{
  "id": "2005.06506",
  "version": "v1",
  "published": "2020-05-13T18:20:12.000Z",
  "updated": "2020-05-13T18:20:12.000Z",
  "title": "A Hellan-Herrmann-Johnson-like method for the stream function formulation of the Stokes equations in two and three space dimensions",
  "authors": [
    "Philip L. Lederer"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan-Herrmann-Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P.L. Lederer, J. Sch\\\"oberl, IMA Journal of numerical Analysis, 2019] that approximates the velocity in an $H(\\operatorname{div})$-conforming space and introduces a new stress-like variable for the approximation of the gradient of the velocity within the function space $H(\\operatorname{curl}\\operatorname{div})$. The properties of the (discrete) de Rham complex allows to extend this method to a stream function formulation in two and three space dimensions. We present a detailed stability analysis in the continuous and the discrete setting where the stream function $\\psi$ and its approximation $\\psi_h$ are elements of $H(\\operatorname{curl})$ and the $H(\\operatorname{curl})$-conforming N\\'ed\\'elec finite element space, respectively. We conclude with an error analysis revealing optimal convergence rates for the error of the discrete velocity $u_h = \\operatorname{curl}(\\psi_h)$ measured in a discrete $H^1$-norm. We present numerical examples to validate our findings and discuss structure-preserving properties such as pressure-robustness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-13T18:20:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N12",
      "65N15",
      "65N30",
      "76D07",
      "76M10"
    ],
    "keywords": [
      "stream function formulation",
      "space dimensions",
      "stokes equations",
      "nedelec finite element space",
      "hellan-herrmann-johnson-like method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}