{
  "id": "2010.10914",
  "version": "v1",
  "published": "2020-10-21T11:51:15.000Z",
  "updated": "2020-10-21T11:51:15.000Z",
  "title": "Convergence and supercloseness in a balanced norm of finite element methods on Bakhvalov-type meshes for reaction-diffusion problems",
  "authors": [
    "Jin Zhang",
    "Xiaowei Liu"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In convergence analysis of finite element methods for singularly perturbed reaction--diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local weighted $L^2$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $L^2$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local weighted $L^2$ projection operator, a vertices-edges-element operator and some corrections on the boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-21T11:51:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite element methods",
      "balanced norm",
      "reaction-diffusion problems",
      "bakhvalov-type meshes",
      "convergence analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}