WIAS Preprint No. 2702, (2020)

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes



Authors

  • Apel, Thomas
  • Kempf, Volker
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian

2010 Mathematics Subject Classification

  • 65N30 65N15 65D05

2008 Physics and Astronomy Classification Scheme

  • 47.10.ad 47.11.Fg

Keywords

  • Anisotropic finite elements, incompressible Navier--Stokes equations, divergence-free methods, pressure-robustness

DOI

10.20347/WIAS.PREPRINT.2702

Abstract

Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case.

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