  We study the convergences of several FFT-based schemes that are widely
applied in computational homogenization for deriving effective coefficients,
and the term "convergence" here means the limiting behaviors as spatial
resolutions going to infinity. Those schemes include Moulinec-Suquent's scheme
[Comput Method Appl M, 157 (1998), pp. 69-94], Willot's scheme [Comptes Rendus
M\'{e}canique, 343 (2015), pp. 232-245], and the FEM scheme [Int J Numer Meth
Eng, 109 (2017), pp. 1461-1489]. Under some reasonable assumptions, we prove
that the effective coefficients obtained by those schemes are all convergent to
the theoretical ones. Moreover, for the FEM scheme, we can present several
convergence rate estimates under additional regularity assumptions.
