{
  "id": "2103.11967",
  "version": "v1",
  "published": "2021-03-22T16:15:51.000Z",
  "updated": "2021-03-22T16:15:51.000Z",
  "title": "Low-order preconditioning of the Stokes equations",
  "authors": [
    "Alexey Voronin",
    "Yunhui He",
    "Scott MacLachlan",
    "Luke N. Olson",
    "Ray Tuminaro"
  ],
  "comment": "In the process of being to submitted to NLA@Wiley",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Low-order finite-element discretizations are well-known to provide effective preconditioners for the linear systems that arise from higher-order discretizations of the Poisson equation. In this work, we show that high-quality preconditioners can also be derived for the Taylor-Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the $\\boldsymbol{ \\mathbb{Q}}_1iso\\boldsymbol{ \\mathbb{Q}}_2/ \\mathbb{Q}_1$ discretization of the Stokes operator as a preconditioner for the $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ system, our ultimate motivation is to apply algebraic multigrid within solvers for $\\boldsymbol{ \\mathbb{Q}}_2/\\mathbb{Q}_1$ systems via the $\\boldsymbol{ \\mathbb{Q}}_1iso\\boldsymbol{ \\mathbb{Q}}_2/ \\mathbb{Q}_1$ discretization, which will be considered in a companion paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-22T16:15:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stokes equations",
      "low-order preconditioning",
      "geometric multigrid",
      "utilize local fourier analysis",
      "low-order finite-element discretizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}