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Homogenization in a class of non-periodically perforated domains

Sylvain Wolf

Dienstag, 15. Juni 2021 14:15Uhr

We consider the deterministic homogenization of the Poisson problem and the Stokes system in a class of non-periodically perforated domains. The size of the perforations is comparable to the distance between two neighbouring holes. The boundary conditions for both problems are of homogeneous Dirichlet type along the holes and the macroscopic boundary. The homogenization of these PDEs when the holes are periodically distributed in space is well-known. We aim at extending these results to local perturbations of the periodic case, that is when the geometry is not periodic but tends to be periodic far from the origin. This setting takes into account local defects that could appear in a pure periodic microstructure. In this talk, we first introduce the conditions imposed on the non-periodic porous medium. We then construct classical objects of the homogenization such as correctors and we obtain convergence rates of the solution to its two scale expansion for both Poisson problem and Stokes system. We finally comment on the optimality of these convergence rates.