  Type Iax supernovae may arise from failed explosions of white dwarfs that
leave behind a bound remnant (i.e., a "postgenitor" star) that could be
identified in wide field surveys. To understand their observational signatures,
we simulate these white dwarf (WD) postgenitors from shortly after explosion
until they move back down the WD cooling track, and we consider several
possible WD masses and explosion energies. To predict the peculiar surface
abundances of the WD postgenitors, our models take into account gravitational
settling and radiative levitation. We find that radiative levitation is
significant at temperatures above a mass-dependent critical temperature,
typically in the range Teff ~ 50-100 * 10^3 K, significantly increasing surface
abundances of iron-group elements. Due to enhanced iron group opacity compared
to normal WDs, the postgenitor peak luminosity and cooling timescale depend
sensitively on mass, with more massive WDs becoming brighter but cooling much
faster. We discuss our results in light of recently discovered hypervelocity
white dwarfs with peculiar surface compositions, finding that our low-mass
postgenitor models match many of their observational characteristics. Finally,
we explore the effects of thermohaline diffusion, tentatively finding that it
strongly suppresses abundance enhancements created by radiative levitation, but
more realistic modeling is required to reach a firm conclusion.
