Boundary layer homogenization for p eriodic oscillating boundaries
Abderrahmane Habbal, Pare Valrose · Control and Cybernetics · 2001
The paper is devoted to the study of the boundary layer behaviour of solutions to partial differential equations occur ring in domains with periodic oscillating boundaries, the frequency and the amplitude of the oscillations being the same. First, the transport method, a classical one from the optimal design theory, is used in order to state the problem in a fixed domain; then, an adapted two-scale boundary layer convergence is developed. Apart from this new hybrid approach, the main difference with related works is consideration of a bounded unit-cell, yielding a simple func tional framework. Convergence, as well &'l a homogeni1.1ed equation for the first order boundary layer term are given, and a first order corrector result is proved. This a priori bounding is very well suited to problems of control, and to numerical implementation considera tions. The difficulty in obtaining higher order correctors due to the bounding of the unit-cell is finally discussed. K eywords: PDEs, boundary layer , periodic oscillating bound aries.