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A numerical approach related to defect-type theories for some weakly random problems in homogenization. (English) Zbl 1233.35014

Summary: We present an approach for computing the homogenized behavior of a medium that is a small random perturbation of a periodic reference material. The random perturbation we consider is, in a sense made precise in our work, a rare event at the microscopic level. It, however, affects the macroscopic properties of the material, and we indeed provide a method to compute the first- and second-order corrections. To this end, we formally establish an asymptotic expansion of the macroscopic properties. Our perturbative approach shares common features with a defect-type theory of solid state physics. The computational efficiency of the approach is demonstrated.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J15 Second-order elliptic equations
35R60 PDEs with randomness, stochastic partial differential equations
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
35C20 Asymptotic expansions of solutions to PDEs