WIAS Preprint No. 2250, (2016)

Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations


  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 76D05 65M60 65M12

2010 Physics and Astronomy Classification Scheme

  • 47.11.Fg


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




Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously.
In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.

Appeared in

  • Comput. Methods Appl. Mech. Engrg., 311 (2016) pp. 304--326.

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