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\title{ Coupling of finite and boundary element methods for  the
transmission problem for the time-harmonic Maxwell equations}
\author{ H.  Ammari}
\address{Center of Applied Mathematics, CNRS UMR 7641,
 Ecole Polytechnique\\
 91128 Palaiseau Cedex, France\\
 E-mail: ammari@cmapx.polytechnique.fr}
\author{
 J.-C.  N\'ed\'elec}
 \address{Center of Applied Mathematics, CNRS UMR 7641,
 Ecole Polytechnique\\
 91128 Palaiseau Cedex, France\\
 E-mail: nedelec@cmapx.polytechnique.fr}

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{\bf Abstract}
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 We describe and analyze a method for computing  
an approximation to the time-harmonic electromagnetic field 
scattered by a bounded dielectric material
surrounding a metallic body. The method is to couple a finite 
element scheme on the dielectric medium with an integral equation on 
its boundary. By proving  that a saddle
point structure holds for the continuous as well as the discretized
problem, we extend the error estimates proved by Bendali for
the scattering problem by a conducting object to the more general
problem of electromagnetics, where we also have dielectric and
metallic objects. Our method is based on the  use of a Hodge
decomposition of the electric field  and a new integral
representation formula for the tangential component of the magnetic
field on the boundary  of the dielectric medium.


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