{
  "id": "1312.1313",
  "version": "v3",
  "published": "2013-12-04T20:20:02.000Z",
  "updated": "2013-12-21T21:14:13.000Z",
  "title": "Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System",
  "authors": [
    "Amanda E. Diegel",
    "Xiaobing H. Feng",
    "Steven M. Wise"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a non-steady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the phase variable is bounded in $L^\\infty \\left(0,T,L^\\infty\\right)$ and the chemical potential is bounded in $L^\\infty \\left(0,T,L^2\\right)$ absolutely unconditionally in two and three dimensions, for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-21T21:14:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M12",
      "65M15",
      "65M60",
      "76D07"
    ],
    "keywords": [
      "mixed finite element method",
      "cahn-hilliard-darcy-stokes system",
      "models phase separation",
      "diblock copolymer melts",
      "non-steady darcy-stokes flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.1313D"
    }
  }
}