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

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, issue.3-4, pp.297-309, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00868281

S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and its Applications, vol.22, 1990.

T. Arbogast, The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal, vol.19, issue.11, pp.1009-1031, 1992.

M. Arioli, E. H. Georgoulis, and D. Loghin, Stopping criteria for adaptive finite element solvers, SIAM J. Sci. Comput, vol.35, issue.3, pp.1537-1559, 2013.

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

M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen, vol.22, issue.4, pp.751-756, 2003.

B. Belgacem, F. Bernardi, C. Blouza, A. Vohralík, and M. , On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments, IMA J. Numer. Anal, vol.32, issue.3, pp.1147-1172, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00461144

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

,

I. Ben-gharbia and J. Gilbert, An algorithmic characterization of P-matricity, SIAM Journal on Matrix Analysis and Applications, vol.34, issue.3, pp.904-916, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00713330

I. Ben-gharbia and J. Gilbert, An algorithmic characterization of P-matricity II: adjustments, refinements, and validation, SIAM Journal on Matrix Analysis and Applications, vol.40, issue.2, pp.800-813, 2019.
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, vol.99, pp.28-36, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00641621

J. F. Bonnans, J. C. Gilbert, C. Lemaréchal, and C. A. Sagastizábal, Numerical optimization, second edn, 2006.

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, issue.5, pp.29-42, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00965384

D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp, vol.77, issue.262, pp.651-672, 2008.

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

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, issue.285, pp.153-188, 2014.

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

Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution, J. Differential Equations, vol.171, issue.2, pp.203-232, 2001.

Z. Chen, Reservoir simulation, CBMS-NSF Regional Conference Series in Applied Mathematics, vol.77, 2007.

Z. Chen, G. Huan, and Y. Ma, Computational methods for multiphase flows in porous media, Society for Industrial and Applied Mathematics (SIAM), vol.2, 2006.

Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems, Numer. Math, vol.84, issue.4, pp.527-548, 2000.

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, issue.1-2, pp.105-129, 1997.

F. H. Clarke, Optimization and nonsmooth analysis, Society for Industrial and Applied Mathematics, vol.5, 1990.

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

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

R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal, vol.19, issue.2, pp.400-408, 1982.

P. Destuynder and B. Métivet, Explicit error bounds in a conforming finite element method, Math. Comp, vol.68, issue.228, pp.1379-1396, 1999.

D. Pietro, D. A. Flauraud, E. Vohralík, M. Yousef, and S. , 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. Pietro, D. A. Vohralík, M. Yousef, and S. , 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, issue.2, pp.393-422, 1994.

A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput, vol.35, issue.4, 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, issue.2, pp.1058-1081, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00921583

R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Handb. Numer. Anal., VII. North-Holland, 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, issue.6, 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 and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, 2003.

F. Facchinei and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, 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, issue.1, pp.433-449, 1992.

Z. Ge, Q. Ni, and X. Zhang, A smoothing inexact Newton method for variational inequalities with nonlinear constraints, J. Inequal. Appl. pp. Paper No, vol.160, p.12, 2017.

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

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

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, issue.2, pp.141-164, 2000.

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, issue.3, pp.1567-1590, 2010.

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.

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, issue.6, pp.2374-2399, 2003.

C. T. Kelley, Iterative methods for linear and nonlinear equations, Society for Industrial and Applied Mathematics, vol.16, 1995.

R. Kornhuber, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl, vol.31, issue.8, pp.49-60, 1996.

D. Kroener and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, vol.55, issue.2, pp.90084-90090, 1984.

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, issue.3, 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. M. Martínez and L. Q. Qi, Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math, vol.60, issue.1-2, 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, issue.2, pp.143-172, 2009.

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

M. Panfilov and I. Panfilova, Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multi-component case, Comput. Geosci, vol.18, issue.5, 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, issue.5, 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, 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.

I. S. Pop, F. Radu, and P. Knabner, Mixed finite elements for the Richards' equation: linearization procedure, J. Comput. Appl. Math, vol.168, issue.1-2, pp.365-373, 2004.

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.

F. A. Radu, K. Kumar, J. M. Nordbotten, and I. S. Pop, A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities, IMA J. Numer. Anal, vol.38, issue.2, pp.884-920, 2018.

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems pp, Lecture Notes in Math, vol.606, pp.292-315, 1977.

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, p.305, 2007.

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

A. Sboui and J. Jaffré, Henry' law and gas phase disappearance, Transport in porous media, vol.82, pp.521-526, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00374073

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

M. Slodi?ka, A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media, SIAM J. Sci. Comput, vol.23, issue.5, pp.1593-1614, 2002.

A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal, vol.39, issue.1, pp.146-167, 2001.

M. Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods, Numer. Math, vol.111, issue.1, pp.121-158, 2008.

M. Vohralík and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for twophase flows, Comput. Geosci, vol.17, issue.5, pp.789-812, 2013.