous Galerkin methods for the obstacle problem, Comput. Methods Appl. Math, vol.14, pp.71-87, 2014.

T. Gudi and K. , A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem, J. Comput. Appl. Math, vol.292, pp.257-278, 2016.

W. Hackbusch, Multigrid methods and applications, vol.4, 1985.

R. Helmig, Multiphase flow and transport processes in the subsurface-A contribution to the modeling of hydrosystems, 1997.

M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim, vol.13, pp.865-888, 2002.

I. Hlavá?ek, J. Haslinger, J. Ne?as, and J. Loví?ek, Solution of variational inequalities in mechanics, Applied Mathematical Sciences, vol.66, 1988.

R. Huber and R. Helmig, Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media, Comput. Geosci, vol.4, pp.141-164, 2000.

J. Jaffré and A. Sboui, Henry' law and gas phase disappearance, Transport in porous media, vol.82, pp.521-526, 2010.

P. Jiránek, Z. Strako?, and M. Vohralík, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput, vol.32, pp.1567-1590, 2010.

C. Johnson, Numerical solution of partial differential equations by the finite element method, 1987.

L. V. Kantorovich, Functional analysis and applied mathematics, NBS Rep. 1509, 1952.

C. Kanzow, An active set-type Newton method for constrained nonlinear systems, Complementarity: applications, algorithms and extensions, vol.50, pp.179-200, 1999.
DOI : 10.1007/978-1-4757-3279-5_9

C. Kanzow, Inexact semismooth Newton methods for large-scale complementarity problems, The First International Conference on Optimization Methods and Software. Part II, vol.19, pp.309-325, 2004.
DOI : 10.1080/10556780310001636369

O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal, vol.41, pp.2374-2399, 2003.

C. T. Kelley, Iterative methods for linear and nonlinear equations, vol.16, 1995.
DOI : 10.1137/1.9781611970944

C. T. Kelley, Solving nonlinear equations with Newton's method, vol.1, 2003.

R. Kornhuber, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl, vol.31, pp.49-60, 1996.
DOI : 10.1016/0898-1221(96)00030-2

URL : https://doi.org/10.1016/0898-1221(96)00030-2

S. Lacroix, Y. Vassilevski, J. Wheeler, and M. Wheeler, Iterative solution methods for modeling multiphase flow in porous media fully implicitly, SIAM J. Sci. Comput, vol.25, pp.905-926, 2003.

P. Ladevèze, Comparaison de modèles de mécanique des milieux continus, 1975.

A. Lauser, C. Hager, R. Helmig, and B. Wohlmuth, A new approach for phase transitions in miscible multi-phase flow in porous media, Advances in Water Resources, vol.68, pp.957-966, 2011.

J. Liesen and Z. Strako?, Krylov subspace methods, Numerical Mathematics and Scientific Computation, 2013.

J. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969.

J. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math, vol.20, pp.493-519, 1967.

F. Louf, J. Combe, and J. Pelle, Constitutive error estimator for the control of contact problems involving friction, Comput. & Structures, vol.81, issue.03, pp.200-201, 2003.
URL : https://hal.archives-ouvertes.fr/hal-01689633

J. Loví?ek, Optimal control of a variational inequality with possibly nonsymmetric linear operator. Application to the obstacle problems in mathematical physics, Acta Math. Univ. Comenian. (N.S.), vol.63, pp.1-23, 1994.

A. Lozinski, M. Picasso, and V. Prachittham, An anisotropic error estimator for the Crank-Nicolson method: application to a parabolic problem, SIAM J. Sci. Comput, vol.31, pp.2757-2783, 2009.

R. Luce and B. I. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal, vol.42, pp.1394-1414, 2004.
DOI : 10.1137/s0036142903433790

URL : https://hal.archives-ouvertes.fr/inria-00343040

J. M. Martínez and L. Q. Qi, Linear/nonlinear iterative methods and verification of solution, J. Comput. Appl. Math, vol.60, pp.127-145, 1993.

D. Meidner, R. Rannacher, and J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations, J. Numer. Math, vol.17, pp.143-172, 2009.

K. Moon, R. H. Nochetto, T. V. Petersdorff, and C. Zhang, A posteriori error analysis for parabolic variational inequalities, ESAIM: Mathematical Modelling and Numerical Analysis -Modélisation Mathéma-tique et Analyse Numérique, vol.41, pp.485-511, 2007.

S. Nicaise and N. Soualem, A posteriori error estimates for a nonconforming finite element discretization of the heat equation, M2AN Math. Model. Numer. Anal, vol.39, pp.319-348, 2005.

J. Niessner and R. Helmig, Multi-scale modeling of three-phase-threecomponent processes in heterogeneous porous media, Advances in Water Resources, vol.30, pp.2309-2325, 2007.

M. A. Olshanskii and E. E. Tyrtyshnikov, Iterative methods for linear systems, Society for Industrial and Applied Mathematics, 2014.

J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, vol.75, pp.658-660, 1968.
DOI : 10.2307/2313800

M. Panfilov and I. Panfilova, Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multicomponent case, Comput. Geosci, vol.18, pp.385-399, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01418288

M. Panfilov and M. Rasoulzadeh, Interfaces of phase transition and disappearance and method of negative saturation for compositional flow with diffusion and capillarity in porous media, Transp. Porous. Med, vol.83, pp.73-98, 2010.

J. Pape?, U. Rüde, M. Vohralík, and B. Wohlmuth, Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. HAL Preprint 01662944, submitted for publication, 2017.

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal, vol.5, pp.286-292, 1960.
DOI : 10.1007/bf00252910

W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math, vol.5, pp.241-269, 1947.

J. Puel, Inéquations variationnelles d'évolution paraboliques du 2ème ordre, Séminaire Equations aux dérivées partielles (Polytechnique), vol.8, pp.1-12, 1974.

A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol.23, 1994.

R. Rankin and B. Rivière, A high order method for solving the black-oil problem in porous media, Advances in Water Resources, vol.78, pp.126-144, 2015.

P. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche, vol.606, pp.292-315, 1975.

S. Repin, A posteriori estimates for partial differential equations, Radon Series on Computational and Applied Mathematics, vol.4, 2008.

S. I. Repin, Functional a posteriori estimates for elliptic variational inequalities, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), vol.348, pp.147-164, 2007.

B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations, vol.35, 2008.

J. E. Roberts and J. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, vol.II, pp.523-639, 1991.
URL : https://hal.archives-ouvertes.fr/inria-00075815

S. M. Robinson and C. To, Local structure of feasible sets in nonlinear programming. III. Stability and sensitivity, Math. Programming Stud. No, vol.30, p.143, 1987.

J. Rodrigues, Obstacle problems in mathematical physics, vol.134, p.114, 1987.

Y. Saad, Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, 2003.

Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput, vol.7, pp.856-869, 1986.

W. T. Sha, Novel porous media formulation for multiphase flow conservation equations, 2011.

G. Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, pp.547-583, 1970.

A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal, vol.39, pp.146-167, 2001.

R. Verfürth, A posteriori error estimates for nonlinear problems: L r (0, T ; W 1,? (?))-error estimates for finite element discretizations of parabolic equations, Numer. Methods Partial Differential Equations, vol.14, pp.487-518, 1998.

R. Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, 2013.

M. Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods, Numer. Math, vol.111, pp.121-158, 2008.

M. Vohralík, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, J. Sci. Comput, vol.46, pp.397-438, 2011.

M. Vohralík and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Comput. Geosci, vol.17, pp.789-812, 2013.

F. Wang, W. Han, and X. Cheng, Discontinuous Galerkin methods for solving elliptic variational inequalities, SIAM J. Numer. Anal, vol.48, pp.708-733, 2010.

S. J. Wright, Primal-dual interior-point methods, Society for Industrial and Applied Mathematics, 1997.

I. Yotov, A multilevel Newton-Krylov interface solver for multiphysics couplings of flow in porous media, Numer. Linear Algebra Appl, vol.8, pp.551-570, 2000.