WIAS Preprint No. 2740, (2020)

A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation


Authors

  • Ahmed, Naveed
    ORCID: 0000-0002-9322-0373
  • Barrenechea, Gabriel R
  • Burman, Erik
  • Guzmán, Johnny
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N30 65N12 76D07

Keywords

  • incompressible Navier--Stokes equations, divergence-free mixed finite element methods, pressure-robustness, convection stabilization, Galerkin least squares, vorticity equation

DOI

10.20347/WIAS.PREPRINT.2740

Abstract

Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.

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