WIAS Preprint No. 2288, (2016)

Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements



Authors

  • Lederer, Philip L.
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
  • Schöberl, Joachim

2010 Mathematics Subject Classification

  • 65N12 65N12 76D07 76D05 76M10

Keywords

  • incompressible Navier--Stokes equations, mixed finite elements, pressure robustness, exact divergence-free velocity reconstruction, flux equilibration

Abstract

Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. How-ever, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations confirm that the new pressure-robust Taylor--Hood and mini elements converge with optimal order and outperform signi--cantly the classical versions of those elements when the continuous pressure is comparably large.

Appeared in

  • SIAM J. Numer. Anal., 55 (2017) pp. 1291--1314.

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