A Comparison of Multiresolution and Classical One-dimensional Homogenization Schemes

Homogenization is a collection of methods for extracting or constructing equations for the coarse-scale behavior of solutions to equations which incorporate many scales. This paper compares the classical method of homogenization with the recently developed multiresolution strategy for a particular class of one-dimensional second-order elliptic equations. We also examine several physical examples which highlight the distinctions between the two methods. 1 Introduction There are many important physical problems which incorporate several scales such as wave propagation through periodic or stratified media. The interactions and the fineness of these scales make solving these problems very difficult and expensive. Often, one would be content with the coarse-scale behavior of the solution but the fine scales affect this behavior so one cannot simply ignore these. Instead, it is useful to find a way of extracting or constructing equations for the coarse behavior of the solution which take in..