Mathematical and numerical approaches for multiscale problems
Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem
We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be –periodic, being the magnitude of the mean thickness of the layer, and a positive parameter describing the degree of roughness. For tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent...
Numerical homogenization: survey, new results, and perspectives
These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the...
A brief introduction to homogenization and miscellaneous applications
This paper is a set of lecture notes for a short introductory course on homogenization. It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the oscillating test function method and two-scale convergence) and briefly describes the main results of the more general theory of − or −convergence. Several applications of the method are given: derivation of Darcy’s law for flows in porous media, derivation of the porosity...
Numerical study of acoustic multiperforated plates
It is rather classical to model multiperforated plates by approximate impedance boundary conditions. In this article we would like to compare an instance of such boundary conditions obtained through a matched asymptotic expansions technique to direct numerical computations based on a boundary element formulation in the case of linear acoustic.
Wall laws for viscous fluids near rough surfaces
In this paper, we review recent results on wall laws for viscous fluids near rough
surfaces, of small amplitude and wavelength
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