A comparison of multiresolution and classical one-dimensional homogenization schemes
The main topic of this paper is an investigation of the multiresolution homogenization method [see \textit{M. Brewster} and \textit{G. Beylkin}, Appl. Comput. Harmon. Anal. 2, 327-349 (1995; Zbl 0840.65047)] for a class of one-dimensional second-order elliptic equations of the form \({d\over dx} \left(k{du\over dx}\right)= f\), where \(k\) is a continuous function on \([0, 1]\), which is periodic with period one, and which is bounded away from zero. The author uses a multiresolution analysis of \(L^2[0, 1]\) and, in most of the paper, the Haar basis of wavelets [see \textit{Y. Meyer}, Rev. Mat. Iberoam. 7, 115-133 (1991; Zbl 0753.42015)]. The multiresolution scheme and the classical theory of homogenization are compared. Several physical examples illustrate the differences between the two methods. In particular, the last example has a continuum of scales and cannot be solved by the classical theory; yet, the multiresolution scheme can be applied.
scientific article; zbMATH DE number 1124816