. Finally, simulating blood ow in the cerebral venous network when subject to a physiological pressure gradient gave relevant and interesting results In view of these ndings and targeting the validation of the results with respect to experiments, we aim at including more data at diierent levels: (i) geometrical description of the network ; (ii) mechanical parameters; (iii) more precise measures at the innow/outtow boundaries, as predominant factors

C. Amrouche, N. El, and H. Seloula, L p -theory for the Navier-Stokes equations with pressure boundary conditions, Discrete Contin, Dyn. Syst. Ser. S, vol.6, issue.5, pp.1113-1137, 2013.

R. E. Bank and T. Dupont, An optimal order process for solving finite element equations, Mathematics of Computation, vol.36, issue.153, pp.35-51, 1981.
DOI : 10.1090/S0025-5718-1981-0595040-2

W. L. Barth and G. F. Carey, On a boundary condition for pressure-driven laminar flow of incompressible fluids, International Journal for Numerical Methods in Fluids, vol.50, issue.11, pp.1313-1325, 2007.
DOI : 10.1002/fld.1427

C. Bègue, C. Conca, F. Murat, and O. Pironneau, A nouveau sur les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, Comptes rendus de l'Académie des sciences, Série Mathématique, vol.1, issue.304 1, pp.23-28, 1987.

C. Bernardi, T. Chacón-rebollo, and D. Yakoubi, Finite Element Discretization of the Stokes and Navier--Stokes Equations with Boundary Conditions on the Pressure, SIAM Journal on Numerical Analysis, vol.53, issue.3, pp.1256-1279, 2015.
DOI : 10.1137/140972299
URL : https://hal.archives-ouvertes.fr/hal-00961653

F. Brezzi and M. Fortin, Mixed and hybrid nite element methods, 2012.

V. Chabannes, M. Ismail, C. Prud-'homme, and M. Szopos, Hemodynamic simulations in the cerebral venous network: A study on the influence of different modeling assumptions, Journal of Coupled Systems and Multiscale Dynamics, vol.3, issue.1, pp.23-37, 2015.
DOI : 10.1166/jcsmd.2015.1062
URL : https://hal.archives-ouvertes.fr/hal-01109767

T. Chacón-rebollo, V. Girault, F. Murat, and O. Pironneau, Analysis of a Coupled Fluid-Structure Model with Applications to Hemodynamics, SIAM Journal on Numerical Analysis, vol.54, issue.2, pp.994-1019, 2016.
DOI : 10.1137/140991509

C. Conca, F. Murat, and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japanese journal of mathematics. New series, vol.20, issue.2, pp.279-318, 1994.

C. Conca, C. Pares, O. Pironneau, and M. Thiriet, Navier-Stokes equations with imposed pressure and velocity fluxes, International Journal for Numerical Methods in Fluids, vol.114, issue.4, pp.267-287, 1995.
DOI : 10.1002/fld.1650200402
URL : http://hdl.handle.net/10533/53222

H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible uid dynamics, p.2014
DOI : 10.1093/acprof:oso/9780199678792.001.0001

A. Ern and J. Guermond, Theory and practice of nite elements, 2013.

L. Formaggia, A. Quarteroni, and A. Veneziani, Cardiovascular mathematics, volume 1 of ms&a. modeling, simulation and applications, 2009.

L. Formaggia and C. Vergara, Prescription of general defective boundary conditions in uid-dynamics, Milan Journal of Mathematics, pp.1-18, 2012.

J. Fouchet-incaux, Artificial boundaries and formulations for the incompressible Navier???Stokes equations: applications to air and blood flows, SeMA Journal, vol.26, issue.7, pp.1-40, 2014.
DOI : 10.1007/s40324-014-0012-y
URL : https://hal.archives-ouvertes.fr/hal-00926273

T. Fries and T. Belytschko, The extended/generalized nite element method: An overview of the method and its applications, International Journal for Numerical Methods in Engineering, vol.84, issue.3, pp.253-304, 2010.

G. Gadda, A. Taibi, F. Sisini, M. Gambaccini, P. Zamboni et al., A new hemodynamic model for the study of cerebral venous outflow, American Journal of Physiology - Heart and Circulatory Physiology, vol.308, issue.3, pp.217-231, 2015.
DOI : 10.1152/ajpheart.00469.2014

V. Girault, Curl-conforming nite element methods for navier-stokes equations with non-standard boundary conditions in â??3, The Navier-Stokes Equations Theory and Numerical Methods, pp.201-218, 1990.

K. P. Gostaf and O. Pironneau, Pressure boundary conditions for blood flows, Chinese Annals of Mathematics, Series B, vol.195, issue.5, pp.829-842, 2015.
DOI : 10.1007/s11401-015-0983-8
URL : https://hal.archives-ouvertes.fr/hal-00865671

R. Rannacher, J. G. Heywood, and S. Turek, Artiicial boundaries and ux and pressure conditions for the incompressible navier-stokes equations, International Journal for Numerical Methods in Fluids, vol.22, pp.325-352, 1996.

S. M. Hosseini and J. J. Feng, Pressure boundary conditions for computing incompressible flows with SPH, Journal of Computational Physics, vol.230, issue.19, pp.7473-7487, 2011.
DOI : 10.1016/j.jcp.2011.06.013

H. Johnston and J. Liu, Finite Difference Schemes for Incompressible Flow Based on Local Pressure Boundary Conditions, Journal of Computational Physics, vol.180, issue.1, pp.120-154, 2002.
DOI : 10.1006/jcph.2002.7079

J. M. Leone and P. M. Gresho, Finite element simulations of steady, two-dimensional, viscous incompressible flow over a step, Journal of Computational Physics, vol.41, issue.1, pp.167-191, 1981.
DOI : 10.1016/0021-9991(81)90086-3

S. Maru?i?, On the Navier???Stokes system with pressure boundary condition, ANNALI DELL'UNIVERSITA' DI FERRARA, vol.182, issue.1, pp.319-331, 2007.
DOI : 10.1007/s11565-007-0024-y

O. Miraucourt, S. Salmon, M. Szopos, and M. Thiriet, Blood flow in the cerebral venous system: modeling and simulation, Computer Methods in Biomechanics and Biomedical Engineering, vol.82, issue.5, pp.471-482, 2017.
DOI : 10.1016/j.jcp.2012.09.016
URL : https://hal.archives-ouvertes.fr/hal-01384285

M. Orlt and A. Sändig, Regularity of viscous Navier-Stokes ows in nonsmooth domains, Boundary value problems and integral equations in nonsmooth domains (Luminy, pp.167-185, 1993.

O. Pironneau, Conditions aux limites sur la pression pour les équations de Stokes et de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math, vol.303, issue.9, pp.403-406, 1986.

C. Vergara, A. Porpora, P. Zunino, and M. Piccinelli, Numerical treatment of boundary conditions to replace lateral branches in hemodynamics, International journal for numerical methods in biomedical engineering, vol.28, issue.12, pp.1165-1183, 2012.

C. Caldini-queiros, V. Chabannes, M. Ismail, G. Pena, C. Prud-'homme et al., Towards large-scale three-dimensional blood ow simulations in realistic geometries, ESAIM Proc, pp.43-195, 2013.

B. Schaller, Physiology of cerebral venous blood flow: from experimental data in animals to normal function in humans, Brain Research Reviews, vol.46, issue.3, pp.243-260, 2004.
DOI : 10.1016/j.brainresrev.2004.04.005

R. Stenberg, On some techniques for approximating boundary conditions in the finite element method, Journal of Computational and Applied Mathematics, vol.63, issue.1-3, pp.139-148, 1995.
DOI : 10.1016/0377-0427(95)00057-7

C. Vergara, Nitsche???s Method for Defective Boundary Value Problems in Incompressibile Fluid-dynamics, Journal of Scientific Computing, vol.198, issue.37???40, pp.100-123, 2011.
DOI : 10.1007/s10915-010-9389-7