  Micromechanical homogenization is often carried out with Fourier-accelerated
methods that are prone to ringing artifacts. We here generalize the
compatibility projection introduced by Vond\v{r}ejc, Zeman & Marek [Comput.
Math. Appl. 68, 156 (2014)] beyond the Fourier basis. In particular, we
formulate the compatibility projection for linear finite elements while
maintaining Fourier-acceleration and the fast convergence properties of the
original method. We demonstrate that this eliminates ringing artifacts and
yields an efficient computational homogenization scheme that is equivalent to
canonical finite-element formulations on fully structured grids.
