. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci, pp.405-432, 2002.

T. Aavatsmark, Ø. Barkve, T. Bøe, and . Mannseth, Discretization on Non-Orthogonal, Curvilinear Grids for Multi-Phase Flow, ECMOR IV, 4th European Conference on the Mathematics of Oil Recovery
DOI : 10.3997/2214-4609.201411179

G. T. Aavatsmark, B. T. Eigestad, J. M. Mallison, E. Nordbotten, and . Øian, A compact multipoint flux approximation method with improved robustness, Numerical Methods for Partial Differential Equations, vol.3, issue.5, 2007.
DOI : 10.1002/num.20320

D. A. Agélas, R. Di-pietro, and . Masson, A symmetric and coercive finite volume scheme for multiphase porous media flow with applications in the oil industry, Finite Volumes for Complex Applications V, pp.35-52, 2008.

D. A. Agélas, J. Di-pietro, and . Droniou, The G method for heterogeneous anisotropic diffusion on general meshes, ESAIM: Mathematical Modelling and Numerical Analysis, vol.44, issue.4, pp.597-625, 2010.
DOI : 10.1051/m2an/2010021

M. Ainsworth and J. Tinsley-oden, A posteriori error estimation in finite element analysis Wiley-Interscience, Pure and Applied Mathematics, 2000.

M. Andreianov, K. H. Bendahmane, and . Karlsen, DISCRETE DUALITY FINITE VOLUME SCHEMES FOR DOUBLY NONLINEAR DEGENERATE HYPERBOLIC-PARABOLIC EQUATIONS, Journal of Hyperbolic Differential Equations, vol.07, issue.01, p.67, 2010.
DOI : 10.1142/S0219891610002062

URL : https://hal.archives-ouvertes.fr/hal-00475752

N. Arnold, An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM Journal on Numerical Analysis, vol.19, issue.4, pp.742-760, 1982.
DOI : 10.1137/0719052

N. Arnold, An Interior Penalty Finite Element Method with Discontinuous Elements, SIAM Journal on Numerical Analysis, vol.19, issue.4, pp.742-760, 1982.
DOI : 10.1137/0719052

N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, SIAM Journal on Numerical Analysis, vol.39, issue.5, pp.1749-1779, 2002.
DOI : 10.1137/S0036142901384162

A. De, D. , and L. Zikatanov, Uniformly convergent iterative methods for discontinuous Galerkin discretizations, J

. Babu?ka, The finite element method with penalty, Mathematics of Computation, vol.27, issue.122, pp.221-228, 1973.
DOI : 10.1090/S0025-5718-1973-0351118-5

A. Baker, Finite element methods for elliptic equations using nonconforming elements, Mathematics of Computation, vol.31, issue.137, pp.3145-3194, 1977.
DOI : 10.1090/S0025-5718-1977-0431742-5

S. Bassi and . Rebay, A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier???Stokes Equations, Journal of Computational Physics, vol.131, issue.2, pp.267-279, 1997.
DOI : 10.1006/jcph.1996.5572

L. Bassi, A. Botti, D. A. Colombo, P. Di-pietro, and . Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, Journal of Computational Physics, vol.231, issue.1, pp.45-65, 2012.
DOI : 10.1016/j.jcp.2011.08.018

URL : https://hal.archives-ouvertes.fr/hal-00562219

C. Bastian, J. Engwer, O. Fahlke, and . Ippisch, An Unfitted Discontinuous Galerkin method for pore-scale simulations of solute transport, Mathematics and Computers in Simulation, vol.81, issue.10, pp.2051-2061, 2011.
DOI : 10.1016/j.matcom.2010.12.024

A. Bonelle and . Ern, Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes, ESAIM: Mathematical Modelling and Numerical Analysis, vol.48, issue.2, 2012.
DOI : 10.1051/m2an/2013104

URL : https://hal.archives-ouvertes.fr/hal-00751284

C. Brenner, Functions, SIAM Journal on Numerical Analysis, vol.41, issue.1, pp.306-324, 2003.
DOI : 10.1137/S0036142902401311

URL : https://hal.archives-ouvertes.fr/hal-01093487

K. Brezzi, M. Lipnikov, and . Shashkov, Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes, SIAM Journal on Numerical Analysis, vol.43, issue.5, pp.1872-1896, 2005.
DOI : 10.1137/040613950

K. Brezzi, V. Lipnikov, and . Simoncini, A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES, Mathematical Models and Methods in Applied Sciences, vol.15, issue.10
DOI : 10.1142/S0218202505000832

K. Brezzi, M. Lipnikov, and . Shashkov, CONVERGENCE OF MIMETIC FINITE DIFFERENCE METHOD FOR DIFFUSION PROBLEMS ON POLYHEDRAL MESHES WITH CURVED FACES, Mathematical Models and Methods in Applied Sciences, vol.16, issue.02, pp.275-298, 2006.
DOI : 10.1142/S0218202506001157

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, 1991.
DOI : 10.1007/978-1-4612-3172-1

P. Burman and . Zunino, A Domain Decomposition Method Based on Weighted Interior Penalties for Advection???Diffusion???Reaction Problems, SIAM Journal on Numerical Analysis, vol.44, issue.4, pp.1612-1638, 2006.
DOI : 10.1137/050634736

M. Cancès, C. L. Cathala, and . Potier, Monotone coercive cell-centered finite volume schemes for anisotropic diffusion equations, 2012.

C. Cancès, . Iuliu-sorin, M. Pop, and . Vohralík, An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow, Math. Comp, 2013.

G. Chainais-hillairet, M. Enchéry, and . Mamaghani, Development of a refinement criterion for adaptive mesh refinement in steam-assisted gravity drainage simulation, Comput. Geosci, vol.15, issue.1, pp.17-34, 2011.

H. Coats, L. K. Thomas, and R. G. Pierson, Compositional and black oil reservoir simulation, SPE Reservoir Evaluation and Engineering, vol.50990, 1998.
DOI : 10.2118/50990-pa

C. Cockburn and . Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp, vol.52, issue.186, pp.411-435, 1989.

C. Cockburn and . Shu, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, vol.25, issue.3, pp.337-361, 1991.
DOI : 10.1051/m2an/1991250303371

A. and D. Pietro, Cell centered Galerkin methods, Comptes Rendus Mathematique, vol.348, issue.1-2, pp.31-34111, 2010.
DOI : 10.1016/j.crma.2009.11.012

URL : https://hal.archives-ouvertes.fr/hal-00398782

A. and D. Pietro, On the conservativity of cell centered Galerkin methods. HAL preprint hal-00781510, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00781510

A. , D. Pietro, and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier?Stokes equations, Math. Comp, vol.79, pp.1303-1330, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00278925

A. , D. Pietro, and A. Ern, Mathematical aspects of discontinuous Galerkin methods, of Mathématiques & Applications, 2011.

A. , D. Pietro, and J. Gratien, Lowest order methods for diffusive problems on general meshes: a unified approach to definition and implementation, Finite Volumes for Complex Applications VI Problems & Perspectives, pp.3-19, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00562500

A. , D. Pietro, and S. Lemaire, An extension of the Crouzeix? Raviart space to general meshes with application to quasiincompressible linear elasticity and Stokes flow, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00753660

A. D. Pietro, A. Ern, and J. Guermond, Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection, SIAM Journal on Numerical Analysis, vol.46, issue.2, pp.805-831, 2008.
DOI : 10.1137/060676106

A. Di-pietro, J. Gratien, and C. , Prud'homme. A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes, BIT Numerical Mathematics, 2012.

A. D. Pietro, M. Vohralík, and S. Yousef, Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem, Mathematics of Computation, vol.84, issue.291, 2012.
DOI : 10.1090/S0025-5718-2014-02854-8

URL : https://hal.archives-ouvertes.fr/hal-00690862

D. A. Pietro, M. Vohralík, and S. Yousef, A posteriori error estimates, stopping criteria, and adaptivity for compositional multi-phase flows, 2013.

R. Droniou and . Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numerische Mathematik, vol.59, issue.1, pp.35-71, 2006.
DOI : 10.1007/s00211-006-0034-1

URL : https://hal.archives-ouvertes.fr/hal-00005565

R. Droniou, T. Eymard, R. Gallouët, and . Herbin, A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS, Mathematical Models and Methods in Applied Sciences, vol.20, issue.02, pp.265-295, 2010.
DOI : 10.1142/S0218202510004222

URL : https://hal.archives-ouvertes.fr/hal-00346077

R. Droniou, T. Eymard, R. Gallouët, and . Herbin, GRADIENT SCHEMES: A GENERIC FRAMEWORK FOR THE DISCRETISATION OF LINEAR, NONLINEAR AND NONLOCAL ELLIPTIC AND PARABOLIC EQUATIONS, Mathematical Models and Methods in Applied Sciences, vol.23, issue.13, 2012.
DOI : 10.1142/S0218202513500358

URL : https://hal.archives-ouvertes.fr/hal-00751551

G. Edwards and C. F. Rogers, A Flux Continuous Scheme for the Full Tensor Pressure Equation, ECMOR IV, 4th European Conference on the Mathematics of Oil Recovery, 1994.
DOI : 10.3997/2214-4609.201411178

L. El-alaoui, A. Ern, and M. Vohralík, Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.37-40, pp.37-40, 2011.
DOI : 10.1016/j.cma.2010.03.024

URL : https://hal.archives-ouvertes.fr/hal-00410471

J. Ern and . Guermond, Discontinuous Galerkin Methods for Friedrichs' Systems. I. General theory, SIAM Journal on Numerical Analysis, vol.44, issue.2, pp.753-778, 2006.
DOI : 10.1137/050624133

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.490.7809

J. Ern and . Guermond, Discontinuous Galerkin Methods for Friedrichs??? Systems. Part II. Second???order Elliptic PDEs, SIAM Journal on Numerical Analysis, vol.44, issue.6, pp.2363-2388, 2006.
DOI : 10.1137/05063831X

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.490.7809

J. Ern and . Guermond, Discontinuous Galerkin Methods for Friedrichs' Systems. Part III. Multifield Theories with Partial Coercivity, SIAM Journal on Numerical Analysis, vol.46, issue.2, pp.776-804, 2008.
DOI : 10.1137/060664045

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.490.7809

A. Ern and M. Vohralík, A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation, SIAM Journal on Numerical Analysis, vol.48, issue.1, pp.198-223, 2010.
DOI : 10.1137/090759008

URL : https://hal.archives-ouvertes.fr/hal-00383692

A. Ern and M. Vohralík, A Unified Framework for a posteriori Error Estimation in Elliptic and Parabolic Problems with Application to Finite Volumes, Finite Volumes for Complex Applications VI, pp.821-837, 2011.
DOI : 10.1007/978-3-642-20671-9_85

A. Ern and M. Vohralík, Adaptive Inexact Newton Methods with A Posteriori Stopping Criteria for Nonlinear Diffusion PDEs, SIAM Journal on Scientific Computing, vol.35, issue.4, 2012.
DOI : 10.1137/120896918

URL : https://hal.archives-ouvertes.fr/hal-00681422

T. Eymard, R. Gallouët, and . Herbin, A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis, Comptes Rendus Mathematique, vol.344, issue.6, pp.403-406, 2007.
DOI : 10.1016/j.crma.2007.01.024

T. Eymard, R. Gallouët, and . Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, vol.30, issue.4, pp.1009-1043, 2010.
DOI : 10.1093/imanum/drn084

D. Falgout and U. M. Yang, hypre: A Library of High Performance Preconditioners, Preconditioners, pp.632-641, 2002.
DOI : 10.1007/3-540-47789-6_66

M. Flemisch, K. Darcis, B. Erbertseder, A. Faigle, K. Lauser et al., DuMux: DUNE for multi-{phase,component,scale,physics,???} flow and transport in porous media, Advances in Water Resources, vol.34, issue.9, pp.1102-1112, 2011.
DOI : 10.1016/j.advwatres.2011.03.007

A. Gallouët, J. C. Larcher, and . Latché, Convergence of a finite volume scheme for the convection-diffusion equation with $\mathrm{L}^{1}$ data, Mathematics of Computation, vol.81, issue.279, pp.1429-1454, 2012.
DOI : 10.1090/S0025-5718-2011-02571-8

B. Grospellier and . Lelandais, The Arcane development framework, Proceedings of the 8th workshop on Parallel/High-Performance Object-Oriented Scientific Computing, POOSC '09, pp.1-4, 2009.
DOI : 10.1145/1595655.1595659

A. Hannukainen, R. Stenberg, and M. Vohralík, A unified framework for a posteriori error estimation for the Stokes problem, Numerische Mathematik, vol.46, issue.272, pp.725-769, 2012.
DOI : 10.1007/s00211-012-0472-x

URL : https://hal.archives-ouvertes.fr/hal-00470131

R. Havé, F. Masson, M. Nataf, T. Szydlarski, and . Zhao, Algebraic Domain Decomposition Methods for Highly Heterogeneous Problems, SIAM Journal on Scientific Computing, vol.35, issue.3, 2011.
DOI : 10.1137/110842648

D. Hilhorst and M. Vohralík, A posteriori error estimates for combined finite volume???finite element discretizations of reactive transport equations on nonmatching grids, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.5-8, pp.5-8597, 2011.
DOI : 10.1016/j.cma.2010.08.017

URL : https://hal.archives-ouvertes.fr/hal-00461154

C. Houston, E. Schwab, and . Süli, -Finite Element Methods for Advection-Diffusion-Reaction Problems, SIAM Journal on Numerical Analysis, vol.39, issue.6
DOI : 10.1137/S0036142900374111

URL : https://hal.archives-ouvertes.fr/hal-00882219

P. Jiránek, Z. Strako?, and M. Vohralík, A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers, SIAM Journal on Scientific Computing, vol.32, issue.3, pp.1567-1590, 2010.
DOI : 10.1137/08073706X

D. A. Pietro and M. Vohralík, A review of discretization methods, a posteriori analysis, and adaptive algorithms 25

. Ladevèze, Comparaison de modèles de milieux continus, 1975.

. Lesaint, Finite element methods for symmetric hyperbolic equations, Numerische Mathematik, vol.13, issue.3, pp.244-25574, 1973.
DOI : 10.1007/BF01436628

. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d'éléments finis, 1975.

P. Lesaint and . Raviart, On a finite element method for solving the neutron transport equation In Mathematical Aspects of Finite Elements in Partial Differential Equations, Math. Res. Center, pp.89-123, 1974.

. Michel, Convergence de schémas volumes finis pour des problèmes de convection diffusion non linéaires, 2001.

S. Neittaanmäki and . Repin, Reliable methods for computer simulation Error control and a posteriori estimates, 70 J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem, 2004.

. Univ and . Hamburg, Collection of articles dedicated to Lothar Collatz on his sixtieth birthday, pp.9-15, 1971.

. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions In The mathematical foundations of the finite element method with applications to partial differential equations, Proc. Sympos, pp.603-627, 1972.

W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quarterly of Applied Mathematics, vol.5, issue.3, pp.241-269, 1947.
DOI : 10.1090/qam/25902

V. Prud-'homme, V. Chabannes, M. Doyeux, A. Ismail, G. Samake et al., Feel++: a computational framework for Galerkin methods and advanced numerical methods
URL : https://hal.archives-ouvertes.fr/hal-00662868

P. Raviart and J. Thomas, A mixed finite element method for 2-nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), pp.292-315, 1975.
DOI : 10.1007/BF01436186

H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, 1973.

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

P. S. Scheichl, L. T. Vassilevski, and . Zikatanov, Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids, SIAM Journal on Numerical Analysis, vol.50, issue.3, pp.1675-1694, 2012.
DOI : 10.1137/100805248

A. Scovazzi, S. S. Gerstenberger, and . Collis, A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media, Journal of Computational Physics, vol.233, pp.373-399, 2013.
DOI : 10.1016/j.jcp.2012.09.003

C. Sun and M. F. Wheeler, L2(H1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. Sci. Comp, vol.205, pp.501-530, 2005.

C. Sun and M. F. Wheeler, Discontinuous Galerkin methods for coupled flow and reactive transport problems, Applied Numerical Mathematics, vol.52, issue.2-3, pp.2-3283, 2005.
DOI : 10.1016/j.apnum.2004.08.035

R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, 1996.

R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, Calcolo, vol.40, issue.3, pp.195-212, 2003.
DOI : 10.1007/s10092-003-0073-2

B. Vohralík and . Wohlmuth, MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS, Mathematical Models and Methods in Applied Sciences, vol.23, issue.05, 2013.
DOI : 10.1142/S0218202512500613

M. Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.2, pp.367-391, 2006.
DOI : 10.1051/m2an:2006013

M. Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods, Numerische Mathematik, vol.59, issue.3, pp.121-158, 2008.
DOI : 10.1007/s00211-008-0168-4

M. Vohralík, Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients, Journal of Scientific Computing, vol.24, issue.2, pp.397-438, 2011.
DOI : 10.1007/s10915-010-9410-1

M. Vohralík and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Computational Geosciences, vol.17, issue.5, 2011.
DOI : 10.1007/s10596-013-9356-0

F. Wheeler, An Elliptic Collocation-Finite Element Method with Interior Penalties, SIAM Journal on Numerical Analysis, vol.15, issue.1, pp.152-161, 1978.
DOI : 10.1137/0715010

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.453.5927

A. Younès, R. Mosé, P. Ackerer, and G. Chavent, A New Formulation of the Mixed Finite Element Method for Solving Elliptic and Parabolic PDE with Triangular Elements, Journal of Computational Physics, vol.149, issue.1, pp.148-167, 1999.
DOI : 10.1006/jcph.1998.6150

C. Zienkiewicz and R. L. Taylor, The finite element methodThe Basis) The date of receipt and acceptance will be, 2000.