L. Ambrosio, Transport equation and Cauchy problem for BV vector fields and applications, Journ??es ??quations aux d??riv??es partielles, vol.158, issue.2, pp.227-260, 2004.
DOI : 10.5802/jedp.1

L. Ambrosio, The flow associated to weakly di?erentiable vector fields: recent results and open problems, in Nonlinear conservation laws and applications, Math. Appl, vol.153, pp.181-193, 2011.

D. N. Arnold, Spaces of finite element di?erential forms, in Analysis and numerics of partial di?erential equations, pp.117-140, 2013.

D. N. Arnold, D. Bo, and F. Bonizzoni, Finite element di?erential forms on curvilinear cubic meshes and their approximation properties, Numer. Math, pp.1-20, 2014.
DOI : 10.1007/s00211-014-0631-3
URL : http://arxiv.org/abs/1212.6559

D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, vol.15, pp.1-155, 2006.
DOI : 10.1017/S0962492906210018

D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bulletin of the American Mathematical Society, vol.47, issue.2, pp.281-354, 2010.
DOI : 10.1090/S0273-0979-10-01278-4

A. Bossavit, On the geometry of electromagnetismGeometrical objects, J. Japan Soc. Appl. Electromagn. and Mech, vol.6, issue.22, pp.114-123, 1998.

F. Bouchut and G. Crippa, LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL, Journal of Hyperbolic Differential Equations, vol.10, issue.02, pp.235-282, 2013.
DOI : 10.1142/S0219891613500100
URL : https://hal.archives-ouvertes.fr/hal-00724586

F. Boyer, Analysis of the upwind finite volume method for general initial- and boundary-value transport problems, IMA Journal of Numerical Analysis, vol.32, issue.4, pp.1404-1439, 2012.
DOI : 10.1093/imanum/drr031
URL : https://hal.archives-ouvertes.fr/hal-00559586

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol.15, issue.3, 2008.

F. Brezzi, J. Douglas, J. , and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, vol.36, issue.2, pp.217-235, 1985.
DOI : 10.1007/BF01389710

F. Brezzi, L. D. Marini, and E. Süli, DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS, Mathematical Models and Methods in Applied Sciences, vol.14, issue.12, pp.1893-1903, 2004.
DOI : 10.1142/S0218202504003866

E. Burman, A. Ern, and M. A. Fernández, Explicit Runge???Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems, SIAM Journal on Numerical Analysis, vol.48, issue.6, pp.2019-2042, 2010.
DOI : 10.1137/090757940
URL : https://hal.archives-ouvertes.fr/hal-00380659

P. G. Ciarlet, The finite element method for elliptic problems, 1978.

G. Crippa and C. De-lellis, Regularity and compactness for the DiPerna-Lions flow, in Hyperbolic problems: theory, numerics, applications, pp.423-430, 2008.

P. A. Davidson, An Introduction to Magnetohydrodynamics, 2001.
DOI : 10.1017/cbo9780511626333

C. and D. Lellis, Notes on hyperbolic systems of conservation laws and transport equations, Handbook of Di?erential Equations: Evolutionary Equations, pp.277-383, 2006.

R. J. Diperna and P. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, vol.307, issue.3, pp.511-547, 1989.
DOI : 10.1007/BF01393835

A. Ern and J. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol.159, 2004.
DOI : 10.1007/978-1-4757-4355-5

K. O. Friedrichs, Symmetric positive linear differential equations, Communications on Pure and Applied Mathematics, vol.35, issue.3, pp.333-418, 1958.
DOI : 10.1002/cpa.3160110306

F. G. Fuchs, A. D. Mcmurry, S. Mishra, N. H. Risebro, and K. Waagan, Abstract, Communications in Computational Physics, vol.3, issue.02, pp.324-362, 2011.
DOI : 10.1016/0021-9991(87)90031-3

H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, 2004.
DOI : 10.1017/CBO9780511616945

H. Heumann, Eulerian and Semi-Lagrangian Methods for Advection-Di?usion of Di?erential Forms, ETH dissertation, 2011.

H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, ESAIM: Mathematical Modelling and Numerical Analysis, vol.47, issue.6, pp.1713-1732, 2013.
DOI : 10.1051/m2an/2013085
URL : https://hal.archives-ouvertes.fr/hal-01108272

H. Heumann and R. Hiptmair, Convergence of Lowest Order Semi-Lagrangian Schemes, Found, Comput. Math, vol.13, issue.2, pp.187-220, 2013.
DOI : 10.1007/s10208-012-9139-3

H. Heumann, R. Hiptmair, and C. Pagliantini, Stabilized Galerkin for Transient Advection of Di?erential Forms, Seminar for Applied Mathematics, ETH Zürich

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer, vol.11, pp.237-339, 2002.
DOI : 10.1017/cbo9780511550140.004

P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numerische Mathematik, vol.169, issue.3, pp.485-518, 2005.
DOI : 10.1007/s00211-005-0604-7

O. A. Karakashian and F. 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.2374-2399, 2003.
DOI : 10.1137/S0036142902405217

T. Kato, Linear and Quasi-Linear Equations of Evolution of Hyperbolic Type, Hyperbolicity, C.I.M.E. Summer Schools, vol.72, pp.125-191, 1976.
DOI : 10.1007/978-3-642-11105-1_4

J. M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, vol.218, issue.2
DOI : 10.1007/978-1-4419-9982-5

D. Levy and E. Tadmor, From Semidiscrete to Fully Discrete: Stability of Runge--Kutta Schemes by The Energy Method, SIAM Review, vol.40, issue.1, pp.40-73, 1998.
DOI : 10.1137/S0036144597316255

S. Mishra, C. Schwab, and J. Sukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, Journal of Computational Physics, vol.231, issue.8, pp.3365-3388, 2012.
DOI : 10.1016/j.jcp.2012.01.011

S. A. Orszag and C. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, Journal of Fluid Mechanics, vol.17, issue.01, pp.129-143, 1979.
DOI : 10.1007/BF01474610

A. Pazy, Semigroups of linear operators and applications to partial di?erential equations, Applied Mathematical Sciences, vol.44, 1983.
DOI : 10.1007/978-1-4612-5561-1

T. E. Peterson, A Note on the Convergence of the Discontinuous Galerkin Method for a Scalar Hyperbolic Equation, SIAM Journal on Numerical Analysis, vol.28, issue.1, pp.133-140, 1991.
DOI : 10.1137/0728006

P. Raviart and J. M. Thomas, A mixed finite element method for 2-nd order elliptic problems, Proc. Conf., Consiglio Naz, pp.292-315, 1975.
DOI : 10.1007/BF01436186

G. Schwarz, Hodge Decomposition?a Method for Solving Boundary Value Problems, Lecture Notes in Mathematics, vol.1607, 1995.
DOI : 10.1007/BFb0095978

N. J. Walkington, Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions, SIAM Journal on Numerical Analysis, vol.42, issue.5, pp.1801-1817, 2005.
DOI : 10.1137/S0036142902412233