WIAS Preprint No. 2402, (2017)

Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations


  • Ahmed, Naveed
    ORCID: 0000-0002-9322-0373
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65M12 665M30 65M15 76D07 76M10


  • incompressible Navier--Stokes equations, mixed finite element methods, pressure-robustness, Helmholtz projector, a priori error estimates




In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly.

Appeared in

  • Comput. Methods Appl. Math., 18 (2018), pp. 353--372 (published online on 18.11.2017), DOI 10.1515/cmam-2017-0047 .

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