[. A. Hervé, Integration of the magnetic components of circular thick coils and disk coils, Proc. of fteenth int. conf. on Magnet Technology, p.12621270, 1997.

M. Bebendorf and J. Ostrowski, Parallel Hierarchical Matrix Preconditioners for the Curl-Curl Operator, Journal of Computational Mathematics, vol.27, issue.5, p.624641, 2009.
DOI : 10.4208/jcm.2009.27.5.014

. Biro, On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems, IEEE Transactions on Magnetics, vol.32, issue.3, pp.651-654, 1996.
DOI : 10.1109/20.497322

. Bo, Mixed Finite Element Methods and Applications, 2013.

. Brezzi, . Fortin, F. Brezzi, and M. Fortin, Mixed and hybrid nite element methods, 1991.

. Brezzi, Two families of mixed nite elements for second order elliptic problems, Numerische Mathematik, vol.47, p.217235, 1985.

. Canuto, Spectral Methods, 2006.
DOI : 10.1002/0470091355.ecm003m

URL : https://hal.archives-ouvertes.fr/hal-01296839

. Canuto, A posteriori error analysis of the reduced basis method for nonane parametrized nonlinear pdes, SIAM Journal on Numerical Analysis, issue.3, p.47, 2009.

V. Chabannes, Vers la simulation des écoulements sanguins, 2013.

P. Ciarlet, The nite element method for elliptic problems, p.24, 1978.

A. Colburn, A method of correlating forced convection heat transfer data and a comparison with uid friction, Trans. AIChE, vol.29, p.174210, 1933.

J. T. Conway, Exact solutions for the magnetic elds of axisymmetric solenoids and current distributions, IEEE Transactions on Magnetics, vol.37, issue.4, p.29772988, 2001.

A. Cotton, Le grand électro-aimant de l'académie des sciences, Recherches et Inventions, 1928.

. Daversin, Full 3d non-linear multi-physics model for high eld polyhelix magnets, 10th International Symposium on Electric and Magnetic Fields, 2016.

. Daversin, . Prud-'homme, C. Daversin, and C. Prud-'homme, Simultaneous empirical interpolation and reduced basis method for non-linear problems, Comptes Rendus Mathematique, vol.353, issue.12, p.11051109, 2015.
DOI : 10.1016/j.crma.2015.08.003

URL : https://hal.archives-ouvertes.fr/hal-01219063

. Daversin, . Prud-'homme, C. Daversin, and C. Prud-'homme, Simultaneous empirical interpolation and reduced basis method for non-linear problems. Model Reduction of Parametrized Systems, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01219063

. Daversin, C. Prud-'homme-]-daversin, and C. Prud-'homme, Simultaneous empirical interpolation and reduced basis method, application to non-linear multiphysics problem, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01332437

. Daversin, Full 3d multiphysics model of high eld polyhelices magnets, 24th International Conference on Magnet Technology, 2015.

. Daversin, Full three-dimensional multiphysics model of high-eld polyhelices magnets, IEEE transactions on applied superconductivity, vol.26, issue.4, 2016.

. Daversin, A Reduced Basis Framework: Application to large scale non-linear multi-physics problems, ESAIM: Proceedings, vol.43, p.225254, 2013.
DOI : 10.1051/proc/201343015

URL : https://hal.archives-ouvertes.fr/hal-00786557

. Debray, Dc high eld magnets at the lncmi, IEEE transactions on applied superconductivity, vol.22, issue.3, 2012.

. Debray, The grenoble high magnetic eld laboratory as a user facility, IEEE transactions on applied superconductivity, vol.12, issue.1, 2002.

[. D. Parkinson-]-d and B. M. Parkinson, The generation of high magnetic elds, 1967.

G. Dular, P. Dular, and C. Geuzaine, Getdp: a general environment for the treatment of discrete problems, 1997.

C. Dumitru, Numerical problems in 3d magnetostatic fem analysis, Advances in Automatic Control, Modelling & Simulation. Proceeding of the 15th International Conference on Automatic Control, Modelling & Simulation, 2013.

. Dutfoy, OpenTURNS, an Open Source initiative to Treat Uncertainties, Risks'N Statistics in a structured industrial approach, 41èmes Journées de Statistique, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00386719

H. Egger and J. Schöberl, A hybrid mixed discontinuous galerkin nite-element method for convectiondiusion problems, IMA Journal of Numerical Analysis, vol.30, issue.4, p.12061234, 2010.

. Falgout, The design and implementation of hypre, a library of parallel high performance preconditioners. Hypre : High performance preconditioners, 2006.

. Geuzaine, C. Remacle-]-geuzaine, and J. F. Remacle, Gmsh: a threedimensional nite element mesh generator with built-in pre-and post-processing facilities, International Journal for Numerical Methods in Engineering, issue.11, p.7913091331, 2009.

R. D. Graglia, On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle, IEEE Transactions on Antennas and Propagation, vol.41, issue.10, p.4114481455, 1993.
DOI : 10.1109/8.247786

C. Greif and D. Schötzau, Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numerical Linear Algebra with Applications, vol.22, issue.4, p.281297, 2007.
DOI : 10.1002/nla.515

X. Hiptmair, R. Hiptmair, and J. Xu, Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces, SIAM Journal on Numerical Analysis, vol.45, issue.6, pp.2483-2509, 2007.
DOI : 10.1137/060660588

. Huynh, A successive constraint linear optimization method for lower bounds of parametric coercivity and infsup stability constants, Comptes Rendus Mathematique, issue.8, p.345473478, 2007.

. Janon, Asymptotic normality and eciency of two Sobol index estimators, ESAIM: Probability and Statistics, p.342364, 2014.

. Janon, Certied reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values, 2013.

. Janon, UNCERTAINTIES ASSESSMENT IN GLOBAL SENSITIVITY INDICES ESTIMATION FROM METAMODELS, International Journal for Uncertainty Quantification, vol.4, issue.1, p.2136, 2014.
DOI : 10.1615/Int.J.UncertaintyQuantification.2012004291

URL : https://hal.archives-ouvertes.fr/inria-00567977

. Kolev, . Vassilevski, T. Kolev, and P. Vassilevski, Parallel Auxiliary Space AMG For H(Curl) Problems, Journal of Computational Mathematics, vol.27, issue.5, p.604623, 2009.
DOI : 10.4208/jcm.2009.27.5.013

. Masserey, Numerical integration of the three-dimensional Green kernel for an electromagnetic problem, Journal of Computational Physics, vol.205, issue.1, p.4871, 2005.
DOI : 10.1016/j.jcp.2004.10.035

J. C. Maxwell, A dynamical theory of the electromagnetic eld, 1865.

J. Nédélec, Mixed nite elements in R 3, Numerische Mathematik, vol.35, p.315341, 1980.

J. Nédélec, A new family of mixed nite elements in R 3, Numerische Mathematik, vol.50, p.5781, 1986.

R. Patera, A. Patera, and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial dierential equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006.

G. Pena, Spectral element approximation of the incompressible Navier-Stokes equations evolving in a moving domain and applications, p.4529, 2009.

C. Prieur, Analyse de sensibilité globale, 2014.

C. Prud-'homme, A domain specic embedded language in C++ for automatic dierentiation, projection, integration and variational formulations, Scientic Programming, p.81110, 2006.

P. Prud-'homme, C. Prud-'homme, and A. Patera, Reduced-basis output bounds for approximately parameterized elliptic coercive partial dierential equations, Computing and Visualization in Science, vol.6, issue.2-3, p.147162, 2004.

. Prud-'homme, Reliable real-time solution of parametrized partial dierential equations: Reduced-basis output bound methods, Journal of Fluids Engineering, vol.124, issue.1, pp.7080-189, 2002.

. Quarteroni, Certied reduced basis approximation for parametrized partial dierential equations and applications, Journal of Mathematics in Industry, vol.1, issue.1, p.149, 2011.

. Raviart, P. Thomas-]-raviart, and J. M. Thomas, Primal hybrid nite element methods for 2nd order elliptic equations, Mathematics of computation, issue.138, p.31391413, 1977.

. Rognes, Ecient assembly of h(div) and h(curl) conforming nite elements, SIAM Journal on Scientic Computing, vol.31, p.41304151, 2009.

. Rozza, Reduced basis approximation and a posteriori error estimation for anely parametrized elliptic coercive partial dierential equations, Archives of Computational Methods in Engineering, vol.15, issue.3, p.147, 2007.

. Sabariego, Fast multipole accelerated finite element-boundary element analysis of shielded induction heaters, IEEE Transactions on Magnetics, vol.42, issue.4, pp.1407-1410, 2006.
DOI : 10.1109/TMAG.2006.871974

I. Sobol, Sensitivity analysis for nonlinear mathematical models, Mathematical Modeling and Computational Experiment, vol.1, pp.407-414, 1993.

B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Reliability Engineering & System Safety, vol.93, issue.7, p.964979, 2008.
DOI : 10.1016/j.ress.2007.04.002

URL : https://hal.archives-ouvertes.fr/hal-01432217

P. Tissot, J. Tissot, and C. Prieur, A randomized Orthogonal Array-based procedure for the estimation of rst-and second-order Sobol' indices, Journal of Statistical Computation and Simulation, p.124, 2014.

. Trophime, Magnet calculations at the grenoble high magnetic eld laboratory, IEEE transactions on applied superconductivity, vol.12, issue.1, 2002.

. Trophime, Magnetic eld homogeneity optimization of the giga-nmr resistive insert, IEEE Transactions on Applied Superconductivity, vol.16, issue.2, p.15091512, 2006.

P. Veroy, K. Veroy, and A. Patera, Certied real-time solution of the parametrized steady incompressible navier-stokes equations: Rigorous reducedbasis a posteriori error bounds, Int. J. Numer. Meth. Fluids, vol.47, p.773788, 2005.

. Veroy, Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds, Comptes Rendus Mathematique, vol.337, issue.9, p.619624, 2003.
DOI : 10.1016/j.crma.2003.09.023

URL : https://hal.archives-ouvertes.fr/hal-01219048

. Veroy, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial dierential equations, 2003.

S. Veys, Un framework de calcul pour la méthode des bases réduites : applications à des problèmes non-linéaires multi-physiques, 2014.

. White, A parallel computer implementation of fast low-rankqr approximation of the biot-savart law, Proceedings of Progress In Electromagnetics Research Symposium, 2006.

S. Zaglmayr, High order nite element methods for electromagnetic eld computation, 2006.

Z. Zienkiewicz, O. C. Zienkiewicz, and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg, vol.33, issue.7, p.13311364, 1992.