Mathematical analysis of a coupling method for the practical computation of homogenized coefficients

Gorynina, Olga; Le Bris, Claude; Legoll, Frédéric

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 44

We present the mathematical study of a computational approach originally introduced by R. Cottereau [Int. J. Numer. Meth. Eng. 95 (2013) 71-90]. The approach aims at evaluating the effective (a.k.a. homogenized) coefficient of a medium with some fine-scale structure. It combines, using the Arlequin coupling method, the original fine-scale description of the medium with an effective description and optimizes upon the coefficient of the effective medium to best fit the response of an equivalent purely homogeneous medium. We prove here that the approach is mathematically well-posed and that it provides, under suitable assumptions, the actual value of the homogenized coefficient of the original medium in the limit of asymptotically infinitely fine structures. The theory presented here therefore usefully complements our numerical developments of Gorynina et al. [SIAM J. Sci. Comput. 43 (2021) A1273-A1304].

Classification : 35J, 35B27, 74Q15
Keywords: Elliptic PDEs, oscillatory coefficients, homogenization, coarse-graining

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