E. Ahmed, S. A. Hassan, C. Japhet, M. Kern, and M. Vohralík, A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01540956

E. Ahmed, J. Jaffré, and J. E. Roberts, A reduced fracture model for two-phase flow with different rock types, Math. Comput. Simulation, pp.49-70, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01119986

E. Ahmed, C. Japhet, and M. Kern, Global-in-time domain decomposition for a nonlinear diffusion problem. HAL Preprint 02263280, the Proceedings of the 25th International Conference on Domain Decomposition Methods, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02263280

E. Ahmed, C. Japhet, and M. Kern, Space-time domain decomposition for two-phase flow between different rock types. HAL Preprint 02275690, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02275690

C. Alboin, J. Jaffré, J. E. Roberts, X. Wang, and C. Serres, Domain decomposition for some transmission problems in flow in porous media, Numerical treatment of multiphase flows in porous media (Beijing, 1999), vol.552, pp.22-34, 2000.

S. A. Hassan, C. Japhet, M. Kern, and M. Vohralík, A posteriori stopping criteria for optimized Schwarz domain decomposition algorithms in mixed formulations, Comput. Methods Appl. Math, vol.18, pp.495-519, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01529532

S. A. Hassan, C. Japhet, and M. Vohralík, A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations, Electron. Trans. Numer. Anal, vol.49, pp.151-181, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01586862

B. Andreianov, K. Brenner, and C. Cancès, Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium, ZAMM Z. Angew. Math. Mech, vol.94, pp.655-667, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00744359

M. Arioli, D. Loghin, and A. J. Wathen, Stopping criteria for iterations in finite element methods, Numer. Math, vol.99, pp.381-410, 2005.

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

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

D. Bennequin, M. J. Gander, and L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp, vol.78, pp.185-223, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00111643

H. Berninger, S. Loisel, and O. Sander, The 2-Lagrange multiplier method applied to nonlinear transmission problems for the Richards equation in heterogeneous soil with cross points, SIAM J. Sci. Comput, vol.36, pp.2166-2198, 2014.

M. Bertsch, R. Passo, and C. J. Van-duijn, Analysis of oil trapping in porous media flow, SIAM J. Math. Anal, vol.35, pp.245-267, 2003.

K. Brenner, C. Cancès, and D. Hilhorst, Finite volume approximation for an immiscible twophase flow in porous media with discontinuous capillary pressure, Comput. Geosci, vol.17, pp.573-597, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00675681

F. Caetano, M. J. Gander, L. Halpern, and J. Szeftel, Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Netw. Heterog. Media, vol.5, pp.487-505, 2010.

C. Cancès, Nonlinear parabolic equations with spatial discontinuities, NoDEA Nonlinear Differential Equations Appl, vol.15, pp.427-456, 2008.

C. Cancès, Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities, M2AN Math. Model. Numer. Anal, vol.43, pp.973-1001, 2009.

C. Cancès, T. Gallouët, and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound, vol.11, pp.239-258, 2009.

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.

G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation, Studies in Mathematics and Its Applications, vol.17, 1986.

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, Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem, Math. Comp, vol.84, pp.153-186, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00690862

G. Enchéry, R. Eymard, and A. Michel, Numerical approximation of a two-phase flow problem in a porous medium with discontinuous capillary forces, SIAM J. Numer. Anal, vol.43, pp.2402-2422, 2006.

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

R. Eymard, T. Gallouët, and R. Herbin, Handbook of Numerical Analysis, vol.VII, pp.713-1020, 2000.

R. Eymard, T. Gallouët, and R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient, Appl. Numer. Math, vol.37, pp.24-30, 2001.

M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal, vol.44, pp.699-731, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00107263

M. J. Gander, L. Halpern, and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal, vol.41, pp.1643-1681, 2003.

B. Ganis, K. Kumar, G. Pencheva, M. F. Wheeler, and I. Yotov, A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model, Multiscale Model. Simul, vol.12, pp.1401-1423, 2014.

F. Haeberlein, L. Halpern, and A. Michel, Newton-Schwarz optimised waveform relaxation Krylov accelerators for nonlinear reactive transport, Domain decomposition methods in science and engineering XX, vol.91, pp.387-394, 2013.

L. Halpern and F. Hubert, A finite volume Ventcell-Schwarz algorithm for advection-diffusion equations, SIAM J. Numer. Anal, vol.52, pp.1269-1291, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01271249

R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, 1997.

T. Hoang, J. Jaffré, C. Japhet, M. Kern, and J. E. Roberts, Space-time domain decomposition methods for diffusion problems in mixed formulations, SIAM J. Numer. Anal, vol.51, pp.3532-3559, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00803796

T. Hoang, C. Japhet, M. Kern, J. E. Roberts, T. Xxii et al., Ventcell conditions with mixed formulations for flow in porous media, Decomposition Methods in Science and Engineering, vol.104, pp.531-540, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01113964

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

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.

H. Li and M. F. Wheeler, Sequential Refinement Solver using Space-Time Domain Decomposition for Non-linear Multiphase Flow Problems, 2019.

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, Comput. Geosci, vol.16, pp.297-322, 2012.

V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math, vol.52, pp.401-428, 2005.

R. H. Nochetto, A. Schmidt, and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp, vol.69, pp.1-24, 2000.

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.

J. Pape?, Z. Strako?, and M. Vohralík, Estimating and localizing the algebraic and total numerical errors using flux reconstructions, Numer. Math, vol.138, pp.681-721, 2018.

G. V. Pencheva, M. Vohralík, M. F. Wheeler, and T. Wildey, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling, SIAM J. Numer. Anal, vol.51, pp.526-554, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00467738

V. Rey, P. Gosselet, and C. Rey, Strict lower bounds with separation of sources of error in nonoverlapping domain decomposition methods, Internat. J. Numer. Methods Engrg, vol.108, pp.1007-1029, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01332674

V. Rey, C. Rey, and P. Gosselet, A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods, Comput. Methods Appl. Mech. Engrg, vol.270, pp.293-303, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00919435

D. Seus, K. Mitra, I. S. Pop, F. A. Radu, and C. Rohde, A linear domain decomposition method for partially saturated flow in porous media, Comput. Methods Appl. Mech. Engrg, vol.333, pp.331-355, 2018.

G. Singh and M. F. Wheeler, A space-time domain decomposition approach using enhanced velocity mixed finite element method, J. Comput. Phys, vol.374, pp.893-911, 2018.

J. O. Skogestad, E. Keilegavlen, and J. M. Nordbotten, Domain decomposition strategies for nonlinear flow problems in porous media, J. Comput. Phys, vol.234, pp.439-451, 2013.

J. O. Skogestad, E. Keilegavlen, and J. M. Nordbotten, Two-scale preconditioning for twophase nonlinear flows in porous media, Transp. Porous Media, vol.114, pp.485-503, 2016.

C. J. Van-duijn, J. Molenaar, and M. J. De-neef, The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media, Transport in Porous Media, vol.21, pp.71-93, 1995.

M. T. Van-genuchten, A closed form for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Amer. J, vol.44, pp.892-898, 1980.

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.

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, vol.7, pp.157-167, 2001.