• Medientyp: Sonstige Veröffentlichung; Bericht; E-Book
  • Titel: Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
  • Beteiligte: Frerichs, Derk [Verfasser:in]; Merdon, Christian [Verfasser:in]
  • Erschienen: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020
  • Ausgabe: published Version
  • Sprache: Englisch
  • DOI: https://doi.org/10.34657/8371; https://doi.org/10.20347/WIAS.PREPRINT.2683
  • ISSN: 2198-5855
  • Schlagwörter: Incompressible Navier--Stokes equations ; mixed virtual element method ; polygonal meshes ; divergence-free velocity reconstruction ; pressure-robustness
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  • Beschreibung: 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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