B. Le-cas, 152 6.2.1 Résultats numériques pour la méthode numérique associée au premier modèle approché, p.153

M. Ainsworth, P. Davies, and D. Duncan, Topics in computational wave propagation : direct and inverse problems. Lecture notes in computational science and engineering, 2003.
DOI : 10.1007/978-3-642-55483-4

G. Allaire, Analyse numérique et optimisation : une introduction à la modélisation mathématique et à la simulation numérique, 2005.

H. Ammari, P. Calmon, and E. Iakovleva, Direct Elastic Imaging of a Small Inclusion, SIAM Journal on Imaging Sciences, vol.1, issue.2, pp.169-187, 2008.
DOI : 10.1137/070696076

URL : https://hal.archives-ouvertes.fr/hal-00654620

H. Ammari, H. Kang, and M. Lim, Polarization tensors and their applications, Journal of Physics: Conference Series, vol.12, issue.1, p.13, 2005.
DOI : 10.1088/1742-6596/12/1/002

P. F. Antonietti, I. Mazzieri, A. Quarteroni, and F. Rapetti, Non-conforming high order approximations of the elastodynamics equation, Computer Methods in Applied Mechanics and Engineering, vol.209, issue.212, pp.209-212212, 2012.
DOI : 10.1016/j.cma.2011.11.004

D. N. Arnold, Discretization by finite elements of a model parameter dependent problem, Numerische Mathematik, vol.5, issue.41, pp.405-421, 1981.
DOI : 10.1007/BF01400318

F. Assous, P. Ciarlet, J. , S. Labrunie, and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, Journal of Computational Physics, vol.191, issue.1, pp.147-176, 2003.
DOI : 10.1016/S0021-9991(03)00309-7

URL : https://hal.archives-ouvertes.fr/hal-00989621

A. Avantaggiati, On compact embedding theorems in weighted Sobolev spaces, Czechoslovak Math. J, vol.29, issue.1044, pp.635-648, 1979.

I. Babuska and M. Suri, Locking effects in the finite element approximation of elasticity problems, Numerische Mathematik, vol.7, issue.1, pp.439-463, 1992.
DOI : 10.1007/BF01396238

I. Babuska and M. Suri, On Locking and Robustness in the Finite Element Method, SIAM Journal on Numerical Analysis, vol.29, issue.5, pp.1261-1293, 1992.
DOI : 10.1137/0729075

C. Bellis, M. Bonnet, and F. Cakoni, cost functionals, Inverse Problems, vol.29, issue.7, p.75012, 2013.
DOI : 10.1088/0266-5611/29/7/075012

URL : https://hal.archives-ouvertes.fr/hal-01323305

A. Bendali, P. Cocquet, and S. Tordeux, Scattering of a scalar time-harmonic wave by N small spheres by the method of matched asymptotic expansions, Numerical Analysis and Applications, vol.5, issue.2, pp.116-123, 2012.
DOI : 10.1134/S1995423912020036

URL : https://hal.archives-ouvertes.fr/hal-00760457

A. Bendali, P. Cocquet, and S. Tordeux, Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering, Archive for Rational Mechanics and Analysis, vol.55, issue.12, 2014.
DOI : 10.1007/s00205-015-0915-5

URL : https://hal.archives-ouvertes.fr/hal-01258966

A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a timeharmonicwave for the helmholtz equation, SIAM Journal of Applied Mathematics, vol.6, issue.5, pp.1664-1693, 1996.

A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a timeharmonicwave by a perfectly conductingmetal coated with a thin dielectric shell, Asymptotic Analysis, vol.57, pp.199-227, 2008.

J. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, vol.114, issue.2, pp.185-200, 1994.
DOI : 10.1006/jcph.1994.1159

V. Bonnaillie-noel, M. Dambrine, S. Tordeux, and G. Vial, INTERACTIONS BETWEEN MODERATELY CLOSE INCLUSIONS FOR THE LAPLACE EQUATION, Mathematical Models and Methods in Applied Sciences, vol.19, issue.10, 2009.
DOI : 10.1142/S021820250900398X

URL : https://hal.archives-ouvertes.fr/hal-00455312

V. Bonnaillie-noël, M. Dambrine, S. Tordeux, and G. Vial, On moderately close inclusions for the Laplace equation, Comptes Rendus Mathematique, vol.345, issue.11, pp.345609-614, 2007.
DOI : 10.1016/j.crma.2007.10.037

S. Boyd and L. Vandenberghe, Convex Optimization, 2004.

H. Brezis, Analyse fonctionnelle. Théorie et applications, 1983.

E. Bécache and P. Joly, On the analysis of B??renger's Perfectly Matched Layers for Maxwell's equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.36, issue.1, pp.87-119
DOI : 10.1051/m2an:2002004

E. Bécache, J. Rodríguez, and C. Tsogka, Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition, ESAIM : M2AN, pp.377-398, 2009.
DOI : 10.1051/m2an:2008047

É. Bécache, P. Joly, M. Kachanovska, and V. Vinoles, Perfectly matched layers in negative index metamaterials and plasmas, ESAIM: Proceedings and Surveys, vol.50, pp.113-132, 2015.
DOI : 10.1051/proc/201550006

Y. Capdeboscq, A. Karrman, and J. Nédélec, Numerical computation of approximate generalized polarization tensors, Applicable Analysis, vol.144, issue.6, pp.1189-1203, 2012.
DOI : 10.1017/CBO9780511613357

URL : https://hal.archives-ouvertes.fr/hal-00770191

M. Cassier and C. Hazard, Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: Mathematical justification of the Foldy???Lax model, Wave Motion, vol.50, issue.1, pp.18-28, 2013.
DOI : 10.1016/j.wavemoti.2012.06.001

URL : https://hal.archives-ouvertes.fr/hal-00849557

J. Chabassier and S. Imperiale, Introduction and study of fourth order theta schemes for linear wave equations, Journal of Computational and Applied Mathematics, vol.245, pp.194-212, 2013.
DOI : 10.1016/j.cam.2012.12.023

URL : https://hal.archives-ouvertes.fr/hal-01051803

L. Chen, T. Rabczuk, S. P. Bordas, G. R. Liu, K. Y. Zeng et al., Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth, Computer Methods in Applied Mechanics and Engineering, vol.209, issue.212, pp.250-265, 2012.
DOI : 10.1016/j.cma.2011.08.013

P. Ciarlet, J. , and S. Labrunie, Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl, vol.3, issue.1, pp.113-155, 2011.

P. G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, 1978.

X. Claeys, Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces, 2008.

X. Claeys and F. Collino, Augmented Galerkin schemes for the numerical solution of scattering by small obstacles, Numerische Mathematik, vol.17, issue.12, pp.243-268, 2010.
DOI : 10.1007/s00211-010-0301-z

URL : https://hal.archives-ouvertes.fr/hal-00976812

G. Cohen, Higher-order numerical methods for transient wave equations, 2001.
URL : https://hal.archives-ouvertes.fr/hal-01166961

G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Lecture Notes in Economic and Mathematical Systems, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01166961

G. Cohen and S. Fauqueux, MIXED FINITE ELEMENTS WITH MASS-LUMPING FOR THE TRANSIENT WAVE EQUATION, Journal of Computational Acoustics, vol.08, issue.01, pp.171-188, 2000.
DOI : 10.1142/S0218396X0000011X

G. Cohen, P. Joly, J. E. Roberts, and N. Tordjman, Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation, SIAM Journal on Numerical Analysis, vol.38, issue.6, pp.2047-2078, 2001.
DOI : 10.1137/S0036142997329554

URL : https://hal.archives-ouvertes.fr/hal-01010373

F. Collino, T. Fouquet, and P. Joly, A conservative space-time mesh refinement method for the 1-d wave equation. part i : Construction, Numerische Mathematik, vol.95, issue.2, pp.197-221, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00989055

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Number 93 in Series of Applied Mathematics, 1992.

B. Delourme, H. Haddar, and P. Joly, ON THE WELL-POSEDNESS, STABILITY AND ACCURACY OF AN ASYMPTOTIC MODEL FOR THIN PERIODIC INTERFACES IN ELECTROMAGNETIC SCATTERING PROBLEMS, Mathematical Models and Methods in Applied Sciences, vol.23, issue.13, pp.12-2013
DOI : 10.1142/S021820251350036X

URL : https://hal.archives-ouvertes.fr/hal-00682357

J. Diaz and M. Grote, Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations, SIAM Journal on Scientific Computing, vol.31, issue.3, pp.1985-2014, 2009.
DOI : 10.1137/070709414

URL : https://hal.archives-ouvertes.fr/inria-00193160

D. G. Duffy, Green's Functions with Applications, Applied Mathematics, vol.38, 2001.
DOI : 10.1201/9781420034790

L. C. Evans, Partial Differential Equations. Graduate studies in mathematics, 2010.

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, vol.144, issue.1-2, pp.3-58, 1995.
DOI : 10.1016/0304-3975(95)00002-E

URL : https://hal.archives-ouvertes.fr/inria-00074307

L. L. Foldy, The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers, Physical Review, vol.67, issue.3-4, pp.107-119, 1945.
DOI : 10.1103/PhysRev.67.107

W. Gander and W. Gautschi, Adaptive quadrature?revisited, Bit Numerical Mathematics, vol.40, issue.1, pp.84-101, 2000.
DOI : 10.1023/A:1022318402393

T. Geveci, On the application of mixed finite element methods to the wave equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.22, issue.2, pp.243-250, 1988.
DOI : 10.1051/m2an/1988220202431

V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations Theory and algorithms, of Springer Series in Computational Mathematics, 1986.

J. Giroire, Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales, Thèse de doctorat dirigée par Raviart, Pierre- Arnaud Sciences, 1987.

R. Glowinski, T. Pan, and J. Periaux, A fictitious domain method for Dirichlet problem and applications, Computer Methods in Applied Mechanics and Engineering, vol.111, issue.3-4, pp.283-303, 1994.
DOI : 10.1016/0045-7825(94)90135-X

H. Ammari, H. Kang, and K. K. , Polarization tensors and effective properties of anisotropic composite materials, Journal of Differential Equations, vol.215, issue.2, pp.401-428, 2005.
DOI : 10.1016/j.jde.2004.09.010

J. Hadamard, Sur les problèmes aux dérivés partielles et leur signification physique, Princeton University Bulletin, vol.13, pp.49-52, 1902.

S. Ham and K. Bathe, A finite element method enriched for wave propagation problems, Computers & Structures, vol.94, issue.95, pp.94-951, 2012.
DOI : 10.1016/j.compstruc.2012.01.001

URL : https://hal.archives-ouvertes.fr/hal-01403646

P. Holnicki, On L2-estimates for discrete-time galerkin approximation to second order hyperbolic equations, Mathematics and Computers in Simulation, vol.23, issue.2, pp.127-132, 1981.
DOI : 10.1016/0378-4754(81)90049-5

K. Huang, K. Solna, and H. Zhao, Generalized Foldy-Lax formulation, Journal of Computational Physics, vol.229, issue.12, pp.4544-4553, 2010.
DOI : 10.1016/j.jcp.2010.02.021

S. Imperiale, Modélisation mathématique et numérique de capteurs piézoélectriques, p.2012

S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, Applied Numerical Mathematics, vol.79, 2013.
DOI : 10.1016/j.apnum.2013.03.011

URL : https://hal.archives-ouvertes.fr/hal-00875811

S. Imperiale, S. Marmorat, N. Leymarie, and S. Chatillon, A complete FE simulation tools for NDT inspections with piezoelectric transducers, Acoustics 2012, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00811151

P. Jeanquartier, Transformation de mellin et développements asymptotiques, Enseignement Mathématique, vol.25, p.285, 1979.

P. Joly, Analyse et approximation de modèles de propagation d'ondes, partie 1, 2001.

P. Joly and A. Semin, Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots, ESAIM: Proceedings, vol.25, issue.12, pp.44-67, 2008.
DOI : 10.1051/proc:082504

URL : https://hal.archives-ouvertes.fr/hal-00976404

P. Joly and S. Tordeux, Matching of asymptotic expansions for wave propagation in media with thin slots i : The asymptotic expansion. Multiscale Modeling and Simulation : A, SIAM Interdisciplinary Journal, vol.5, issue.1, pp.304-336, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00527588

S. Karaa, Abstract, Advances in Applied Mathematics and Mechanics, vol.22, issue.01, pp.181-203, 2011.
DOI : 10.4208/aamm.10-m1018

J. B. Keller, Removing Small Features from Computational Domains, Journal of Computational Physics, vol.113, issue.1, pp.148-150, 1994.
DOI : 10.1006/jcph.1994.1124

V. A. Kozlov, V. G. Maz-'ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs . AMS, vol.52, 1997.
DOI : 10.1090/surv/052

A. Kufner and A. Sändig, Some applications of weighted Sobolev spaces, 1987.

H. Liang, M. Z. Liu, and W. Lv, Stability of <mml:math altimg="si44.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>??</mml:mi></mml:math>-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments, Applied Mathematics Letters, vol.23, issue.2, pp.198-206, 2010.
DOI : 10.1016/j.aml.2009.09.012

V. Mazya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, 2000.

V. G. Maz-'ya, S. A. Nazarov, and B. A. Plamenevskï, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, 2000.

V. G. Maz-'ya, S. A. Nazarov, B. A. Plamenevskï, G. Heinig, and C. Posthoff, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Volume I, volume 111 of Operator theory, advances and applications, 2000.

G. E. Moore, Cramming More Components Onto Integrated Circuits, Proceedings of the IEEE, vol.86, issue.1, 1938.
DOI : 10.1109/JPROC.1998.658762

S. Nazarov, Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions, Proceedings of the St, pp.77-125, 1999.
DOI : 10.1090/trans2/193/05

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

R. L. Pitangueira and R. R. Silva, Numerical Characterization of Concrete Heterogeneity, Materials Research, vol.5, issue.3, pp.309-314, 2002.
DOI : 10.1590/S1516-14392002000300015

Q. Qi and T. L. Geers, Evaluation of the Perfectly Matched Layer for Computational Acoustics, Journal of Computational Physics, vol.139, issue.1, pp.166-183, 1998.
DOI : 10.1006/jcph.1997.5868

P. A. Raviart and J. M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles, 2004.

S. Rienstra and A. Hirschberg, An introduction to acoustics, p.19, 2003.

W. Rudin, Functional Analysis. International series in pure and applied mathematics, 1991.

K. Schmidt and A. Chernov, A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model, SIAM Journal on Applied Mathematics, vol.73, issue.6, pp.731980-2003, 2013.
DOI : 10.1137/120901398

K. Schmidt and R. Hiptmair, Asymptotic boundary element methods for thin conducting sheets, Discrete Contin. Dyn. Syst. Ser. S, vol.8, issue.3, pp.619-647, 2015.

L. Slimane and Y. Renard, The treatment of the locking phenomenon for a general class of variational inequalities, Journal of Computational and Applied Mathematics, vol.170, issue.1, pp.121-143, 2004.
DOI : 10.1016/j.cam.2003.12.044

URL : https://hal.archives-ouvertes.fr/hal-00690589

F. Trèves, Basic Linear Partial Differential Equations, 1975.

F. Trèves, Topological vector spaces, distributions and kernels, 2006.

G. Vial, Analyse multi-échelle et conditions aux limites approchées pour un prolème avec couche mince dans un domaine à coin, 2003.

E. Weinan, Principles of Multiscale Modeling, 2011.