  Iterative Fast Fourier Transform methods are useful for calculating the
fields in composite materials and their macroscopic response. By iterating back
and forth until convergence, the differential constraints are satisfied in
Fourier space, and the constitutive law in real space. The methods correspond
to series expansions of appropriate operators and to series expansions for the
effective tensor as a function of the component moduli. It is shown that the
singularity structure of this function can shed much light on the convergence
properties of the iterative Fast Fourier Transform methods. We look at a model
example of a square array of conducting square inclusions for which there is an
exact formula for the effective conductivity (Obnosov). Theoretically some of
the methods converge when the inclusions have zero or even negative
conductivity. However, the numerics do not always confirm this extended range
of convergence and show that accuracy is lost after relatively few iterations.
There is little point in iterating beyond this. Accuracy improves when the grid
size is reduced, showing that the discrepancy is linked to the discretization.
Finally, it is shown that none of the three iterative schemes investigated
over-performs the others for all possible microstructures and all contrasts.
