{
  "id": "1805.07835",
  "version": "v1",
  "published": "2018-05-20T22:51:08.000Z",
  "updated": "2018-05-20T22:51:08.000Z",
  "title": "An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation",
  "authors": [
    "Thomas Führer",
    "Norbert Heuer",
    "Antti H. Niemi"
  ],
  "comment": "Accepted for publication in Mathematics of Computation",
  "categories": [
    "math.NA"
  ],
  "abstract": "We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(\\mathrm{div\\,Div})$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(\\mathrm{div\\,Div})$. They are essential to construct basis functions for an approximation of $H(\\mathrm{div\\,Div})$. To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-20T22:51:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "74S05",
      "74K20",
      "35J35",
      "65N30",
      "35J67"
    ],
    "keywords": [
      "kirchhoff-love plate bending model",
      "ultraweak formulation",
      "dpg approximation",
      "construct basis functions",
      "variational formulation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}