{
  "id": "2007.14068",
  "version": "v1",
  "published": "2020-07-28T08:53:30.000Z",
  "updated": "2020-07-28T08:53:30.000Z",
  "title": "Nonconforming finite element Stokes complexes in three dimensions",
  "authors": [
    "Xuehai Huang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Two nonconforming finite element Stokes complexes ended with the nonconforming $P_1$-$P_0$ element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order $\\boldsymbol H(\\textrm{grad}~\\textrm{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\\boldsymbol H(\\textrm{grad}~\\textrm{curl})$-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem is decoupled into two mixed methods of the Maxwell equation and the nonconforming $P_1$-$P_0$ element method for the Stokes equation, based on which a fast solver is developed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-28T08:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65N12",
      "65N22"
    ],
    "keywords": [
      "nonconforming finite element stokes complexes",
      "dimensions",
      "stokes equation",
      "quad-curl problem",
      "combining nonconforming finite element stokes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}