Corrector theory for MsFEM and HMM in random media

TL;DR

This paper develops a fluctuation theory for MsFEM and HMM in random media, analyzing their ability to capture solution fluctuations in heterogeneous environments with different correlation structures.

Contribution

It introduces a fluctuation theory for multi-scale algorithms in random media and compares their effectiveness in capturing solution fluctuations.

Findings

MsFEM captures fluctuations for both short-range and long-range correlations.

HMM captures long-range fluctuations but amplifies short-range fluctuations.

A modified scheme balances computational cost and fluctuation capture.

Abstract

We analyze the random fluctuations of several multi-scale algorithms such as the multi-scale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential equations with highly heterogeneous coefficients. Such multi-scale algorithms are often shown to correctly capture the homogenization limit when the highly oscillatory random medium is stationary and ergodic. This paper is concerned with the random fluctuations of the solution about the deterministic homogenization limit. We consider the simplified setting of the one dimensional elliptic equation, where the theory of random fluctuations is well understood. We develop a fluctuation theory for the multi-scale algorithms in the presence of random environments with short-range and long-range correlations. What we find is that the computationally more expensive method MsFEM captures the random fluctuations both for short-range and long-range oscillations in the medium. The less expensive method HMM correctly captures the fluctuations for long-range oscillations and strongly amplifies their size in media with short-range oscillations. We present a modified scheme with an intermediate computational cost that captures the random fluctuations in all cases.