Homogenization: in Mathematics or Physics?

TL;DR

This paper explores the differences between mathematical and physical homogenization, emphasizing the importance of scale considerations and providing error estimates to validate the homogenized models in real-world scenarios.

Contribution

It clarifies the distinction between mathematical and physical homogenization and offers new error estimates for validating homogenized models against actual physical problems.

Findings

Homogenization in physics differs from mathematical homogenization due to fixed micro-scale in real media.

Error estimates are provided to justify the use of homogenized models in physical contexts.

Results include homogenization of problems with weakly compacted source terms in $H^{-1}$.

Abstract

Homogenization appeared more than 100 years ago. It is an approach to study the macro-behavior of a medium by its micro-properties. In mathematics, homogenization theory considers the limitations of the sequences of the problems and its solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size in the micro scale to that in the macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But for the real situations in physics or engineering, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what means we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.