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

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.

Research highlights▶ Energy structure exploiting FE-schemes give stable discretizations of Stefan problem. ▶ Schemes avoid mesh distortions and lead to well-conditioned discretizations. ▶ Discretization of anisotropic and nearly crystalline Gibbs-Thomson laws possible. ▶ Dendritic and snow crystal growth simulations in 2D and 3D are presented. ▶ Quasi-static variants such as the Mullins-Sekerka problem can be treated.