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

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 ω^1−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. L ²-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

1991 Mathematics Subject Classifications: :

• 34E05
• 34E10
• 35J05

Acknowledgement

The original author thanks very warmly O. Lafitte for his well-considered advice and suggestions.

Notes

†As we said above, Leontovitch condition has been shown only for , whereas here with γ∈[0,1].