{
  "id": "2212.03735",
  "version": "v1",
  "published": "2022-12-07T15:59:05.000Z",
  "updated": "2022-12-07T15:59:05.000Z",
  "title": "$hp$-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem",
  "authors": [
    "Zhaonan Dong",
    "Lorenzo Mascotto"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We prove $hp$-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $H^2$ piecewise polynomial approximants with $hp$-optimal approximation properties over the given meshes. The $hp$-optimality is also discussed for $\\mathcal C^0$-IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $p$-suboptimality occurs in presence of singular essential boundary conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-07T15:59:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N12",
      "65N30",
      "65N50"
    ],
    "keywords": [
      "interior penalty discontinuous galerkin methods",
      "optimal interior penalty discontinuous galerkin",
      "biharmonic problem",
      "singular essential boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}