Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

Author: Poignard, Clair

Source: Applicable Analysis, Volume 86, Number 12, December 2007 , pp. 1549-1568(20)

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Abstract:

Consider a conducting disk surrounded by a thin dielectric layer submitted to an electric field at the pulsation ω. The conductivity of the layer grows like ω1-γ, γ∈[0,1], when the pulsation ω tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane in the high frequency limit for small thickness of the layer. L2-error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau (Lafitte O. Lebeau G. 1993, Équations de Maxwell et opérateur d'impédance sur le bord d'un obstacle convexe absorbant. Comptes Rendus de l ' Académic dis Science, Paris, Série I, Mathématiques, 316(11), 1177-1182); (Lafitte O.D., 1999, Diffraction in the high frequency regime by a thin layer of dielectric material. I. The equivalent impedance boundary condition. SIAM Journal on Applied Mathematics, 59(3), 1028-1052 (electronic)) in which γ identically equals zero.

Keywords: Asymptotics; Helmholtz equation; Thin layer; Generalized impedance boundary conditions; Fourier analysis

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00036810701714172

Affiliations: 1: Centre de Mathématiques Appliquées, 91128 Palaiseau Cedex, France

Publication date: 2007-12-01

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