  A large fraction of white dwarf stars shows photospheric chemical composition
polluted by heavy elements accreted from a debris disk. Such debris disks
result from the tidal disruption of rocky planetesimals which had survived to
whole stellar evolution from the main sequence to the final white dwarf stage.
Determining the accretion rate of this material is an important step towards
estimating the mass of the planetesimals and towards understanding the ultimate
fate of the planetary systems. The accretion of heavy material with a mean
molecular weight, $\mu$, higher than the mean molecular weight of the white
dwarf outer layers, induces a double-diffusive instability producing fingering
convection and extra-mixing. As a result, the accreted material is diluted deep
into the star. We explore the effect of this extra-mixing on the abundance
evolution of Mg, O, Ca, Fe and Si in the cases of the two well studied polluted
DAZ white dwarfs: GD~133 and G~29-38. We performed numerical simulations of the
accretion of material with a chemical composition similar to the bulk Earth
one. We considered accretion rates from $10^{4}$~g/s to $10^{10}$~g/s. The
double-diffusive instability develops on a very short time scale. The accretion
rate needed to reproduce the observed abundances exceeds by more than 2 orders
of magnitude the rate estimated by neglecting the fingering convection in the
case of GD~133, and by approximately 1.7 dex in the case of G~29-38. Our
numerical simulations show that fingering convection is an efficient mechanism
to mix the accreted material and that it must be taken into account in the
determination of accretion rates.
