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

TL;DR

This paper introduces a velocity reconstruction operator to modify classical pressure-robust finite element methods for Stokes equations, improving accuracy especially with large pressures.

Contribution

It extends pressure-robust discretizations to continuous pressure elements like Taylor-Hood and mini, using a novel divergence-free reconstruction operator.

Findings

Reconstruction operator achieves optimal convergence rates.

Enhanced pressure-robustness over classical elements.

Significant performance improvements with large pressures.

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. 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 $H(\mathrm{div})$-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.