On numerical averaging of the conductivity coefficient using two-scale extensions

TL;DR

This paper compares solutions of elliptic problems with rapidly oscillating conductivity coefficients to their homogenized counterparts using two-scale extensions, analyzing the approximation error across various test cases.

Contribution

It introduces a numerical comparison framework for elliptic problems with oscillatory coefficients and assesses the impact of averaging size on approximation accuracy.

Findings

Error decreases with larger averaging size

Homogenized solutions approximate oscillatory problems effectively

Performance varies with oscillation intensity

Abstract

In this article we compare solutions to elliptic problems having rapidly oscillated conductivity (permeability, etc) coefficient with solutions to corresponding homogenized problems obtained from two-scale extensions of the initial coefficient. The comparison is done numerically on several one and two dimensional test problems with randomly generated coefficients for different intensities of oscillation. The dependency of the approximation error on the size of averaging is investigated.