{
  "id": "2008.01801",
  "version": "v1",
  "published": "2020-08-04T19:59:13.000Z",
  "updated": "2020-08-04T19:59:13.000Z",
  "title": "On the Sobolev and $L^p$-Stability of the $L^2$-projection",
  "authors": [
    "Lars Diening",
    "Johannes Storn",
    "Tabea Tscherpel"
  ],
  "comment": "34 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees greater than one. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T19:59:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65N50",
      "65N12",
      "65M60"
    ],
    "keywords": [
      "projection",
      "lagrange finite element spaces",
      "space dimension",
      "polynomial degrees greater",
      "newest vertex bisection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}