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

TL;DR

This paper investigates the accuracy of a heterogeneous multi-scale finite element method in estimating random fluctuations of solutions to elliptic equations with random potentials, highlighting conditions for correct estimation and effects of patch size.

Contribution

It provides a detailed analysis of how patch size and interaction range of the random potential affect the accuracy of fluctuation estimates in multi-scale schemes.

Findings

Correct estimation of fluctuations when patches cover the entire domain.

Amplification of variance for short-range interactions with smaller patches.

Accurate fluctuation capture for long-range interactions regardless of patch size.

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. 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.