{
  "id": "1610.01660",
  "version": "v1",
  "published": "2016-10-05T21:35:49.000Z",
  "updated": "2016-10-05T21:35:49.000Z",
  "title": "Cut Finite Element Methods for Partial Differential Equations on Embedded Manifolds of Arbitrary Codimensions",
  "authors": [
    "Erik Burman",
    "Peter Hansbo",
    "Mats G. Larson",
    "Andre Massing"
  ],
  "comment": "30 pages, 4 figures, 3 tables",
  "categories": [
    "math.NA"
  ],
  "abstract": "We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\\mathbb{R}^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-05T21:35:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "65N85",
      "58J05"
    ],
    "keywords": [
      "partial differential equations",
      "arbitrary codimension",
      "embedded manifolds",
      "background mesh",
      "stabilizing form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}