{
  "id": "2205.08764",
  "version": "v1",
  "published": "2022-05-18T07:22:07.000Z",
  "updated": "2022-05-18T07:22:07.000Z",
  "title": "Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes",
  "authors": [
    "Carsten Carstensen",
    "Rekha Khot",
    "Amiya K. Pani"
  ],
  "comment": "38 pages, 21 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\\in V:=H^2_0(\\Omega)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\\in V^*=H^{-2}(\\Omega)$. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator $J:V_h\\to V$ from the nonconforming virtual element space $V_h$. The operator $J$ is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on $u\\in V$. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T07:22:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biharmonic equation",
      "morley degrees",
      "polygonal meshes",
      "convergence rates",
      "posteriori error"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}