Multiscale Modeling &Simulation, Volume 18, Issue 3, Page 1179-1209, January 2020.
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 [Cancès 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 [Cancès 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.

中文翻译：

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均质的嵌入式校正器问题。第一部分：理论

多尺度建模与仿真，第18卷，第3期，第1179-1209页，2020年1月。
本文是两项研究的第一部分，其目的是在随机均化的情况下对新的近似校正器问题进行理论分析和数值研究。在这里，我们提出了三个新的替代方案，用于对具有高振荡系数的扩散问题的均化矩阵进行逼近。这些不同的近似值都依赖于嵌入式校正器问题的使用（我们先前在[Cancès等人，CR Math。Acad。Sci。Paris，353（2015），第801--806页]中介绍过）由高度振荡的材料制成的有限尺寸区域嵌入到均匀的无限介质中，该介质必须适当确定其扩散系数。伴随文章[Cancèset al。，J. 计算 Phys。，407（2020），109254]，其中提出了一种非常有效的算法来解决特定异质材料的此类问题。在本文中，我们证明当嵌入域的大小达到无穷大时，我们引入的三种不同逼近收敛到介质的均质矩阵。