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Finite Element Pressure Stabilizations for Incompressible Flow Problems

Chapter
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

Abstract

The simulation of incompressible flow problems with pairs of velocity-pressure finite element spaces that do not satisfy the discrete inf-sup condition requires a so-called pressure stabilization. This chapter provides a survey of available methods which are presented for the Stokes problem to concentrate on the main ideas and to avoid additional difficulties originating from more complicated models. The methods can be divided into residual-based stabilizations and stabilizations that utilize only the pressure. For the first class, a comprehensive numerical analysis is presented, whereas for the second class, the presentation is more concise except for a detailed analysis of a local projection stabilization method. Connections of various pressure stabilizations to inf-sup stable discretizations with velocity spaces enriched by bubble functions are also discussed. Numerical studies compare several of the available pressure stabilizations.

Keywords

Stokes equations Discrete inf-sup condition Pressure-stabilized Petrov–Galerkin (PSPG) method Galerkin least squares (GLS) method Douglas–Wang method Local projection stabilization (LPS) method Velocity finite element spaces with bubble functions 

MSC 2010

65N30 65N12 65N15 

Notes

Acknowledgements

The work of P. Knobloch was supported through the grant No. 16-03230S of the Czech Science Foundation.

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Authors and Affiliations

  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Charles UniversityPragueCzech Republic

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