An embedded corrector problem for homogenization. Part II: Algorithms and discretization Abstract   This contribution is the numerically oriented companion article of the work  9 . We focus here on the numerical resolution of the embedded corrector problem introduced in  8 ,  9  in the context of homogenization of diffusion equations. Our approach consists in considering a correctortype problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In  9 , we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problem. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases.  We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrixvector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.

Stochastic homogenization of multicontinuum heterogeneous flows We consider a multicontinuum model in porous media applications, which is described as a system of coupled flow equations. The coupling between different continua depends on many factors and its modeling is important for porous media applications. The coefficients depend on particle deposition that is described in term of a stochastic process solution of an SDE. The stochastic process is considered to be faster than the flow motion and we introduce timespace scales to model the problem. Our goal is to pass to the limit in time and space and to find an associated averaged system. This is an averaginghomogenization problem, where the averages are computed in terms of the invariant measure associated to the fast motion and the spatial variable. We use the techniques developed in our previous paper to model the interactions between the continua and derive the averaged model problem that can be used in many applications.