Introduced to the field of image-based computational micromechanics by Moulinec and Suquet [1], spectral trigonometric solvers have been widely adopted in the field due to their computational efficiency, low memory requirements, and ease of implementation. In this talk, we focus on the finite element (FE)-based reformulation of the original scheme pioneered by Schneider et al. [2] and Leuschner and Fritzen [3], and (1) provide a linear algebra perspective on their developments, and (2) relate them to recent advances in Laplace preconditioning of elliptic partial differential equations. Our approach [6] rests on preconditioning the periodic cell problem by a (discrete) Green's operator derived from a reference problem with constant data. On the theoretical side, this implies that the eigenvalues of the preconditioned matrix can be bounded from above and below by coefficients of the original and the reference problem, which means that the system can be solved by the preconditioned conjugate gradient method in a number of steps that is (almost) independent of the grid size. On the implementation side, we exploit the fact that, for a generic arbitrary regular mesh, the system matrix of the preference problem exhibits a block-diagonal structure in the Fourier space and can be efficiently inverted using the Fast Fourier Transform (FFT) techniques. Consequently, the computational complexity of the scheme is dominated by the FFT, making it equivalent to that of spectral solvers. However, in contrast to trigonometric spectral solvers, the proposed scheme works with arbitrary FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We conclude the talk by discussing the potential of this framework for error control in FFT-based homogenization solvers. References [1] Moulinec, H. & Suquet, P. C R Acad Sci Paris 318, 1417–1423 (1994). [2] Schneider, M., Merkert, D. & Kabel, M. Int J Numer Methods Eng 109, 1461–1489 (2017). [3] Leuschner, M. & Fritzen, F. Comput Mech 62, 359–392 (2018). [4] Ladecký, M. et al. Appl Math Comput 446, 127835 (2023).