  We propose a matrix-free finite element (FE) homogenization scheme that is
considerably more efficient than generic FE implementations. The efficiency of
our scheme follows from a preconditioned well-scaled reformulation allowing for
the use of the conjugate gradient or similar iterative solvers. The
geometrically-optimal preconditioner -- a discretized Green's function of a
periodic homogeneous reference problem -- has a block-diagonal structure in the
Fourier space which permits its efficient inversion using the fast Fourier
transform (FFT) techniques for generic regular meshes. This implies that the
scheme scales as $\mathcal{O}(n \log(n))$ like FFT, rendering it equivalent to
spectral solvers in terms of computational efficiency. However, in contrast to
classical spectral solvers, the proposed scheme works with FE shape functions
with local supports and is free of the Fourier ringing phenomenon. We showcase
that the scheme achieves the number of iterations that are almost independent
of spatial discretisation and scales mildly with the phase contrast.
Additionally, we discuss the equivalence between our displacement-based scheme
and the recently proposed strain-based homogenization technique with
finite-element projection.
