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

Naveed Ahmed, Gabriel R. Barrenechea, Erik Burman, Johnny Guzmán , Alexander Linke, Christian Merdon

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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 pressure-independent 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.
Original languageEnglish
Pages (from-to)2746–2774
Number of pages29
JournalSIAM Journal on Numerical Analysis
Volume59
Issue number5
Early online date28 Oct 2021
DOIs
Publication statusPublished - 31 Oct 2021

Keywords

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

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