{
  "id": "2407.11482",
  "version": "v1",
  "published": "2024-07-16T08:18:31.000Z",
  "updated": "2024-07-16T08:18:31.000Z",
  "title": "An implementation of hp-FEM for the fractional Laplacian",
  "authors": [
    "Björn Bahr",
    "Markus Faustmann",
    "Jens Markus Melenk"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-16T08:18:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "implementation",
      "integral dirichlet fractional laplacian",
      "preserve root exponential convergence",
      "shape regular meshes",
      "finite element spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}