This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches.
These results are consistent with those obtained by the authors for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence less computationally intensive, algorithms.
An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation
École des Ponts ParisTech（巴黎路桥大学）；INRIA Rocquencourt（法国国家信息与自动化研究所罗坎库尔中心）；Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire J.-L. Lions（索邦大学、巴黎第六大学及法国国家科学研究中心J.-L.李翁斯实验室）
A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems
Beijing Institute of Technology（北京理工大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth
INRIA Paris-Rocquencourt（法国国家信息与自动化研究所巴黎罗克安古尔分部）；Université Libre de Bruxelles (ULB)（布鲁塞尔自由大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔北欧分部）；EPFL - Institut de mathématiques（洛桑联邦理工学院数学研究所）；CMAP École polytechnique（巴黎综合理工学院计算与应用数学中心）
Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization
Université Libre de Bruxelles (ULB)（布鲁塞尔自由大学）；Team MEPHYSTO, Inria Lille - Nord Europe（In里尔-北欧洲MEPHYSTO团队）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进研究所数学系）
A Stable SPH Discretization of the Elliptic Operator with Heterogeneous Coefficients
Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics（代尔夫特理工大学，电气、数学与计算机科学学院，应用数学研究所）
$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part II. Application to the Nonconforming Heterogeneous Multiscale Method
