Homogenization: In mathematics or physics?
Shixin Xu, Xingye Yue, Changrong Zhang · Discrete and Continuous Dynamical Systems - S · 2016
In mathematics, homogenization theory considers the limitations ofthe sequences of the problems and their solutions when a parametertends to zero. This parameter is regarded as the ratio of thecharacteristic size between the micro scale and macro scale.So what is considered is a sequence of problems in a fixed domainwhile the characteristic size in micro scale tends to zero. But inthe real physics or engineering situations, the micro scale of amedium is fixed and can not be changed. In the process ofhomogenization, it is the size in macro scale which becomes largerand larger and tends to infinity. We observe that the homogenizationin physics is not equivalent to the homogenization in mathematics upto some simple rescaling. With some direct error estimates, weexplain in what sense we can accept the homogenized problem as thelimitation of the original real physical problems. As a byproduct, we present some results on the mathematicalhomogenization 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 isalso given to show the validation of our observation and results.