{
  "id": "2301.07260",
  "version": "v1",
  "published": "2023-01-18T01:58:03.000Z",
  "updated": "2023-01-18T01:58:03.000Z",
  "title": "Additive Schwarz methods for fourth-order variational inequalities",
  "authors": [
    "Jongho Park"
  ],
  "comment": "22 pages, 2 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We consider additive Schwarz methods for fourth-order variational inequalities. While most existing works on Schwarz methods for fourth-order variational inequalities deal with auxiliary linear problems instead of the original ones, we deal with the original ones directly by using a nonlinear subspace correction framework for convex optimization. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\\delta$ measures the overlap among the subdomains. To the best of our knowledge, the proposed two-level method is the first scalable Schwarz method for fourth-order variational inequalities. An efficient numerical method to solve coarse problems in the two-level method is also presented. Our theoretical results are verified by numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-18T01:58:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N55",
      "65K15",
      "65N30",
      "49M27"
    ],
    "keywords": [
      "two-level method",
      "nonlinear subspace correction framework",
      "fourth-order variational inequalities deal",
      "two-level additive schwarz methods",
      "first scalable schwarz method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}