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We generalize and provide a linear algebra-based perspective on a finite element (FE) ho-mogenization scheme, pioneered by Schneider et al. (2017)[1] and Leuschner and Fritzen (2018)[2]. The efficiency of the scheme is based on a preconditioned, well-scaled refor-mulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner-a discretized Green's function of a periodic homo-geneous reference problem-has a block-diagonal structure in the Fourier space which per-mits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as O(n log(n)), like FFT, rendering it equiva-lent to spectral solvers in terms of computational efficiency. However, in contrast to clas-sical spectral solvers, the proposed scheme works with FE shape functions with local sup-ports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection. (c) 2023 Published by Elsevier Inc.