  Guaranteed upper-lower bounds on homogenized coefficients, arising from the
periodic cell problem, are calculated in a scalar elliptic setting. Our
approach builds on the recent variational reformulation of the Moulinec-Suquet
(1994) Fast Fourier Transform (FFT) homogenization scheme by Vond\v{r}ejc et
al. (2014), which is based on the conforming Galerkin approximation with
trigonometric polynomials. Upper-lower bounds are obtained by adjusting the
primal-dual finite element framework developed independently by Dvo\v{r}\'{a}k
(1993) and Wieckowski (1995) to the FFT-based Galerkin setting. We show that
the discretization procedure differs for odd and non-odd number of grid points.
Thanks to the Helmholtz decomposition inherited from the continuous
formulation, the duality structure is fully preserved for the odd
discretizations. In the latter case, a more complex primal-dual structure is
observed due to presence of the trigonometric polynomials associated with the
Nyquist frequencies. These theoretical findings are confirmed with numerical
examples. To conclude, the main advantage of the FFT-based approach over
conventional finite-element schemes is that the primal and the dual problems
are treated on the same basis, and this property can be extended beyond the
scalar elliptic setting.
