.. Numerical-results, 164 6.3.1 Flow simulation on cerebrovenous system, p.169

P. R. Amestoy, A. Guermouche, J. , and S. Pralet, Hybrid scheduling for the parallel solution of linear systems, Parallel Computing, vol.32, issue.2, pp.136-156, 2006.
DOI : 10.1016/j.parco.2005.07.004

URL : https://hal.archives-ouvertes.fr/hal-00358623

C. Amrouche, N. El, and H. , Stokes equations and elliptic systems with nonstandard boundary conditions, Comptes Rendus Mathematique, vol.349, issue.11-12, pp.703-708, 2011.
DOI : 10.1016/j.crma.2011.04.007

A. Arseneev, Global existence of a weak solution of vlasov's system of equations, USSR Computational Mathematics and Mathematical Physics, vol.15, issue.1, pp.131-143, 1975.
DOI : 10.1016/0041-5553(75)90141-X

S. Balay, W. D. Gropp, and B. F. , SMITH : Modern software tools in scientific computing, pp.163-202, 1997.

N. Ben and . Abdallah, Weak solutions of the initial-boundary value problem for the vlasov?poisson system, Mathematical methods in the applied sciences, pp.451-476, 1994.

M. Bennoune, M. Lemou, and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier???Stokes asymptotics, Journal of Computational Physics, vol.227, issue.8, pp.3781-3803, 2008.
DOI : 10.1016/j.jcp.2007.11.032

URL : https://hal.archives-ouvertes.fr/hal-00348598

A. Bers and J. L. , DELCROIX : Physique des plasmas, EDP Sciences, 1994.

C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation, 1991.
DOI : 10.1887/0750301171

M. Bostan, Asymptotic behavior of weak solutions for the relativistic Vlasov???Maxwell equations with large light speed, Journal of Differential Equations, vol.227, issue.2, pp.444-498, 2006.
DOI : 10.1016/j.jde.2005.10.018

M. Bostan, The Vlasov?Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptotic Analysis, vol.61, issue.2, pp.91-123, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00139666

M. Bostan, Gyrokinetic Vlasov Equation in Three Dimensional Setting. Second Order Approximation, Multiscale Modeling & Simulation, vol.8, issue.5, pp.1923-1957, 2010.
DOI : 10.1137/090777621

URL : https://hal.archives-ouvertes.fr/hal-00431289

M. Bostan, Gyrokinetic Vlasov Equation in Three Dimensional Setting. Second Order Approximation, Multiscale Modeling & Simulation, vol.8, issue.5, pp.1923-1957, 2010.
DOI : 10.1137/090777621

URL : https://hal.archives-ouvertes.fr/hal-00431289

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, Journal of Differential Equations, vol.249, issue.7, pp.1620-1663, 2010.
DOI : 10.1016/j.jde.2010.07.010

URL : https://hal.archives-ouvertes.fr/hal-00595157

M. Bostan, Transport of Charged Particles Under Fast Oscillating Magnetic Fields, SIAM Journal on Mathematical Analysis, vol.44, issue.3, pp.1415-1447, 2012.
DOI : 10.1137/100797400

URL : https://hal.archives-ouvertes.fr/hal-01266545

M. Bostan, C. Caldini, and . Queiros, Finite larmor radius approximation for collisionnal magnetic confinement. part I : The linear boltzmann equation, Quaterly of Applied Mathematics, 2012.

M. Bostan, C. Caldini, and . Queiros, Finite larmor radius approximation for collisionnal magnetic confinement. part II : The fokker?planck?landau equation, 2012.

M. Bostan and C. Negulescu, Mathematical models for strongly magnetized plasmas with mass disparate particles. Discrete and Continuous Dynamical Systems-Series B, pp.513-544, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00617226

F. Bouchut, F. Golse, and C. , PALLARD : Classical solutions and the glasseystrauss theorem for the 3d vlasov-maxwell system. Archive for rational mechanics and analysis, pp.1-15, 2003.

A. J. Brizard and T. S. , Foundations of nonlinear gyrokinetic theory, Reviews of Modern Physics, vol.79, issue.2, p.421, 2007.
DOI : 10.1103/RevModPhys.79.421

C. Cercignani, R. Illner, and M. Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol.106, 1994.
DOI : 10.1007/978-1-4419-8524-8

V. Chabannes, Vers la simulation des écoulements sanguins, Thèse de doctorat, 2013.

V. Chabannes, G. Pena, and C. Prud, HOMME : High order fluid structure interaction in 2d and 3d application to blood flow in arteries, Journal of Computational and Applied Mathematics, 2012.

F. F. Chen, Introduction to plasma physics and controlled fusion, 1984.

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.

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits, Kinetic and Related Models, vol.4, issue.2, pp.441-477, 2011.
DOI : 10.3934/krm.2011.4.441

URL : https://hal.archives-ouvertes.fr/hal-00533327

N. Crouseilles, M. Mehrenberger, and E. Sonnendrücker, Conservative semi-Lagrangian schemes for Vlasov equations, Journal of Computational Physics, vol.229, issue.6, pp.1927-1953, 2010.
DOI : 10.1016/j.jcp.2009.11.007

URL : https://hal.archives-ouvertes.fr/hal-00363643

P. Degond and H. Neunzert, Local existence of solutions of the vlasovmaxwell equations and convergence to the vlasov-poisson equations for infinite light velocity, Mathematical methods in the applied sciences, pp.533-558, 1986.

P. Degond and P. Raviart, An asymptotic analysis of the onedimensional vlasov?poisson system : the child?langmuir law, Asymptotic Analysis, vol.4, issue.3, pp.187-214, 1991.

L. Desvillettes, Plasma kinetic models : the fokker-planck-landau equation , modelling and computational methods for kinetic equations, pp.171-193

L. Desvillettes and C. , On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness, Communications in Partial Differential Equations, vol.1, issue.1-2, pp.179-259, 2000.
DOI : 10.1007/BF02183613

L. Desvillettes and C. , On the spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications, Communications in Partial Differential Equations, vol.315, issue.1-2, pp.261-298, 2000.
DOI : 10.1007/s002050050106

R. J. Diperna and P. , Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics, vol.2, issue.6, pp.729-757, 1989.
DOI : 10.1002/cpa.3160420603

V. Doyeux, Y. Guyot, V. Chabannes, C. Prud-'homme, and M. Ismail, Simulation of two-fluid flows using a finite element/level set method. Application to bubbles and vesicle dynamics, Journal of Computational and Applied Mathematics, vol.246, 2012.
DOI : 10.1016/j.cam.2012.05.004

URL : https://hal.archives-ouvertes.fr/hal-01345573

L. Formaggia and A. Quarteroni, VENEZIANI : Cardiovascular Mathematics : Modeling and simulation of the circulatory system, 2009.

E. Frénod and E. Sonnendrücker, Homogenization of the vlasov equation and of the vlasov?poisson system with a strong external magnetic field, Asymptotic Analysis, vol.18, issue.3, pp.193-213, 1998.

E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation, SIAM Journal on Mathematical Analysis, vol.32, issue.6, pp.1227-1247, 2001.
DOI : 10.1137/S0036141099364243

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptotic Analysis, vol.46, issue.1, pp.1-28, 2006.

X. Garbet, G. Dif-pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard et al., Neoclassical equilibrium in gyrokinetic simulations, Physics of Plasmas, vol.16, issue.6, p.62503, 2009.
DOI : 10.1063/1.3153328

C. Geuzaine and J. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, vol.69, issue.4, pp.791309-1331, 2009.
DOI : 10.1002/nme.2579

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Communications in Mathematical Physics, vol.33, issue.4, pp.459-473, 1985.
DOI : 10.1007/BF01210740

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities. Archive for rational mechanics and analysis, pp.59-90, 1986.

F. Golse, P. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a Transport Equation, Journal of Functional Analysis, vol.76, issue.1, pp.110-125, 1988.
DOI : 10.1016/0022-1236(88)90051-1

F. Golse and L. Saint-raymond, The Vlasov???Poisson System with Strong Magnetic Field, Journal de mathématiques pures et appliquées, pp.791-817, 1999.
DOI : 10.1016/S0021-7824(99)00021-5

C. Greengard and P. , A boundary-value problem for the stationary vlasov-poisson equations: The plane diode, Communications on Pure and Applied Mathematics, vol.3, issue.4, pp.473-507, 1990.
DOI : 10.1002/cpa.3160430404

F. Hubert and J. H. Hubbard, Calcul scientifique : Équations différentielles et équations aux dérivées partielles, Vuibert, vol.2, 2006.

S. Jin, L. Pareschi, and G. Toscani, Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations, SIAM Journal on Numerical Analysis, vol.38, issue.3, pp.913-936, 2000.
DOI : 10.1137/S0036142998347978

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.1966

A. Klar, An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit, SIAM Journal on Numerical Analysis, vol.35, issue.3, pp.1073-1094, 1998.
DOI : 10.1137/S0036142996305558

A. Klar, An Asymptotic Preserving Numerical Scheme for Kinetic Equations in the Low Mach Number Limit, SIAM Journal on Numerical Analysis, vol.36, issue.5, pp.1507-1527, 1999.
DOI : 10.1137/S0036142997321765

M. Lemou, Relaxed micro???macro schemes for kinetic equations, Comptes Rendus Mathematique, vol.348, issue.7-8, pp.455-460, 2010.
DOI : 10.1016/j.crma.2010.02.017

URL : https://hal.archives-ouvertes.fr/hal-00521523

M. Lemou, MIEUSSENS : A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit

E. Marchandise, C. Geuzaine, and J. Remacle, Cardiovascular and lung mesh generation based on centerlines. International journal for numerical methods in biomedical engineering, 2013.

B. Maury, The Respiratory System in Equations, 2013.
DOI : 10.1007/978-88-470-5214-7

URL : https://hal.archives-ouvertes.fr/hal-00929739

A. Mouton and E. Frénod, Two-dimensional finite larmor radius approximation in canonical gyrokinetic coordinates, J. Pures Appl. Math. Adv. Appl, issue.4, pp.135-169, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00531361

G. Pena and C. Prud-'homme, QUARTERONI : High order methods for the approximation of the incompressible navier?stokes equations in a moving domain, Computer Methods in Applied Mechanics and Engineering, vol.209, pp.197-211, 2012.

F. Poupaud, Runaway Phenomena and Fluid Approximation Under High Fields in Semiconductor Kinetic Theory, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.308, issue.8, pp.72359-372, 1992.
DOI : 10.1002/zamm.19920720813

F. Poupaud and C. Schmeiser, Charge transport in semiconductors with degeneracy effects, Mathematical methods in the applied sciences, pp.301-318, 1991.
DOI : 10.1002/mma.1670140503

C. Prud-'homme, Life : Overview of a unified c++ implementation of the finite and spectral element methods in 1d, 2d and 3d, Applied Parallel Computing. State of the Art in Scientific Computing, pp.712-721, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00319983

C. Prud-'homme, V. Chabannes, V. Doyeux, M. Ismail, A. Samakae et al., Feel++ : A computational framework for galerkin methods and advanced numerical methods, ESAIM : Proceedings, pp.429-455, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00662868

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

M. C. Reed and B. Simon, Methods of modern mathematical physics : Functional analysis, 1980.

L. Saint-raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, Journal de mathématiques pures et appliquées, pp.379-399, 2002.
DOI : 10.1016/S0021-7824(01)01245-4

S. Salmon, S. Sy, and M. Szopos, Cerebral blood flow simulations in realistic geometries, ESAIM : Proceedings
DOI : 10.1051/proc/201235028

B. Smith, P. Bjorstad, and W. Gropp, Domain Decomposition, 2004.
DOI : 10.1007/978-3-540-70529-1_411

A. R. Gourlay and J. L. Morris, A multistep formulation of the optimized Lax-Wendroff method for nonlinear hyperbolic systems in two space variables, Mathematics of Computation, vol.22, issue.104, pp.715-719, 1968.
DOI : 10.1090/S0025-5718-1968-0251931-7

X. Q. Xu and M. N. , Numerical simulation of ion???temperature???gradient???driven modes, Physics of Fluids B: Plasma Physics, vol.3, issue.3, pp.627-643, 1991.
DOI : 10.1063/1.859862

. Landau, On se place pour cela dans le cas du régime du rayon de Larmor fini Avant de réaliser les calculs sur l'opérateur de Fokker-Planck-Landau, qui contient des convolutions et des termes de diffusion, il semble raisonnable de calculer la moyenne de l'opérateur de relaxation de Boltzmann, dont l'expression est plus simple. On se place ensuite dans le cas du régime centre-guide et on présente un schéma numérique basé sur une décomposition micro-macro de la fonction de distribution des particules qui provient