{
  "id": "2312.05511",
  "version": "v1",
  "published": "2023-12-09T09:18:47.000Z",
  "updated": "2023-12-09T09:18:47.000Z",
  "title": "Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates",
  "authors": [
    "Alessandro Contri",
    "Balázs Kovács",
    "André Massing"
  ],
  "comment": "28 pages, 3 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fern\\'andez [\\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-09T09:18:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M12",
      "65M15",
      "65M60",
      "76M10"
    ],
    "keywords": [
      "transient stokes problem",
      "error analysis",
      "pressure estimates",
      "time-stepping methods",
      "mild inverse cfl-type condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}