{"paper":{"title":"Stable Phase Field Approximations of Anisotropic Solidification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.mtrl-sci","physics.comp-ph"],"primary_cat":"math.NA","authors_text":"Harald Garcke, John W. Barrett, Robert N\\\"urnberg","submitted_at":"2012-10-25T10:59:39Z","abstract_excerpt":"We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs--Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by {align*} \\vartheta\\,w_t + \\lambda\\,\\varrho(\\varphi)\\,\\varphi_t & = \\nabla \\,.\\, (b(\\varphi)\\,\\nabla\\, w) \\,, \\cPsi\\,\\tfrac{a}\\alpha\\,\\varrho(\\varphi)\\,w & = \\epsilon\\,\\tfrac\\rho\\alpha\\,\\mu(\\nabla\\,\\varphi)\\,\\varphi_t -\\epsilon\\,\\nabla \\,.\\, A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6791","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}