FINITE-ELEMENT APPROXIMATION OF ONE-SIDED STEFAN PROBLEMS WITH ANISOTROPIC, APPROXIMATELY CRYSTALLINE, GIBBS–THOMSON LAW John W. Barrett Department of Mathematics, Imperial College London, London, SW7 2AZ, UK Harald Garcke Fakult¨ at f¨ ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany Robert N¨ urnberg Department of Mathematics, Imperial College London, London, SW7 2AZ, UK Abstract. 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 approxi- mation 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 re- alistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in ap- proximately crystalline evolutions can be observed. 1. Introduction Pattern formation during crystal growth is one of the most fascinating areas in physics and materials science. Furthermore, crystallisation is a fun- damental phase transition, and a good understanding is crucial for many applications. In this paper we will concentrate on a mathematical model based on the one-sided Stefan and Mullins–Sekerka problems, for which we will introduce a new numerical method of approximation. The numerical so- lutions presented here are tailored for the description of snow crystal growth. However, we note that with minor modifications our approach can be used AMS Subject Classifications: 80A22, 74N05, 65M60, 35R37, 65M12, 80M10. 1 arXiv:1201.1802v2 [physics.comp-ph] 18 Jan 2013