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Title: Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods
Authors: Lederer, Philip LukasMerdon, ChristianSchöberl, Joachim
Publishers Version: https://doi.org/10.20347/WIAS.PREPRINT.2462
Issue Date: 2017
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2462, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the NavierStokes equations. The novel error estimators only take the curl of the righthand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the TaylorHood and mini finite element methods.
Keywords: Incompressible Navier–Stokes equations; mixed finite elements; pressure-robustness; a posteriori error estimators; adaptive mesh refinement
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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