Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

Abstract

International audienceWe study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi $\eps$--periodic function, $\eps$ being a small parameter, while the mean thickness of the layer is of magnitude $\eps^\beta$, where $\beta\in(0,1)$. Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and Ciuperca \textsl{et al.} (Research report INRIA RR-6812), in which $\beta\geq1$

Influence of a Rough Thin Layer on the Potential

“…2 The notation means that goes to zero faster than as goes to zero. We refer to [2]- [4] for a precise description of the involved norms and the accuracy of the convergence.…”

“…According to [4], satisfies the following problem: (5) and the following transmission conditions on :…”

“…We denote by the solution to Problem (2). 4 The characteristic function of a set is the function equal to 1 if and 0 if . …”

Explicit characterization of the polarization tensor for rough thin layers

“…According to (3.1) the polarization tensor is a full 2 × 2 matrix, whereas for smooth and for very rough thin layers the respective polarization tensors are diagonal 2 × 2 matrices [1,3,4]. This feature can be seen as a coupling between the thickness of the layer and the period of the oscillations that are in the same order of magnitude.…”

Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem

“…We aim at revisiting these results using a general framework that allows to treat similarly the three cases α < 1, α = 1 and α > 1. We emphasize that for α > 1 only weak results have been obtained in [13] by using two-scale convergence techniques, in the sense that no error estimate has been provided for the approximation. In this paper we push forward the analysis by defining the boundary layer corrector for α > 1 and by proving error estimates for α ∈ (0, 2): far from the layer, we recover the results proved in [13], while in the neighborhood of the roughness the boundary layer corrector provides an accurate description of the potential.…”