Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements
- Lederer, Philip L.
- Linke, Alexander
- Merdon, Christian
- Schöberl, Joachim
2010 Mathematics Subject Classification
- 65N12 65N12 76D07 76D05 76M10
- incompressible Navier--Stokes equations, mixed finite elements, pressure robustness, exact divergence-free velocity reconstruction, flux equilibration
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.
- SIAM J. Numer. Anal., 55 (2017) pp. 1291--1314.