Divergence-free Reconstruction Operators for Pressure-Robust Stokes Discretizations with Continuous Pressure Finite Elements

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. However, 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 significantly the classical versions of those elements when the continuous pressure is comparably large.