{
  "id": "1811.05429",
  "version": "v1",
  "published": "2018-11-13T17:42:22.000Z",
  "updated": "2018-11-13T17:42:22.000Z",
  "title": "Improved $L^2$ and $H^1$ error estimates for the Hessian discretisation method",
  "authors": [
    "Devika Shylaja"
  ],
  "comment": "29 pages, 4 figures, 7 tables",
  "categories": [
    "math.NA"
  ],
  "abstract": "The framework of Hessian discretisation method (HDM) for fourth order linear elliptic equations enables us to develop a study that encompasses numerous classical methods such as finite element methods (conforming and non-conforming), finite volume methods, and others such as methods based on gradient recovery operators. A generic error estimate has been established for the HDM in $L^2$, discrete $H^1$ and $H^2$ norms based on three properties in literature. In this paper, we derive improved $L^2$ and $H^1$ error estimates in the framework of HDM and apply it to various schemes. Since an improved $L^2$ estimate is not expected in general for finite volume method (FVM), here we consider a modified FVM by changing the quadrature of the source term and prove a superconvergence result for this modified FVM. We also show that the non-conforming Morley element fit into the framework of HDM.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-13T17:42:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N08",
      "65N12",
      "65N15",
      "65N30"
    ],
    "keywords": [
      "hessian discretisation method",
      "error estimate",
      "finite volume method",
      "order linear elliptic equations enables",
      "fourth order linear elliptic equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}