{
  "id": "2010.04132",
  "version": "v1",
  "published": "2020-10-08T17:21:16.000Z",
  "updated": "2020-10-08T17:21:16.000Z",
  "title": "Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification",
  "authors": [
    "Aleš Wodecki",
    "Pavel Strachota",
    "Michal Beneš"
  ],
  "comment": "21 pages, 2 figures. Initial submission that has yet to have the cross-references to another related arXiv paper fixed",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We present a convergence result for the finite volume method applied to a particular phase field problem suitable for simulation of pure substance solidification. The model consists of the heat equation and the phase field equation with a general form of the reaction term which encompasses a variety of existing models governing dendrite growth and elementary interface tracking problems. We apply the well known compact embedding techniques in the context of the finite volume method on admissible unstructured polyhedral meshes. We develop the necessary interpolation theory and derive an a priori estimate to obtain boundedness of the key terms. Based on this estimate, we conclude the convergence of all of the terms in the equation system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-08T17:21:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M20",
      "80A22",
      "35K51",
      "35K57",
      "80M12"
    ],
    "keywords": [
      "finite volume method",
      "3d phase field model",
      "models governing dendrite growth",
      "unstructured meshes",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}