Volume 48, Number 2, March-April 2014
Multiscale problems and techniques
|Page(s)||387 - 409|
|Published online||20 February 2014|
Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential
1 Department of Applied Physics and
Applied Mathematics, Columbia University, 10027
2 Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France.
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  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.
Mathematics Subject Classification: 35R60 / 65N30 / 65C99
Key words: Equations with random coefficients / multi-scale finite element method / heterogeneous multi-scale method / corrector test / long-range correlations
© EDP Sciences, SMAI, 2014
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.