WIAS Preprint No. 2683, (2020)

Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem



Authors

  • Frerichs, Derk
  • Merdon, Christian

2010 Mathematics Subject Classification

  • 65N12 65N30 76D07 76D05 76M10

2008 Physics and Astronomy Classification Scheme

  • 47.10.ad 47.11.Fg

Keywords

  • Incompressible Navier--Stokes equations, mixed virtual element method, pressure-robustness, divergence-free velocity reconstruction, polygonal meshes

DOI

10.20347/WIAS.PREPRINT.2683

Abstract

Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

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