Phase transition problems on curved surfaces can lead to a panopticon of fascinating patterns. In this paper we consider finite element approximations of phase field models with a spatially inhomogeneous and anisotropic surface energy density. The problems are either posed in $\mathbb R^3$ or on a two-dimensional hypersurface in $\mathbb R^3$. In the latter case, a fundamental choice regarding the anisotropic energy density has to be made. Our numerical method can be employed both situations, where for the problems on hypersurfaces the algorithm uses parametric finite elements. We prove an unconditional stability result for our schemes and present several numerical experiments, including for the modelling of ice crystal growth on a sphere.

翻译：曲面上的相变问题可呈现出多种迷人图案。本文考虑具有空间非均匀性与各向异性表面能量密度的相场模型的有限元近似。问题可置于$\mathbb R^3$中或$\mathbb R^3$中的二维超曲面上。对于后者，需就各向异性能量密度做出基本选择。我们的数值方法可同时适用于这两种情形，其中对于超曲面上的问题，该算法采用参数化有限元。我们证明了方案的无条件稳定性结果，并展示了若干数值实验，包括球面上冰晶生长的建模。