{
  "id": "2305.02057",
  "version": "v1",
  "published": "2023-05-03T11:55:49.000Z",
  "updated": "2023-05-03T11:55:49.000Z",
  "title": "Extraction and application of super-smooth cubic B-splines over triangulations",
  "authors": [
    "Jan Grošelj",
    "Hendrik Speleers"
  ],
  "journal": "Computer Aided Geometric Design 103, art. 102194 (2023)",
  "doi": "10.1016/j.cagd.2023.102194",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The space of $C^1$ cubic Clough-Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative $C^1$ cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell-Sabin refined triangulation and present different types of $C^2$ super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough-Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-03T11:55:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "super-smooth cubic b-splines",
      "triangulation",
      "b-spline basis",
      "fourth order boundary value problems",
      "extraction"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}