{
  "id": "1311.4254",
  "version": "v2",
  "published": "2013-11-18T03:33:42.000Z",
  "updated": "2014-02-25T16:05:32.000Z",
  "title": "A Mixed Variational Formulation for the Wellposedness and Numerical Approximation of a PDE Model Arising in a 3-D Fluid-Structure Interation",
  "authors": [
    "George Avalos",
    "Thomas J. Clark"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. The wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\\mathcal{O}$ being coupled to a fourth order plate equation, possibly with rotational inertia parameter $\\rho >0$, which evolves on a flat portion $\\Omega$ of the boundary of $\\mathcal{O}$. The coupling on $\\Omega$ is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on this elastic portion of the boundary. We note here that inasmuch as the Stokes fluid velocity does not vanish on $\\Omega$, the pressure variable cannot be eliminated by the classic Leray projector; instead, the pressure is identified as the solution of a certain elliptic boundary value problem. Eventually, wellposedness of this fluid-structure dynamics is attained through a certain nonstandard variational (``inf-sup\") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a certain mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-25T16:05:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mixed variational formulation",
      "pde model arising",
      "fluid-structure interation",
      "wellposedness",
      "numerical approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4254A"
    }
  }
}