2011/02/18 by Ronan Costaouec, Costaouec, Ronan
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA) #Scientific Research and Discoveries
paper · pdf · doi:10.48550/arxiv.1102.3804
openalex publication_date 2011/02/18 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
This article studies some numerical approximations of the homogenized matrix\nfor stochastic linear elliptic partial differential equations in divergence\nform. We focus on the case when the underlying random field is a small\nperturbation of a reference periodic tensor. The size of such a perturbation is\nencoded by a real parameter eta. In this case, it has already been\ntheoretically shown in the literature that the exact homogenized matrix\npossesses an expansion in powers of the parameter eta for both models\nconsidered in this article, the coefficients of which are deterministic. In\npractice, one cannot manipulate the exact terms of such an expansion. All\nobjects are subjected to a discretization approach. Thus we need to derive a\nsimilar expansion for the approximated random homogenized matrix. In contrast\nto the expansion of the exact homogenized matrix, the expansion of the\napproximated homogenized matrix contains intrinsically random coefficients. In\nparticular, the second order term is random in nature. The purpose of this work\nis to derive and study this expansion in function of the parameters of the\napproximation procedure (size of the truncated computational domain used,\nmeshsize of the finite elements approximation).\n