The issue of "First wall materials & compatibility with ITER /DEMO relevant plasmas" is among the R&D missions for possible new European plasma fusion devices that the FAST project will address. FAST can operate with ITER relevant values of P/R (up to 22 MW/m, against the ITER 24 MW/m, inclusive of the particles power), thanks to its compactness; thus it can investigate the physics of large heat loads on divertor plates. The FAST divertor will be made of bulk W tiles, for basic operations, but also fully toroidal divertor targets made of liquid lithium (L-Li) are foreseen. To have reliable predictions of the thermal loads on the divertor plates and of the core plasma purity a number of numerical self-consistent simulations have been made for the H-mode and steady-state scenario by using the code COREDIV. This code, already validated in the past on experimental data (namely JET, FTU, Textor), is able to describe self-consistently the core and edge plasma in a tokamak device by imposing the continuity of energy and particle fluxes and of particle densities and temperatures at the separatrix. In the present work the results of such calculations will be illustrated, including heat loads on the divertor. The overall picture shows that, marginally in the intermediate and, necessarily in the high density H-mode scenarios (=2 and 5·10 20 m -3 respectively), impurity seeding should be foreseen with W as target material: however, only a small amount of Ar (0.03% atomic concentration), not affecting the core purity, is sufficient to maintain the divertor peak loads below 18 MW/m 2 , that represents the safety limit for the W monoblock technology, presently accepted for the ITER divertor tiles. Li always needs additional impurities for decreasing divertor heat loads, the Z eff value being than 1.8. At low plasma densities (but 1.3·10 20 m -3 ), typical of steady state regimes, W by alone is effective in dissipating the input power by radiative losses, without excessive core contamination. Impurity seeding would lead to excessive W sputtering by Ar and too high Z eff . The impact of the ELMs on the divertor in the case of a good H-mode with low pedestal dimensionless collisionality will be discussed too.