. Dans-le-cas-où-l-'objet-diffractant-est-filiforme, on devrait aussi considérer une équation intégrale bidimensionnelle puisqu'en toute rigueur la surface d'un fil est bidimensionnelle. Mais comme l'épaisseur d'un fil est très petite vis-à-vis de la longueur d'onde ( par définition ! ) Pocklington a proposé dans ce cas une équation intégrale 1D posée sur la ligne médiane ( voir, pp.72-73

Y. Achdou, O. Pironneau, and F. Valentin, Effective Boundary Conditions for Laminar Flows over Periodic Rough Boundaries, Journal of Computational Physics, vol.147, issue.1, pp.187-218, 1998.
DOI : 10.1006/jcph.1998.6088

S. Agmon, B. F. Jones, J. With, G. W. Batten, and J. , Lectures on elliptic boundary value problems Prepared for publication, Van Nostrand Mathematical Studies, issue.2, 1965.

Y. Amirat and R. Touzani, ASYMPTOTIC BEHAVIOR OF THE INDUCTANCE COEFFICIENT FOR THIN CONDUCTORS, Mathematical Models and Methods in Applied Sciences, vol.12, issue.02, pp.273-289, 2002.
DOI : 10.1142/S0218202502001647

URL : https://hal.archives-ouvertes.fr/hal-00471580

H. Ammari and J. Nédélec, Full low-frequency asymptotics for the reduced wave equation, Applied Mathematics Letters, vol.12, issue.1, pp.127-131, 1999.
DOI : 10.1016/S0893-9659(98)00137-2

H. Ammari and C. Latiri-grouz, Conditions aux limites approch??es pour les couches minces p??riodiques, ESAIM: Mathematical Modelling and Numerical Analysis, vol.33, issue.4, pp.673-693, 1999.
DOI : 10.1051/m2an:1999157

URL : http://archive.numdam.org/article/M2AN_1999__33_4_673_0.pdf

H. Ammari, M. S. Vogelius, and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations, Journal de Math??matiques Pures et Appliqu??es, vol.80, issue.8, pp.80769-814, 2001.
DOI : 10.1016/S0021-7824(01)01217-X

H. Ammari and D. Volkov, Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter, Asymptot. Anal, vol.30, pp.3-4331, 2002.

M. Artola and M. Cessenat, Diffraction d'une onde électromagnétique par une couche composite mince accolée à un corps conducteur épais. II. Cas d'inclusions (ou d'une couche) de forte permitivité et de forte perméabilité, C. R. Acad. Sci. Paris Sér. I Math, issue.6, pp.313381-385, 1991.

E. Bécache, J. Rodríguez, and C. Tsogka, A Fictitious Domain Method with Mixed Finite Elements for Elastodynamics, SIAM Journal on Scientific Computing, vol.29, issue.3, pp.1244-1267, 2007.
DOI : 10.1137/060655821

A. Bendali and K. Lemrabet, The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation, SIAM Journal on Applied Mathematics, vol.56, issue.6, pp.1664-1693, 1996.
DOI : 10.1137/S0036139995281822

A. Bonnet-bendhia and E. Lunéville, Résolution numérique des équations aux dérivées partielles, Cours ENSTA, 2003.

M. Bourlard, M. Dauge, . Mbaro-saman, S. Lubuma, and . Nicaise, Coefficients of the Singularities for Elliptic Boundary Value Problems on Domains with Conical Points. III: Finite Element Methods on Polygonal Domains, SIAM Journal on Numerical Analysis, vol.29, issue.1, pp.136-155, 1992.
DOI : 10.1137/0729009

M. Bourlard, M. Dauge, and S. Nicaise, Error estimates on the coeficients obtained by the singular function method, Numerical Functional Analysis and Optimization, vol.32, issue.11-12, pp.1077-1113, 1989.
DOI : 10.1007/BF01085326

H. Brezis, Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise, Théorie et applications, 1983.

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, of Springer Series in Computational Mathematics, 1991.
DOI : 10.1007/978-1-4612-3172-1

L. Brillouin, The antenna problem, Quarterly of Applied Mathematics, vol.1, issue.3, pp.201-214, 1943.
DOI : 10.1090/qam/9540

P. Oscar, M. C. Bruno, and . Haslam, Regularity theory and superalgebraic solvers for wire antenna problems, SIAM J. Sci. Comput, vol.29, issue.4, pp.1375-1402, 2007.

G. Caloz, M. Costabel, M. Dauge, and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal, vol.50, issue.1 2, pp.121-173, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00001555

W. Carpes, G. Ferreira, A. Raizer, L. Pichon, and A. Razek, TLM and FEM methods applied in the analysis of electromagnetic coupling, IEEE Transactions on Magnetics, vol.36, issue.4, pp.982-985, 2000.
DOI : 10.1109/20.877606

J. Casado-díaz, M. Luna-laynez, and F. Murat, Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities, Comptes Rendus Mathematique, vol.338, issue.12, pp.338975-980, 2004.
DOI : 10.1016/j.crma.2004.02.020

Z. Chen and X. Yue, Numerical Homogenization of Well Singularities in the Flow Transport through Heterogeneous Porous Media, Multiscale Modeling & Simulation, vol.1, issue.2, pp.260-303, 2003.
DOI : 10.1137/S1540345902413322

P. Ciarlet, J. Jung, S. Kaddouri, S. Labrunie, and J. Zou, The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains, Numerische Mathematik, vol.10, issue.3, pp.423-450, 2005.
DOI : 10.1007/s00211-005-0621-6

URL : https://hal.archives-ouvertes.fr/hal-00094285

P. G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.40, 2002.

A. Cimetière, G. Geymonat, H. L. Dret, A. Raoult, and Z. Tutek, Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods, Journal of Elasticity, vol.18, issue.18, pp.111-161, 1988.
DOI : 10.1007/BF00040890

X. Claeys, H. Haddar, and P. Joly, Etude d'un problème modèle pour la diffraction par des fils minces par développements asymptotiques raccordés cas 2d, 2007.

F. Collino and F. Millot, Fils et méthodes d' éléments finis pour les équations de maxwell. le modèle de holland revisit'e, 1998.

F. Collino, P. Joly, and F. Millot, Fictitious Domain Method for Unsteady Problems:, Journal of Computational Physics, vol.138, issue.2, pp.907-938, 1997.
DOI : 10.1006/jcph.1997.5849

URL : https://hal.archives-ouvertes.fr/inria-00073735

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, of Applied Mathematical Sciences, 1998.

L. David, R. Colton, and . Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics, 1983.

M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Comm. Partial Differential Equations, vol.21, issue.1112, pp.1919-1949, 1996.

M. Costabel and M. Dauge, Singularities of Electromagnetic Fields??in Polyhedral Domains, Archive for Rational Mechanics and Analysis, vol.151, issue.3, pp.221-276, 2000.
DOI : 10.1007/s002050050197

M. Dambrine and G. Vial, A multiscale correction method for local singular perturbations of the boundary, ESAIM: Mathematical Modelling and Numerical Analysis, vol.41, issue.1, pp.111-127, 2007.
DOI : 10.1051/m2an:2007012

URL : https://hal.archives-ouvertes.fr/hal-00085376

M. Dauge, Elliptic boundary value problem on corner domains, 1988.
DOI : 10.1007/BFb0086682

M. Dauge, S. Nicaise, M. Bourlard, J. Mbaro-saman, and . Lubuma, Coefficients des singularit??s pour des probl??mes aux limites elliptiques sur un domaine ?? points coniques. II : Quelques op??rateurs particuliers, ESAIM: Mathematical Modelling and Numerical Analysis, vol.24, issue.3, pp.343-367, 1990.
DOI : 10.1051/m2an/1990240303431

M. Dauge, S. Nicaise, M. Bourlard, and J. Lubuma, Coefficients des singularit??s pour des probl??mes aux limites elliptiques sur un domaine ?? points coniques. I : R??sultats g??n??raux pour le probl??me de Dirichlet, ESAIM: Mathematical Modelling and Numerical Analysis, vol.24, issue.1, pp.27-52, 1990.
DOI : 10.1051/m2an/1990240100271

R. Dautray and J. Lions, Mathematical analysis and numerical methods for science and technology Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, 1990.

P. J. Davies, D. B. Duncan, and S. A. Funken, Accurate and Efficient Algorithms for Frequency Domain Scattering from a Thin Wire, Journal of Computational Physics, vol.168, issue.1, pp.155-183, 2001.
DOI : 10.1006/jcph.2000.6688

M. Perdigão and . Carmo, Riemannian geometry Mathematics : Theory & Applications, Birkhäuser Boston Inc, 1992.

M. Durufle, Intégration Numérique et éléments finis d'ordre élevé appliqués aux équations de Maxwell en régime harmonique, 2006.

F. Edelvik, A new technique for accurate and stable modeling of arbitrarily oriented thin wires in the fdtd method, IEEE Transactions on Electromagnetic Compatibility, vol.45, issue.2, pp.416-423, 2003.
DOI : 10.1109/TEMC.2003.811294

B. Engquist and J. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers, Ecole Polytechnique (France), 1993.

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp, issue.139, pp.31629-651, 1977.

E. Faou, Développements asymptotiques dans les coques minces linéairement élastiques, 2000.

M. V. Fedoryuk, Asymptotics of the solution of the dirichlet problem for the laplace and helmholtz equations in the exterior of a slender cylinder, Izv. Akad. Nauk SSSR Ser. Mat, 1981.

M. V. Fedoryuk, The Dirichlet problem for the Laplace operator in the exterior of a thin body of revolution, Theory of Cubature Formulas and the Applications of Functionnal Analysis to Problems of Mathematical Physics, number 2 in 126, 1985.
DOI : 10.1090/trans2/126/07

R. R. Gadyl and ?. Shin, The method of matching asymptotic expansions in a singularly perturbed boundary value problem for the Laplace operator, Sovrem. Mat. Prilozh. Asimptot. Metody Funkts. Anal, issue.5, pp.3-32, 2003.

S. Garces, Une méthode de domaines fictifs pour la modélisation des structures rayonnantes tridimensionnelles Ecole nationale supérieure de l'aéronautique et de l'espace, 1997.

J. Geer, The Scattering of a Scalar Wave by a Slender Body of Revolution, SIAM Journal on Applied Mathematics, vol.34, issue.2, pp.348-370, 1978.
DOI : 10.1137/0134029

J. Geer, Electromagnetic Scattering by a Slender Body of Revolution: Axially Incident Plane Wave, SIAM Journal on Applied Mathematics, vol.38, issue.1, pp.93-102, 1980.
DOI : 10.1137/0138007

V. Girault and R. Glowinski, Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan Journal of Industrial and Applied Mathematics, vol.33, issue.3, pp.487-514, 1995.
DOI : 10.1007/BF03167240

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

D. Givoli, Non-reflecting boundary conditions, Journal of Computational Physics, vol.94, issue.1, pp.1-29, 1991.
DOI : 10.1016/0021-9991(91)90135-8

R. Glowinski and S. Lapin, Solution of a Wave Equation by a Mixed Finite Element - Fictitious Domain Method, Computational Methods in Applied Mathematics, vol.4, issue.4, pp.431-444, 2004.
DOI : 10.2478/cmam-2004-0024

R. Glowinski, T. Pan, and J. Périaux, Fictitious domain/domain decomposition methods for partial differential equations. In Domain-based parallelism and problem decomposition methods in computational science and engineering, SIAM, pp.177-192, 1995.

I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators, of Operator Theory : Advances and Applications, 1990.

P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol.24, 1985.
DOI : 10.1137/1.9781611972030

H. Haddar, Asymptotic models in ferromagnetism : thin layer approximations and homogeneization, 2000.

H. Haddar, P. Joly, and H. Nguyen, GENERALIZED IMPEDANCE BOUNDARY CONDITIONS FOR SCATTERING BY STRONGLY ABSORBING OBSTACLES: THE SCALAR CASE, Mathematical Models and Methods in Applied Sciences, vol.15, issue.08, pp.1273-1300, 2005.
DOI : 10.1142/S021820250500073X

URL : https://hal.archives-ouvertes.fr/hal-00743895

H. Haddar, P. Joly, and H. Nguyen, GENERALIZED IMPEDANCE BOUNDARY CONDITIONS FOR SCATTERING BY STRONGLY ABSORBING OBSTACLES: THE SCALAR CASE, Mathematical Models and Methods in Applied Sciences, vol.15, issue.08, pp.1273-1300, 2005.
DOI : 10.1142/S021820250500073X

URL : https://hal.archives-ouvertes.fr/hal-00743895

A. Richard, J. B. Handelsman, and . Keller, Axially symmetric potential flow around a slender body, J. Fluid Mech, vol.28, pp.131-147, 1967.

E. Heikkola, Y. A. Kuznetsov, and K. N. Lipnikov, FICTITIOUS DOMAIN METHODS FOR THE NUMERICAL SOLUTION OF THREE-DIMENSIONAL ACOUSTIC SCATTERING PROBLEMS, Journal of Computational Acoustics, vol.07, issue.03, pp.161-183, 1999.
DOI : 10.1142/S0218396X99000126

E. Heikkola, Y. A. Kuznetsov, P. Neittaanmäki, and J. Toivanen, Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems, Journal of Computational Physics, vol.145, issue.1, pp.89-109, 1998.
DOI : 10.1006/jcph.1998.6014

E. W. Hobson, The theory of spherical and ellipsoidal harmonics, 1955.

R. Holland and L. Simpson, Finite-Difference Analysis of EMP Coupling to Thin Struts and Wires, IEEE Transactions on Electromagnetic Compatibility, vol.23, issue.2, 1981.
DOI : 10.1109/TEMC.1981.303899

. Il-'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, volume 102 of Translation of Mathematical Monographs, 1992.

A. M. Il, ?. In, A. R. Danilin, and S. V. Zakharov, Application of the method of matching asymptotic expansions to the solution of boundary value problems, Sovrem. Mat. Prilozh. Asimptot. Metody Funkts. Anal, issue.5, pp.33-78, 2003.

E. Jamelot, Résolution des équations de Maxwell avec des éléments finis de Galerkin continus, 2005.

P. Jeanquartier, Transformation de Mellin et développements asymptotiques, Enseign. Math, vol.25, issue.23-4, pp.285-308, 1979.

P. Joly and S. Tordeux, Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.1, pp.63-97, 2006.
DOI : 10.1051/m2an:2006008

URL : https://hal.archives-ouvertes.fr/inria-00527590

D. S. Jones, A thin Wire Approximation, SIAM Journal on Applied Mathematics, vol.50, issue.2, pp.547-558, 1990.
DOI : 10.1137/0150033

D. S. Jones, Methods in electromagnetic wave propagation, volume 40 of Oxford Engineering Science Series, 1994.

D. S. Jones, Note on the integral equation for a straight wire antenna, IEE Proceedings H Microwaves, Optics and Antennas, vol.128, issue.2, pp.114-130, 1981.
DOI : 10.1049/ip-h-1.1981.0018

R. Kleinman and B. Va?-inberg, Full low-frequency asymptotic expansion for second-order elliptic equations in two dimensions, Mathematical Methods in the Applied Sciences, vol.8, issue.12, pp.989-1004, 1994.
DOI : 10.1002/mma.1670171207

V. A. Kondrat-'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Ob??, vol.16, pp.209-292, 1967.

V. A. Kozlov, V. G. Maz, ?. , and J. Rossmann, Elliptic boundary value problems in domains with point singularities, volume 52 of Mathematical Surveys and Monographs, 1997.

V. Kozlov, V. Maz, and ?. Ya, Differential equations with operator coefficients with applications to boundary value problems for partial differential equations, 1999.

Y. A. Kuznetsov and K. N. Lipnikov, The method of fictitious domains for solving the Helmholtz wave equation in an unbounded domain, Numerical methods and mathematical modeling (Russian ), pp.56-70, 1992.

J. Lang, H. Shuanshui, and S. Kemin, Separation of the helmholtz equation in prolate spheroidal coordinates, Journal of applied physics, vol.56, issue.5, pp.1532-1535, 1984.

S. Lang, Fundamentals of differential geometry, volume 191 of Graduate Texts in Mathematics, 1999.

J. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, 1968.

L. Alexandre, F. Madureira, and . Valentin, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal, vol.38, issue.5, pp.1450-147307, 2006.

G. F. Maslennikova, A neumann problem for the helmholtz operator in the exterior to a thin body of revolution. Differential equations, pp.316-324, 1984.

A. Mazari, Détermination par une méthode d'équations intégrales du champ électromagnétique rayonné par une structure filiforme, 1991.

V. Maz-'ya, S. Nazarov, and B. Plamenevskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol II. Number 111 in Operator theory :Advances and Applications, 2000.

J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete , Band LXXI, 1954.

P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation, 2003.

S. A. Nazarov, Averaging of boundary value problems in a domain that contains a thin cavity with a periodically changing cross section, Trudy Moskov. Mat. Obshch, vol.53, pp.98-129, 1990.

J. Nédélec, Acoustic and Electromagnetic Equations, 2001.

J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Mathematics of Computation, vol.28, issue.128, pp.937-958, 1974.
DOI : 10.1090/S0025-5718-1974-0373325-9

J. Nédélec, Mixed finite elements in ?3, Numerische Mathematik, vol.12, issue.3, pp.315-356, 1980.
DOI : 10.1007/BF01396415

F. W. Olver, Asymptotics and special functions, Computer Science and Applied Methematics, 1974.

J. Paul, C. Christopoulos, D. Thomas, and X. Liu, Time-Domain Modeling of Electromagnetic Wave Interaction With Thin-Wires Using TLM, IEEE Transactions on Electromagnetic Compatibility, vol.47, issue.3, pp.447-455, 2005.
DOI : 10.1109/TEMC.2005.852217

H. C. Pocklington, Electrical oscillations in wires. Pro. of the Cambridge Philosophical Society, 1897.

P. Raviart and J. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson, 1983.

F. Rogier, Problèmes mathématiques et numériques lies à l'approximation de la géométrie d'un corps diffractant dans les équations de l'électromagnétisme, 1989.

W. Rudin, Real and complex analysis, 1966.

W. Rudin, Functional analysis. International Series in Pure and Applied Mathematics, 1991.

B. P. Rynne, The well-posedness of the integral equations for thin wire antennas, IMA Journal of Applied Mathematics, vol.49, issue.1, 1992.
DOI : 10.1093/imamat/49.1.35

B. P. Rynne, On the Well-Posedness of Pocklington's Equation for a Straight Wire Antenna and Convergence of Numerical Solutions, Journal of Electromagnetic Waves and Applications, vol.12, issue.11, pp.1489-1503, 2000.
DOI : 10.1163/156939300X00257

A. Sellier, Asymptotic solution for the electrostatic field around a slender conducting body, IMA Journal of Applied Mathematics, vol.62, issue.2, pp.167-193, 1999.
DOI : 10.1093/imamat/62.2.167

A. Sellier, Asymptotic solution of 2D and 3D boundary integral equations arising in Fluid Mechanics and Electrostatics, Computational Mechanics, vol.25, issue.6, pp.600-612, 2000.
DOI : 10.1007/s004660050507

URL : https://hal.archives-ouvertes.fr/hal-01025357

T. B. Senior and J. Volakis, Approximate boundary conditions in electromagnetics. IEE Electromagnetic waves series, 1995.

S. Steinberg, Meromorphic families of compact operators, Archive for Rational Mechanics and Analysis, vol.124, issue.5, pp.372-379, 1968.
DOI : 10.1007/BF00251419

S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes, 2004.

S. Tordeux, G. Vial, and M. Dauge, Matching and multiscale expansions for a model singular perturbation problem, Comptes Rendus Mathematique, vol.343, issue.10, pp.343637-642, 2006.
DOI : 10.1016/j.crma.2006.10.010

URL : https://hal.archives-ouvertes.fr/hal-00453366

K. Umashankar, A. Taflove, and B. Beker, Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity, IEEE Transactions on Antennas and Propagation, vol.35, issue.11, pp.351248-1257, 1987.
DOI : 10.1109/TAP.1987.1144000

M. Van-dyke, Perturbation Method in Fluid Mechanics, Journal of Applied Mechanics, vol.43, issue.1, 1964.
DOI : 10.1115/1.3423785

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

S. Vial and G. Tordeux, Matching of asymptotic expansions and multiscale expansion for the rounded corner problem, 2006.

N. Ja, Vilenkin. Special functions and the theory of group representations. Translated from the Russian by, Translations of Mathematical Monographs, vol.22, 1968.

C. Wagschal, Dérivation, intégration. Collection Méthodes, 1999.

G. Watson, A Treatise on the Theory of Bessel Functions, The Mathematical Gazette, vol.18, issue.231, 1944.
DOI : 10.2307/3605513

S. Zhang and J. Jin, Computation of special functions. A Wiley-Interscience Publication, 1996.

G. V. Zhdanova, Dirichlet problem for the helmholtz operator in the exterior of a thin body of revolution, Differential Equations, vol.20, issue.8, pp.1403-1411, 1984.