In this talk, we consider the variant of the classical theory of stochastic homogenization introduced by [Blanc, Le Bris and Lions, CRAS 2006]. We consider a linear elliptic equation in divergence form. The coefficient of the equation is obtained by composing a periodic matrix with a random, rapidly oscillating, diffeomorphism, the gradient of which is stationary. The coefficient of the equation is hence periodic, up to a random change of variables.
In the one-dimensional case, we study how the residu, namely the difference between the exact random solution and the homogenized solution, behaves as epsilon goes to 0.
We next present some numerical approaches for approximating the stochastic problem when the random character is only a perturbation of an underlying deterministic model.
This is joint work with Ronan Costaouec, Claude Le Bris and Florian Thomines.
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