{
  "id": "2301.03007",
  "version": "v1",
  "published": "2023-01-08T09:33:25.000Z",
  "updated": "2023-01-08T09:33:25.000Z",
  "title": "Averaging-based local projections in finite element exterior calculus",
  "authors": [
    "Martin W. Licht"
  ],
  "comment": "27 pages. Submitted. Comments welcome",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We develop projection operators onto finite element differential forms over simplicial meshes. Our projection is locally bounded in Lebesgue and Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover, it incorporates homogeneous boundary conditions and satisfies a local broken Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond projection and a modified Cl\\'ement-type interpolant with the projection property. The latter seems to be a new result even for Lagrange elements. This projection operator immediately enables an equivalence result on local- and global-best approximations. We combine techniques for the Scott-Zhang and Ern-Guermond projections and adopt the framework of finite element exterior calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini, N\\'ed\\'elec, and Raviart-Thomas elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-08T09:33:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30"
    ],
    "keywords": [
      "finite element exterior calculus",
      "averaging-based local projections",
      "finite element differential forms",
      "ern-guermond projection",
      "local broken bramble-hilbert estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}