  We study the structural, ferroelectric, and magnetic properties of the
potentially multiferroic Aurivillius phase material Bi$_5$FeTi$_3$O$_{15}$
using first principles electronic structure calculations. Calculations are
performed both with PBE and PBEsol exchange correlation functionals. We
conclude that PBE systematically overestimates the lattice constants and the
magnitude of the ferroelectric distortion, whereas PBEsol leads to good
agreement with available experimental data. We then assess a potential site
preference of the Fe$^{3+}$ cation by comparing 10 different distributions of
the perovskite $B$-sites. We find a slight preference for the "inner" site,
consistent with recent experimental observations. We obtain a large value of
$\sim$55 $\mu$C/cm$^2$ for the spontaneous electric polarization, which is
rather independent of the specific Fe distribution. Finally, we calculate the
strength of the magnetic coupling constants and find strong antiferromagnetic
coupling between Fe$^{3+}$ cations in nearest neighbor positions, whereas the
coupling between further neighbors is rather weak. This poses the question
whether magnetic long range order can occur in this system in spite of the low
concentration of magnetic ions.
