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

Guillaume Bal; Wenjia Jing

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 387-409
  • ISSN: 0764-583X

Abstract

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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 multi-scale 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 second-order 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.

How to cite

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Bal, Guillaume, and Jing, Wenjia. "Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 387-409. <http://eudml.org/doc/273278>.

@article{Bal2014,
abstract = {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 multi-scale 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 second-order 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.},
author = {Bal, Guillaume, Jing, Wenjia},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {equations with random coefficients; multi-scale finite element method; heterogeneous multi-scale method; corrector test; long-range correlations; elliptic equations with random potential},
language = {eng},
number = {2},
pages = {387-409},
publisher = {EDP-Sciences},
title = {Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential},
url = {http://eudml.org/doc/273278},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Bal, Guillaume
AU - Jing, Wenjia
TI - Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 387
EP - 409
AB - 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 multi-scale 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 second-order 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.
LA - eng
KW - equations with random coefficients; multi-scale finite element method; heterogeneous multi-scale method; corrector test; long-range correlations; elliptic equations with random potential
UR - http://eudml.org/doc/273278
ER -

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