Methods for the approximate computation of homogenized coefficients.

摘要

The problem of homogenization is that of replacing one partial differential equation, which has finely-detailed coefficients and hence is difficult to solve, with another equation which is simpler to solve, the latter ideally having constant coefficients. In the isotropic case, the single coefficient of the homogenized equation is called the “effective” coefficient. The solutions of the two equations should agree up to a certain level of accuracy. This thesis addresses the problem of the homogenization of second-order periodic elliptic partial differential equations. In the first few chapters, variational techniques are used to prove the iterated homogenization theorem and also to find a first-order estimate for the effective coefficient for one important equation in three dimensions. In the second part of the thesis, wavelet analysis is used to devise a numerical method of computing the effective coefficient in two-dimensional problems.