Commun. Comput. Phys., 16 (2014), pp. 213-238.


Thin Layer Models for Electromagnetism

Marc Durufle 1, Victor Peron 2, Clair Poignard 3*

1 Team Magique-3D, INRIA Bordeaux-Sud-Ouest, Institut de Mathematiques de Bordeaux, CNRS UMR 5251 & Universite de Bordeaux1, 351 cours de la Liberation, 33405 Talence Cedex, France.
2 Team Magique-3D, INRIA Bordeaux-Sud-Ouest, LMAP CNRS UMR 5142 & Universite de Pau et des Pays de l'Adour, France.
3 Team MC2, INRIA Bordeaux-Sud-Ouest, Institut de Mathematiques de Bordeaux, CNRS UMR 5251 & Universite de Bordeaux1, 351 cours de la Liberation, 33405 Talence Cedex, France.

Received 12 August 2013; Accepted (in revised version) 10 January 2014
Available online 16 April 2014
doi:10.4208/cicp.120813.100114a

Abstract

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer. These models appear as first order approximations of the electromagnetic field. They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer, that makes possible to replace the thin layer by approximate conditions. We present the advantages and the drawbacks of several approximations together with numerical validations and simulations. The main motivation of this work concerns the computation of electromagnetic field in biological cells. The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane, which provides a specific behavior of the electromagnetic field at low frequencies.

AMS subject classifications: 34E05, 34E10, 58J37

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Key words: Asymptotics, time-harmonic Maxwell's equations, finite element method, edge elements.

*Corresponding author.
Email: marc.durufle@inria.fr (M. Durufle), victor.peron@inria.fr (V. Peron), clair.poignard@inria.fr (C. Poignard)
 

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