  Fourier solvers have become efficient tools to establish structure-property
relations in heterogeneous materials. Introduced as an alternative to the
Finite Element (FE) method, they are based on fixed-point solutions of the
Lippmann-Schwinger type integral equation. Their computational efficiency
results from handling the kernel of this equation by the Fast Fourier Transform
(FFT). However, the kernel is derived from an auxiliary homogeneous linear
problem, which renders the extension of FFT-based schemes to non-linear
problems conceptually difficult. This paper aims to establish a link between
FE- and FFT-based methods, in order to develop a solver applicable to general
history- and time-dependent material models. For this purpose, we follow the
standard steps of the FE method, starting from the weak form, proceeding to the
Galerkin discretization and the numerical quadrature, up to the solution of
non-linear equilibrium equations by an iterative Newton-Krylov solver. No
auxiliary linear problem is thus needed. By analyzing a two-phase laminate with
non-linear elastic, elasto-plastic, and visco-plastic phases, and by
elasto-plastic simulations of a dual-phase steel microstructure, we demonstrate
that the solver exhibits robust convergence. These results are achieved by
re-using the non-linear FE technology, with the potential of further extensions
beyond small-strain inelasticity considered in this paper.
