  We critically compare the practicality and accuracy of numerical
approximations of phase field models and sharp interface models of
solidification. Particular emphasis is put on Stefan problems, and their
quasi-static variants, with applications to crystal growth. New approaches with
a high mesh quality for the parametric approximations of the resulting free
boundary problems and new stable discretizations of the anisotropic phase field
system are taken into account in a comparison involving benchmark problems
based on exact solutions of the free boundary problem.
