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

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Date
2016
Volume
2288
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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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 (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.

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Keywords
Incompressible Navier–Stokes equations, mixed finite elements, pressure robustness, exact divergence-free velocity reconstruction, flux equilibration
Citation
Lederer, P. L., Linke, A., Merdon, C., & Schöberl, J. (2016). Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements (Vol. 2288). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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