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.
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