An embedded corrector problem for homogenization. Part I: Theory

This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Cances et al., C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 801-806]), 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. The motivation for considering such embedded corrector problems is made clear in the companion article [Cances et al., J. Comput. Phys., 407 (2020), 109254], where a very efficient algorithm is presented for the resolution of such problems for particular heterogeneous materials. In the present article, 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.