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

  • Authors:

  • John W. Barrett;Harald Garcke;Robert Nürnberg

  • Affiliations:

  • Department of Mathematics, Imperial College London, London SW7 2AZ, UK;NWF I, Mathematik, Universität Regensburg, 93040 Regensburg, Germany;Department of Mathematics, Imperial College London, London SW7 2AZ, UK

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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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.