J. E. Aarnes and Y. Efendiev, An Adaptive Multiscale Method for Simulation of Fluid Flow in Heterogeneous Porous Media, Multiscale Modeling & Simulation, vol.5, issue.3, pp.918-939, 2006.
DOI : 10.1137/050645117

B. Achchab, A. Agouzal, J. Baranger, and A. J. Maître, Estimateur d'erreur a posteriori hi???rarchique. Application aux ???l???ments finis mixtes, Numerische Mathematik, vol.80, issue.2, pp.159-179, 1998.
DOI : 10.1007/s002110050364

Y. Achdou, C. Bernardi, and A. F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy?s equations, Numerische Mathematik, vol.96, issue.1, pp.17-42, 2003.
DOI : 10.1007/s00211-002-0436-7

V. I. Agoshkov and V. I. Lebedev, Poincaré-Steklov operators and methods of partition of the domain in variational problems, in Computational processes and systems, Nauka, issue.2, pp.173-227, 1985.

E. Ahmed, J. Jaffré, and A. J. Roberts, A reduced fracture model for two-phase flow with different rock types, Mathematics and Computers in Simulation, vol.137, pp.49-70, 2017.
DOI : 10.1016/j.matcom.2016.10.005

URL : https://hal.archives-ouvertes.fr/hal-01249322

E. Ahmed, C. Japhet, and A. M. Kern, A finite volume Schwarz algorithm for two-phase immiscible flow with different rock types

M. Ainsworth, A Posteriori Error Estimation for Lowest Order Raviart???Thomas Mixed Finite Elements, SIAM Journal on Scientific Computing, vol.30, issue.1, pp.189-204, 2007.
DOI : 10.1137/06067331X

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

M. Ainsworth and R. Rankin, Fully Computable Error Bounds for Discontinuous Galerkin Finite Element Approximations on Meshes with an Arbitrary Number of Levels of Hanging Nodes, SIAM Journal on Numerical Analysis, vol.47, issue.6, pp.47-4112, 2010.
DOI : 10.1137/080725945

S. and A. Hassan, A posteriori error estimates and stopping criteria for solvers using domain decomposition method and with local time stepping, 2017.
URL : https://hal.archives-ouvertes.fr/tel-01672977

G. Allaire, Analyse numérique et optimisation, Editions Ecole Polytechnique, 2005.

A. Alonso, Error estimators for a mixed method, Numerische Mathematik, vol.74, issue.4, pp.385-395, 1996.
DOI : 10.1007/s002110050222

B. Andreianov, K. Brenner, and A. C. Cancès, Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.38, issue.2, pp.655-667, 2014.
DOI : 10.1137/S0036142999363668

URL : https://hal.archives-ouvertes.fr/hal-00744359

T. Arbogast-and-z and . Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp, vol.64, pp.943-972, 1995.

M. Arioli, D. Loghin, and A. A. Wathen, Stopping criteria for iterations in finite element methods, Numerische Mathematik, vol.15, issue.3, pp.381-410, 2005.
DOI : 10.1145/321406.321416

URL : ftp://ftp.numerical.rl.ac.uk/pub/reports/arlowaRAL0309.ps.gz

D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, vol.19, issue.1, pp.7-32, 1985.
DOI : 10.1007/BF01396493

URL : http://www.esaim-m2an.org/articles/m2an/pdf/1985/01/m2an1985190100071.pdf

K. Aziz and A. Settari, Petroleum Reservoir Simulation, 1979.

J. Bear, Dynamics of Fluids in Porous Media, Soil Science, vol.120, issue.2, 1988.
DOI : 10.1097/00010694-197508000-00022

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

D. Bennequin, M. J. Gander, and A. L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Mathematics of Computation, vol.78, issue.265, pp.185-223, 2009.
DOI : 10.1090/S0025-5718-08-02145-5

URL : https://hal.archives-ouvertes.fr/hal-00111643

C. Bernardi and F. Hecht, Error indicators for the mortar finite element discretization of the Laplace equation, Mathematics of Computation, vol.71, issue.240, pp.1371-1403, 2002.
DOI : 10.1090/S0025-5718-01-01401-6

C. Bernardi, T. C. Rebollo, F. Hecht, and A. Z. Mghazli, Mortar finite element discretization of a model coupling Darcy and Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.17, issue.3, pp.375-410, 2008.
DOI : 10.1142/S0218202507001899

URL : https://hal.archives-ouvertes.fr/hal-00139167

H. Berninger, S. Loisel, and A. O. Sander, The 2-Lagrange Multiplier Method Applied to Nonlinear Transmission Problems for the Richards Equation in Heterogeneous Soil with Cross Points, SIAM Journal on Scientific Computing, vol.36, issue.5, pp.2166-2198, 2014.
DOI : 10.1137/120901064

H. Berninger-and-o and . Sander, Substructuring of a Signorini-type problem and Robin???s method for the Richards equation in heterogeneous soil, Computing and Visualization in Science, vol.44, issue.2, pp.187-205, 2010.
DOI : 10.2136/sssaj1980.03615995004400050002x

P. M. Berthe, C. Japhet, and A. P. Omnes, Space-Time Domain Decomposition with Finite Volumes for Porous Media Applications, in Domain decomposition methods in science and engineering XXI, Lect. Notes Comput. Sci. Eng, vol.98, pp.483-490, 2014.

P. E. Bjørstad, J. Braekhus, and A. A. Hvidsten, Parallel substructuring algorithms in structural analysis, direct and iterative methods, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, pp.321-340, 1990.

E. Blayo, L. Debreu, and A. F. Lemarié, Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients . Part 1: the constant coefficients case, pp.40-170, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00661978

E. Blayo, L. Halpern, and A. C. Japhet, Optimized Schwarz Waveform Relaxation Algorithms with Nonconforming Time Discretization for Coupling Convection-diffusion Problems with Discontinuous Coefficients, Lect. Notes Comput. Sci. Eng, vol.55, pp.267-274, 2007.
DOI : 10.1007/978-3-540-34469-8_31

URL : https://hal.archives-ouvertes.fr/inria-00187555

J. Bourgat, R. Glowinski, P. Le-tallec, and A. M. Vidrascu, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain decomposition methods, pp.3-16, 1988.

D. Braess-and-j and . Schöberl, Equilibrated residual error estimator for edge elements, Mathematics of Computation, vol.77, issue.262, pp.651-672, 2008.
DOI : 10.1090/S0025-5718-07-02080-7

D. Braess-and-r and . Verfürth, A Posteriori Error Estimators for the Raviart???Thomas Element, SIAM Journal on Numerical Analysis, vol.33, issue.6, pp.2431-2444, 1996.
DOI : 10.1137/S0036142994264079

J. H. Bramble, J. E. Pasciak, and A. A. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math, Comp, pp.47-103, 1986.

J. H. Bramble and . Xu, A Local Post-Processing Technique for Improving the Accuracy in Mixed Finite-Element Approximations, SIAM Journal on Numerical Analysis, vol.26, issue.6, pp.1267-1275, 1989.
DOI : 10.1137/0726073

K. Brenner, C. Cancès, and A. D. Hilhorst, Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure, Computational Geosciences, vol.21, issue.4, pp.573-597, 2013.
DOI : 10.1007/BF00615335

URL : https://hal.archives-ouvertes.fr/hal-00675681

F. Brezzi-and-m and . Fortin, Mixed and hybrid finite element methods, of Springer Series in Computational Mathematics, 1991.
DOI : 10.1007/978-1-4612-3172-1

E. A. Burman and . Ern, Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations, Mathematics of Computation, vol.76, issue.259, pp.1119-1140, 2007.
DOI : 10.1090/S0025-5718-07-01951-5

URL : http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01951-5/S0025-5718-07-01951-5.pdf

F. Caetano, M. J. Gander, L. Halpern, and A. J. Szeftel, Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Networks and Heterogeneous Media, vol.5, issue.3, pp.487-505, 2010.
DOI : 10.3934/nhm.2010.5.487

URL : http://www.aimsciences.org/journals/doIpChk.jsp?paperID=5314&mode=full

C. Cancès, Nonlinear Parabolic Equations with Spatial Discontinuities, Nonlinear Differential Equations and Applications NoDEA, vol.15, issue.4-5, pp.427-456, 2008.
DOI : 10.1007/s00030-008-6030-7

C. Cancès, T. Gallouët, and A. A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound, pp.239-258, 2009.

C. Cancès, I. S. Pop, and A. M. Vohralík, An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow, Mathematics of Computation, vol.83, issue.285, pp.153-188, 2014.
DOI : 10.1090/S0025-5718-2013-02723-8

C. Carstensen, A posteriori error estimate for the mixed finite element method, Mathematics of Computation, vol.66, issue.218, pp.465-476, 1997.
DOI : 10.1090/S0025-5718-97-00837-5

G. J. Chavent and . Jaffré, Mathematical models and finite elements for reservoir simulation, North-Holland, Studies in Mathematics and Its Applications, 1986.

G. J. Chavent and . Roberts, A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems, Advances in Water Resources, vol.14, issue.6, pp.329-348, 1991.
DOI : 10.1016/0309-1708(91)90020-O

Y. Chen and W. Liu, A posteriori error estimates of mixed methods for miscible displacement problems, International Journal for Numerical Methods in Engineering, vol.93, issue.3, pp.331-343, 2008.
DOI : 10.1007/978-1-4612-3172-1

Z. Chen, Analysis of mixed methods using conforming and nonconforming finite element methods, ESAIM: Mathematical Modelling and Numerical Analysis, vol.27, issue.1, pp.9-34, 1993.
DOI : 10.1051/m2an/1993270100091

Z. Chen, G. Huan, and A. Y. Ma, Computational methods for multiphase flows in porous media, Society for Industrial and Applied Mathematics, 2006.
DOI : 10.1137/1.9780898718942

L. C. Cowsar, J. Mandel, and A. M. Wheeler, Balancing domain decomposition for mixed finite elements, Mathematics of Computation, vol.64, issue.211, pp.989-1015, 1995.
DOI : 10.1090/S0025-5718-1995-1297465-9

URL : ftp://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR93454.ps.gz

E. Creusé-and-s and . Nicaise, A posteriori error estimations of a coupled mixed and standard Galerkin method for second order operators, Journal of Computational and Applied Mathematics, vol.213, issue.1, pp.35-55, 2008.
DOI : 10.1016/j.cam.2006.12.027

F. Cuvelier, C. Japhet, and A. G. Scarella, Personal communication. [52] , An efficient way to assemble finite element matrices in vector languages, BIT, pp.56-833, 2016.

T. A. Davis, Direct methods for sparse linear systems of Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), vol.2, 2006.

Y. De-roeck, A. P. Le, and . Tallec, Analysis and test of a local domain-decomposition preconditioner, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, pp.112-128, 1990.

D. A. Di-pietro, E. Flauraud, M. Vohralík, and A. S. Yousef, A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media, Journal of Computational Physics, vol.276, pp.163-187, 2014.
DOI : 10.1016/j.jcp.2014.06.061

URL : https://hal.archives-ouvertes.fr/hal-00839487

D. A. Di-pietro, M. Vohralík, and A. 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, pp.153-186, 2015.
DOI : 10.1090/S0025-5718-2014-02854-8

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

V. Dolean, P. Jolivet, and A. F. Nataf, An introduction to domain decomposition methods, Algorithms, theory, and parallel implementation, 2015.
DOI : 10.1137/1.9781611974065

URL : https://hal.archives-ouvertes.fr/cel-01100932

V. Dolej?í, A. Ern, and A. M. , $hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems, SIAM Journal on Scientific Computing, vol.38, issue.5, pp.3220-3246, 2016.
DOI : 10.1137/15M1026687

J. J. Douglas, P. J. Paes-leme, J. E. Roberts, and A. J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numerische Mathematik, vol.35, issue.1, pp.65-95, 1993.
DOI : 10.1007/BF01385742

M. Dryja, Substructuring methods for parabolic problems, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp.264-271, 1991.

R. Durán and C. Padra, An error estimator for nonconforming approximations of a nonlinear problem, in Finite element methods, Lecture Notes in Pure and Appl. Math, vol.164, pp.201-205, 1993.

L. El, . A. Alaoui, and . 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.

G. Enchéry, R. Eymard, and A. A. Michel, Numerical Approximation of a Two-phase Flow Problem in a Porous Medium with Discontinuous Capillary Forces, SIAM Journal on Numerical Analysis, vol.43, issue.6, pp.2402-2422, 2006.
DOI : 10.1137/040602936

A. Ern, S. Nicaise, and A. M. Vohralík, An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems, C. R. Math. Acad. Sci. Paris, pp.345-709, 2007.

A. Ern, I. Smears, and A. M. Vohralík, Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems, SIAM Journal on Numerical Analysis, vol.55, issue.6, 2016.
DOI : 10.1137/16M1097626

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

A. Ern, A. F. Stephansen, and A. M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection???diffusion???reaction problems, Journal of Computational and Applied Mathematics, vol.234, issue.1, pp.114-130, 2010.
DOI : 10.1016/j.cam.2009.12.009

URL : https://hal.archives-ouvertes.fr/hal-00193540

A. Ern-and-m and . 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.

R. Eymard, T. Gallouët, and A. R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, pp.713-1020, 2000.

R. Eymard, T. Gallouët, and A. R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient, Applied Numerical Mathematics, vol.37, issue.1-2, pp.31-53, 2001.
DOI : 10.1016/S0168-9274(00)00024-6

M. J. Gander, Optimized Schwarz Methods, SIAM Journal on Numerical Analysis, vol.44, issue.2, pp.699-731, 2006.
DOI : 10.1137/S0036142903425409

URL : https://hal.archives-ouvertes.fr/hal-00107263

M. J. Gander-and-o and . Dubois, Optimized Schwarz methods for a diffusion problem with discontinuous coefficient, Numerical Algorithms, vol.13, issue.2, pp.109-144, 2015.
DOI : 10.1137/0913032

M. J. Gander-and-l and . Halpern, Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems, SIAM Journal on Numerical Analysis, vol.45, issue.2, pp.666-697, 2007.
DOI : 10.1137/050642137

M. J. Gander, L. Halpern, and A. M. Kern, A Schwarz Waveform Relaxation Method for Advection???Diffusion???Reaction Problems with Discontinuous Coefficients and Non-matching Grids, Lect. Notes Comput. Sci. Eng, vol.55, pp.283-290, 2007.
DOI : 10.1007/978-3-540-34469-8_33

URL : https://hal.archives-ouvertes.fr/hal-01111940

M. J. Gander, L. Halpern, and A. F. Nataf, Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Relaxation, Proceedings of the 11th International Conference on Domain Decomposition Methods, pp.27-36, 1999.
DOI : 10.1007/3-540-26825-1_23

URL : http://www.unige.ch/~gander/Preprints/ganderrohde.pdf

M. J. Gander and C. Japhet, Algorithm 932, ACM Transactions on Mathematical Software, vol.40, issue.1, pp.40-65, 2013.
DOI : 10.1145/2513109.2513115

URL : https://hal.archives-ouvertes.fr/hal-00933643

M. J. Gander, C. Japhet, Y. Maday, and A. F. Nataf, A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case, Lect. Notes Comput . Sci. Eng, vol.40, pp.259-266, 2005.
DOI : 10.1007/3-540-26825-1_24

URL : https://hal.archives-ouvertes.fr/hal-00112937

M. J. Gander and A. M. Stuart, Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation, SIAM Journal on Scientific Computing, vol.19, issue.6, pp.2014-2031, 1998.
DOI : 10.1137/S1064827596305337

M. J. Gander-and-x and . Tu, On the Origins of Iterative Substructuring Methods, Lect. Notes Comput. Sci. Eng, vol.98, pp.597-605, 2014.
DOI : 10.1007/978-3-319-05789-7_57

B. Ganis, K. Kumar, G. Pencheva, M. F. Wheeler, and A. I. Yotov, A Global Jacobian Method for Mortar Discretizations of a Fully Implicit Two-Phase Flow Model, Multiscale Modeling & Simulation, vol.12, issue.4, pp.1401-1423, 2014.
DOI : 10.1137/140952922

L. Gastaldi, A domain decomposition for the transport equation, Contemp. Math., Amer. Math. Soc, vol.157, pp.97-102, 1992.
DOI : 10.1090/conm/157/1410

S. Ghanemi, P. Joly, and A. F. Collino, Domain decomposition method for harmonic wave equations, Third international conference on mathematical and numerical aspect of wave propagation, pp.663-672, 1995.

E. Giladi and H. B. Keller, Space-time domain decomposition for parabolic problems, Numerische Mathematik, vol.93, issue.2, pp.279-313, 2002.
DOI : 10.1007/s002110100345

URL : ftp://softlib.rice.edu/pub/CRPC-TRs/reports/CRPC-TR97701.ps.gz

F. Haeberlein, Time Space Domain Decomposition Methods for Reactive Transport ? Application to CO2 Geological Storage, 2011.
DOI : 10.1016/j.procs.2010.04.081

URL : https://hal.archives-ouvertes.fr/tel-00634507

F. Haeberlein, L. Halpern, A. A. Michel, and N. -. , Newton-Schwarz Optimised Waveform Relaxation Krylov Accelerators for Nonlinear Reactive Transport, Lect. Notes Comput. Sci. Eng, vol.91, pp.387-394
DOI : 10.1007/978-3-642-35275-1_45

H. Hajibeygi-and-p and . Jenny, Adaptive iterative multiscale finite volume method, Journal of Computational Physics, vol.230, issue.3, pp.628-643, 2011.
DOI : 10.1016/j.jcp.2010.10.009

L. Halpern, Artificial boundary conditions for the linear advection diffusion equation, Mathematics of Computation, vol.46, issue.174, pp.425-438, 1986.
DOI : 10.1090/S0025-5718-1986-0829617-8

L. Halpern and F. Hubert, A Finite Volume Ventcell-Schwarz Algorithm for Advection-Diffusion Equations, SIAM Journal on Numerical Analysis, vol.52, issue.3, pp.1269-1291, 2014.
DOI : 10.1137/130919799

URL : https://hal.archives-ouvertes.fr/hal-01271249

L. Halpern and C. Japhet, Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation for Heterogeneous Problems, Lecture Notes in Computational Science and Engineering, vol.60, pp.211-219, 2008.
DOI : 10.1007/978-3-540-75199-1_23

URL : http://www.ricam.oeaw.ac.at/dd17/proceedings/pdf/japhet_m04.pdf

L. Halpern, C. Japhet, and A. P. Omnes, Nonconforming in time domain decomposition method for porous media applications, Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010, 2010.

L. Halpern, C. Japhet, and A. J. Szeftel, Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation, Domain Decomposition Methods in Science and Engineering XIX Lect. Notes Comput. Sci. Eng, vol.78, pp.133-140, 2011.
DOI : 10.1007/978-3-642-11304-8_13

T. T. Hoang, Space-time domain decomposition methods for mixed formulations of flow and transport problems in porous media, 2013.
URL : https://hal.archives-ouvertes.fr/tel-00922325

T. Hoang, J. Jaffré, C. Japhet, M. Kern, and A. J. Roberts, Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations, SIAM Journal on Numerical Analysis, vol.51, issue.6, pp.51-3532, 2013.
DOI : 10.1137/130914401

URL : https://hal.archives-ouvertes.fr/hal-00803796

T. T. Hoang, C. Japhet, M. Kern, and A. J. Roberts, Space???time domain decomposition for advection???diffusion problems in mixed formulations, Mathematics and Computers in Simulation, vol.137, 2016.
DOI : 10.1016/j.matcom.2016.11.002

URL : https://hal.archives-ouvertes.fr/hal-01296348

T. Dickopf, M. Gander, L. Halpern, R. Krause, and L. , Ventcell conditions with mixed formulations for flow in porous media, Decomposition Methods in Science and Engineering XXII, pp.531-540, 2016.

U. Hornung and E. , Homogenization and porous media, Interdisciplinary Applied Mathematics, vol.6, 1997.
DOI : 10.1007/978-1-4612-1920-0

W. J. Hundsdorfer and . Verwer, Numerical solution of time-dependent advectiondiffusion-reaction equations, 2010.
DOI : 10.1007/978-3-662-09017-6

C. Japhet, Méthode de décomposition de domaine et conditions aux limites artificielles en mécanique des fluides: méthode optimisée d'ordre 2, 1998.

C. Japhet-and-f and . Nataf, The best interface conditions for domain decomposition methods: absorbing boundary conditions, in Absorbing Boundaries and Layers, Domain Decomposition Methods, pp.348-373, 2001.

C. Japhet, F. Nataf, and A. F. Rogier, The Optimized Order 2 Method. Application to Convection-Diffusion Problems, Future Gener. Comp. Sy, vol.18, pp.17-30, 2001.
DOI : 10.1007/978-1-4615-5205-5_12

C. Japhet-and-p and . Omnes, Optimized Schwarz Waveform Relaxation for Porous Media Applications, Lect. Notes Comput. Sci. Eng, vol.91, pp.585-592
DOI : 10.1007/978-3-642-35275-1_69

P. Jiránek, Z. Strako?, and A. 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

O. A. Karakashian-and-f and . Pascal, A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems, SIAM Journal on Numerical Analysis, vol.41, issue.6, pp.41-2374, 2003.
DOI : 10.1137/S0036142902405217

K. Y. Kim, A posteriori error analysis for locally conservative mixed methods, Mathematics of Computation, vol.76, issue.257, pp.43-66, 2007.
DOI : 10.1090/S0025-5718-06-01903-X

URL : http://www.ams.org/mcom/2007-76-257/S0025-5718-06-01903-X/S0025-5718-06-01903-X.pdf

V. Kippe, J. E. Aarnes, and A. K. Lie, A comparison of multiscale methods for elliptic problems in porous media flow, Computational Geosciences, vol.11, issue.4, pp.377-398, 2008.
DOI : 10.2118/92965-PA

R. Kirby, Residual a posteriori error estimates for the mixed finite element method, Computational Geosciences, vol.7, issue.3, pp.197-214, 2003.
DOI : 10.1023/A:1025518113877

P. Ladevèze-and-d and . Leguillon, Error Estimate Procedure in the Finite Element Method and Applications, SIAM Journal on Numerical Analysis, vol.20, issue.3, pp.485-509, 1983.
DOI : 10.1137/0720033

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.21-24, pp.2313-2324, 2007.
DOI : 10.1016/j.cma.2006.08.019

K. Lie, S. Krogstad, I. S. Ligaarden, J. R. Natvig, H. M. Nilsen et al., Open-source MATLAB implementation of consistent discretisations on complex grids, Computational Geosciences, vol.13, issue.2, pp.297-322, 2012.
DOI : 10.2118/106254-PA

J. Lions-and-e and . Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, 1968.

P. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp.1-42, 1988.

C. Lovadina-and-r and . Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Mathematics of Computation, vol.75, issue.256, pp.1659-1674, 2006.
DOI : 10.1090/S0025-5718-06-01872-2

R. I. Luce-and-b and . Wohlmuth, A Local A Posteriori Error Estimator Based on Equilibrated Fluxes, SIAM Journal on Numerical Analysis, vol.42, issue.4, pp.1394-1414, 2004.
DOI : 10.1137/S0036142903433790

J. Mandel, Balancing domain decomposition, Communications in Numerical Methods in Engineering, vol.13, issue.3, pp.233-241, 1993.
DOI : 10.1137/1.9781611971057.ch5

URL : http://ftp.ccs.uky.edu/mgnet/www/mgnet/www/mgnet/papers/Mandel/bdd.ps.gz

J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients, Mathematics of Computation, vol.65, issue.216, pp.1387-1401, 1996.
DOI : 10.1090/S0025-5718-96-00757-0

V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Applied Numerical Mathematics, vol.52, issue.4, pp.401-428, 2005.
DOI : 10.1016/j.apnum.2004.08.022

T. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol.61, 2008.
DOI : 10.1007/978-3-540-77209-5

F. Nataf, F. Rogier, A. E. De, and . Sturler, Domain Decomposition Methods for Fluid Dynamics, Navier-Stokes Equations and Related Nonlinear Analysis, pp.367-376, 1995.
DOI : 10.1007/978-1-4899-1415-6_30

R. H. Nochetto, A. Schmidt, and A. C. , A posteriori error estimation and adaptivity for degenerate parabolic problems, Mathematics of Computation, vol.69, issue.229, pp.1-24, 2000.
DOI : 10.1090/S0025-5718-99-01097-2

M. Ohlberger, A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems, Multiscale Modeling & Simulation, vol.4, issue.1, pp.88-114, 2005.
DOI : 10.1137/040605229

J. E. Pasciak, Domain Decomposition Preconditioners for Elliptic Problems in Two and Three Dimensions, First International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, pp.62-72, 1987.
DOI : 10.1007/978-1-4684-6357-6_10

G. V. Pencheva, M. Vohralík, M. F. Wheeler, and A. T. Wildey, Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling, SIAM Journal on Numerical Analysis, vol.51, issue.1, pp.526-554, 2013.
DOI : 10.1137/110839047

URL : https://hal.archives-ouvertes.fr/hal-00467738

I. S. Pop, J. Bogers, and A. K. Kumar, Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition, Vietnam Journal of Mathematics, vol.219, issue.1-2, pp.77-102, 2017.
DOI : 10.1098/rspa.1953.0139

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

J. Przemieniecki, MATRIX STRUCTURAL ANALYSIS OF SUBSTRUCTURES, AIAA Journal, vol.96, issue.1, pp.138-147, 1963.
DOI : 10.1145/1455292.1455305

A. A. Quarteroni and . Valli, Theory and Application of Steklov-Poincar?? Operators for Boundary-Value Problems, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, pp.58-81, 1990.
DOI : 10.1007/978-94-009-1908-2_14

A. A. Quarteroni and . Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, 1999.

P. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], 1983.

S. Repin, S. Sauter, and A. A. Smolianski, Two???Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems, SIAM Journal on Numerical Analysis, vol.45, issue.3, pp.928-945, 2007.
DOI : 10.1137/050641533

S. I. Repin, A posteriori estimates for partial differential equations, of Radon Series on Computational and Applied Mathematics, 2008.
DOI : 10.1515/9783110203042

S. I. Repin and . Smolianski, Functional-type a posteriori error estimates for mixed finite element methods, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.339, issue.3, pp.365-382, 2005.
DOI : 10.1002/nme.874

V. Rey, P. Gosselet, and A. C. Rey, Strict lower bounds with separation of sources of error in non-overlapping domain decomposition methods, International Journal for Numerical Methods in Engineering, vol.155, issue.1-2, pp.1007-1029, 2016.
DOI : 10.1016/S0045-7825(97)00146-1

URL : https://hal.archives-ouvertes.fr/hal-01332674

V. Rey, C. Rey, and A. P. Gosselet, A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods, Computer Methods in Applied Mechanics and Engineering, vol.270, pp.293-303, 2014.
DOI : 10.1016/j.cma.2013.12.001

URL : https://hal.archives-ouvertes.fr/hal-00919435

J. E. Roberts and J. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, pp.523-639, 1991.

Y. Saad, Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, 2003.
DOI : 10.1137/1.9780898718003

H. A. Schwarz, G. Mathematische-abhandlungen-bronx, and N. Y. , Band I, II, 1972.

J. O. Skogestad, E. Keilegavlen, and A. J. Nordbotten, Domain decomposition strategies for nonlinear flow problems in porous media, Journal of Computational Physics, vol.234, pp.439-451, 2013.
DOI : 10.1016/j.jcp.2012.10.001

R. Stenberg, Postprocessing schemes for some mixed finite elements, ESAIM: Mathematical Modelling and Numerical Analysis, vol.25, issue.1, pp.151-167, 1991.
DOI : 10.1051/m2an/1991250101511

URL : https://hal.archives-ouvertes.fr/inria-00075751

J. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes, 1977.

V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 1997.
DOI : 10.1007/978-3-662-03359-3

A. Toselli-and-o and . Widlund, Domain decomposition methods?algorithms and theory, 2005.
DOI : 10.1007/b137868

C. J. Van-duijn, J. Molenaar, A. M. De, and . Neef, The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media, Transport in Porous Media, pp.71-93, 1995.

R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, pp.40-195, 2003.

R. Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation
DOI : 10.1093/acprof:oso/9780199679423.001.0001

M. Vohralík, A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations, SIAM Journal on Numerical Analysis, vol.45, issue.4, pp.1570-1599, 2007.
DOI : 10.1137/060653184

M. Vohralík and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Computational Geosciences, vol.17, issue.5, pp.789-812, 2013.
DOI : 10.1002/nla.742

M. F. Wheeler and I. Yotov, A Posteriori Error Estimates for the Mortar Mixed Finite Element Method, SIAM Journal on Numerical Analysis, vol.43, issue.3, pp.1021-1042, 2005.
DOI : 10.1137/S0036142903431687

O. B. Widlund, Iterative substructuring methods: algorithms and theory for elliptic problems in the plane, First International Symposium on Domain Decomposition Methods for Partial Differential Equations SIAM, pp.113-128, 1987.

B. I. Wohlmuth, A residual based error estimator for mortar finite element discretizations, Numerische Mathematik, vol.84, issue.1, pp.143-171, 1999.
DOI : 10.1007/s002110050467

URL : http://wwwhoppe.math.uni-augsburg.de/~wohlmuth/UNIAUG_370.ps.gz

B. I. Wohlmuth and R. H. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Mathematics of Computation, vol.68, issue.228, pp.1347-1378, 1999.
DOI : 10.1090/S0025-5718-99-01125-4

I. Yotov, A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow, East-West J. Numer . Math, vol.5, pp.211-230, 1997.

I. Yotov, Interface solvers and preconditioners of domain decomposition type for multiphase flow in multiblock porous media, Scientific computing and applications, pp.157-167, 2000.