On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth

Abstract

We introduce a parametric finite element approximation for the Stefan problem
with the Gibbs–Thomson law and kinetic undercooling, which mimics the underlying
energy structure of the problem. The proposed method is also applicable to
certain quasi-stationary variants, such as the Mullins–Sekerka problem. In addition,
fully anisotropic energies are easily handled. The approximation has good
mesh properties, leading to a well-conditioned discretization, even in three space
dimensions. Several numerical computations, including for dendritic growth and
for snow crystal growth, are presented.