T. , S. Nicaise, and A. D. Pardo,

R. A. Adams and J. J. Fournier, Sobolev Spaces, 2003.

A. Abubakar, T. Habashy, V. Druskin, L. Knizhnerman, and D. Alumbaugh, 2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements, Geophysics, vol.73, pp.165-177, 2008.

D. B. Avdeev, Three-dimensional electromagnetic modelling and inversion from theory to application, Surv. Geophys, vol.26, pp.767-799, 2005.

I. Babu?ka, G. Caloz, and J. E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal, vol.31, pp.945-981, 1994.

G. E. Backus, Long-wave anisotropy produced by horizontal layering, J. Geophys. Res, vol.67, pp.4427-4440, 1962.

H. Barucq, T. Chaumont-frelet, and C. Gout, Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation, Math. Comp, vol.86, pp.2129-2157, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01408934

A. Bonito, J. L. Guermond, and F. Luddens, Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl, vol.408, pp.498-512, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01024489

M. Bourlard, M. Dauge, M. Lubuma, and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domain with conical points. III: Finite element methods on polygonal domains, SIAM J. Numer. Anal, vol.29, pp.136-155, 1992.

S. C. Brenner, F. Li, and L. Y. Sung, A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations, Math. Comp, vol.76, pp.573-595, 2006.

Y. Capdeville, L. Guillot, and J. Marigo, 2-D non-periodic homogenization to upscale elastic media for P-SV waves, Geophys. J. Int, vol.182, pp.903-922, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00490536

T. Chaumont-frelet, Finite Element Approximation of Helmholtz Problems with Application to Seimsic Wave Propagation, 2015.

T. Chaumont-frelet, On high order methods for the heterogeneous Helmholtz equation, Comput, Math. Appl, vol.72, pp.2203-2225, 2016.

T. Chaumont-frelet, H. Barucq, H. Calandra, and C. Gout, A Multiscale Medium Approximation Method for the Propagation P-Waves in Highly Heterogeneous Geophysical Media, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01706454

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math, vol.40, 2002.

P. Ciarlet, S. Fliss, and C. Stohrer, On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell's equations, Comput. Math. Appl, vol.73, pp.1900-1919, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01176476

S. Constable and L. J. Srnka, An introduction to marine controlled-source electromagnetic methods for hydrocarbon exploration, Geophysics, vol.72, pp.3-12, 2007.

M. Costabel, M. Dauge, and S. Nicaise, Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal, vol.33, pp.627-649, 1999.

D. V. Ellis and J. M. Singer, Well Logging for Earth Scientists, 2007.

E. Engström and D. Sjöberg, A Comparison of Two Numerical Methods for Homogenization of Maxwell's Equations, 2004.

A. Ern and J. L. Guermond, Finite element quasi-interpolation and best approximation, M2AN Math. Model. Numer. Anal, vol.51, pp.1367-1385, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01155412

A. Ern and J. L. Guermond, Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions, Comput. Math. Appl, vol.75, pp.918-932, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01531940

V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, 1986.

T. M. Habashy and A. Abubakar, A general framework for constraint minimization for the inversion of electromagnetic measurements, Prog. Electromag. Res, vol.46, pp.265-312, 2004.

T. M. Habashy and A. Abubakar, A generalized material averaging formulation for modelling of the electromagnetic fields, J. Electromag. Waves Appl, vol.21, pp.1145-1159, 2007.

P. Henning, M. Ohlberger, and B. Verfürth, A new heterogeneous multiscale method for time-harmonic Maxwell's equations, SIAM J. Numer. Anal, vol.54, pp.3493-3522, 2016.
DOI : 10.1137/15m1039225

T. Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys, vol.134, pp.169-189, 1997.
DOI : 10.1006/jcph.1997.5682

F. Ihlenburg and I. Babu?ka, Finite element solution of the Helmholtz equation with high wave number part I: The h-version of the FEM, Comput. Math. Appl, vol.30, pp.9-37, 1995.

P. Ciarlet, On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields, Comput. Math. Appl, vol.71, pp.85-104, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01176476

P. Ciarlet, J. , and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numer. Math, vol.82, pp.193-219, 1999.

J. B. Keller, A theorem on the conductivity of a composite medium, J. Math. Phys, vol.5, pp.548-549, 1964.

J. Li, J. M. Melenk, B. Wohlmuth, and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math, vol.60, pp.19-37, 2010.
DOI : 10.1016/j.apnum.2009.08.005

J. L. , Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 1972.

P. Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math, vol.63, pp.243-261, 1992.

P. Monk, Finite Element Methods for Maxwell's Equations, 2003.

S. Moskow, V. Druskin, T. Habashy, P. Lee, and S. Davydycheva, A finite difference scheme for elliptic equations with rough coefficients using a Cartesian grid nonconforming to interfaces, SIAM J. Numer. Anal, vol.36, pp.442-464, 1999.

J. Nédélec, Mixed finite elements in R 3, Numer. Math, vol.35, pp.315-341, 1980.

N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations, J. Comput. Phys, vol.230, pp.7151-7175, 2011.

S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations, SIAM J. Numer. Anal, vol.39, pp.784-816, 2001.

D. Pardo, L. Demkowicz, C. Torres-verdín, and M. Paszynski, Two-dimensional highaccuracy simulation of resistivity logging-while-drilling (LWD) measurements using a selfadaptive goal-oriented hp finite element method, SIAM J. Appl. Math, vol.66, pp.2085-2106, 2006.

D. Pardo, M. Paszynski, C. Torres-verdín, and L. Demkowicz, Simulation of 3D DC borehole resistivity measurements with a goal-oriented hp finite-element method. Part I: Laterolog and LWD, J. Serbian Soc. Comput. Mech, vol.1, pp.62-73, 2007.

D. Pardo, C. Torres-verdín, and L. Demkowicz, Feasibility study for 2D frequencydependent electromagnetic sensing through casing, Geophysics, vol.72, pp.111-118, 2007.

G. Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul, vol.1, pp.485-503, 2003.

J. Schöberl, Commuting Quasi-Interpolation Operators for Mixed Finite Elements, 2001.

F. Simpson and K. Bahr, Practical Magnetotellurics, 2005.

M. J. Wilt, D. L. Alumbaught, H. F. Morrison, A. Becker, K. H. Lee et al., Crosswell electromagnetic tomography: System design considerations and field results, vol.60, pp.871-885, 1995.
DOI : 10.1190/1.1443823