Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential

Guillaume Bal Department of Applied Physics and Applied Mathematics, Columbia University, 10027 New York, USA. Wenjia Jing D´epartement de Math´ematiques et Applications, Ecole Normale Sup´erieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France

Analysis of PDEs Numerical Analysis and Scientific Computing Probability mathscidoc:2206.03008

ESAIM: Mathematical Modelling & Numerical Analysis, 48, 387–409, 2014.3
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. Several multiscale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a secondorder elliptic equations with random potential in two dimensions of space. 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 in [9] for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence computationally less intensive, algorithms.
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  title={Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential},
  author={Guillaume Bal, and Wenjia Jing},
  booktitle={ESAIM: Mathematical Modelling & Numerical Analysis},
Guillaume Bal, and Wenjia Jing. Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential. 2014. Vol. 48. In ESAIM: Mathematical Modelling & Numerical Analysis. pp.387–409.
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