  In this paper, we first introduce the reader to the Basic Scheme of Moulinec
and Suquet in the setting of quasi-static linear elasticity, which takes
advantage of the fast Fourier transform on homogenized microstructures to
accelerate otherwise time-consuming computations. By means of an asymptotic
expansion, a hierarchy of linear problems is derived, whose solutions are
looked at in detail. It is highlighted how these generalized homogenization
problems depend on each other. We extend the Basic Scheme to fit this new
problem class and give some numerical results for the first two problem orders.
