Garcke, Harald and Nuernberg, Robert (2025) A Finite Element Method for Anisotropic Crystal Growth on Surfaces. INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, 22 (5). ISSN 1705-5105
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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 R3 or on a two-dimensional hypersurface in R3. In the latter case, a fundamental choice regarding the anisotropic energy density has to be made. One possibility is to use a density defined in the ambient space R3. However, we propose and advocate for an alternative, where a density is defined on a fixed chosen tangent space, and is then moved along geodesics to the other tangent spaces. Our numerical method can be employed in all of the above 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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | PHASE-FIELD MODEL; GEOMETRIC EVOLUTION-EQUATIONS; CAHN-HILLIARD; NUMERICAL APPROXIMATION; MEAN-CURVATURE; LIMIT; DYNAMICS; MOTION; Crystal growth; hypersurface; phase field; anisotropy; finite elements; stability |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Harald Garcke |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 17 Jun 2026 05:17 |
| Last Modified: | 17 Jun 2026 05:17 |
| URI: | https://pred.uni-regensburg.de/id/eprint/66154 |
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