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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
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
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