  We present a finite-element approximation for the one-sided Stefan problem
and the one-sided Mullins--Sekerka problem, respectively. The problems feature
a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our
approximation, which couples a parametric approximation of the moving boundary
with a finite element approximation of the bulk quantities, can be shown to
satisfy a stability bound, and it enjoys very good mesh properties which means
that no mesh smoothing is necessary in practice. In our numerical computations
we concentrate on the simulation of snow crystal growth. On choosing realistic
physical parameters, we are able to produce several distinctive types of snow
crystal morphologies. In particular, facet breaking in approximately
crystalline evolutions can be observed.
