Two-scale homogenization of the Poisson equation with friction boundary condition in a perforated domain

Abstract

We study the homogenization of the Poisson equation in a periodically perforated domain, of period e> 0, with a friction type boundary condition on the holes' boundary. This non-linear condition allows the solution to be non-zero on the periodic boundary if some conditions are satisfied. Using two-scale convergence results we prove that the solution of the mixed variational formulation converges, as e goes to 0, to the solution of a two-scale mixed problem. We also prove that this homogenized problem is well-posed. A numerical test is done, using the Finite Element Method and a quadratic programming algorithm, in order to compare the heterogeneous and homogenized solutions.