Time-harmonic Maxwell equations in biological cells. The differential form formalism to treat the thin layer

Marc Duruflé (1, 2) , Victor Péron (1, 3) , Clair Poignard (4, 1)

1 IMB - Institut de Mathématiques de Bordeaux
2 BACCHUS - Parallel tools for Numerical Algorithms and Resolution of essentially Hyperbolic problems
3 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
4 MC2 - Modélisation, contrôle et calcul

We study the behavior of the electromagnetic field in a biological cell modelled by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain the first order conditions in a simple way, and numerical simulations validate the theoretical results. Asymptotic transmission conditions at any order are given in the last section of the paper.

Keywords: asymptotic expansion; time-harmonic Maxwell's equations; differential forms on manifolds; finite element method; edge elements

Domains: Analysis of PDEs [math.AP]

Marc Duruflé, Victor Péron, Clair Poignard. Time-harmonic Maxwell equations in biological cells. The differential form formalism to treat the thin layer. Confluentes Mathematici, 2011, 3 (2), pp.325-357. ⟨10.1142/S1793744211000345⟩. ⟨hal-00651510⟩