An embedded corrector problem for homogenization

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Résumé : The objective of this study is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of homogenization. We present three new alternatives for the approximation of the homogenized matrix (tensor) for diffusion problems (linear elasticity resp.) with highly-oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem, where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. We prove that the three different approximations we introduce converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics to compute the resulting matrix-vector 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.