  This paper is designed to explore the formation and evolution of
hydrogen-deficient post-AGB white dwarfs. To this end, we compute the complete
evolution of an initially 2.7 M_sun star from the zero-age main sequence
through the thermally pulsing and mass-loss phases to the white dwarf stage.
Particular attention is given to the chemical abundance changes during the
whole evolution. A time-dependent scheme for the simultaneous treatment of
abundance changes caused by nuclear reactions, diffusive overshooting, salt
fingers and convection is considered. We employed the double-diffusive
mixing-length theory of convection for fluids with composition gradients
(Grossman & Taam 1996). The study can therefore be considered as a test of its
performance in low-mass stars. Also, time-dependent element diffusion for
multicomponent gases is taken into account during the white dwarf evolution.
The evolutionary stages corresponding to the last helium thermal pulse on the
early white-dwarf cooling branch and the following born-again episode are
carefully explored. Relevant aspects for PG1159 stars and DB white dwarf
evolution are studied in the frame of these new evolutionary models that take
into account the history of the white dwarf progenitor. The scope of the
calculations is extended to the domain of the helium-rich, carbon-contaminated
DQ white dwarfs with the aim of exploring the plausibility of the evolutionary
connection PG1159-DB-DQ. In this regard, the implications for the
double-layered chemical structure in pulsating DB white dwarfs is investigated.
Another aspect of the investigation concerned the consequences of mass-loss
episodes during the PG1159 stage for the chemical stratification of the outer
layer of DB and DQ white dwarfs.
