P. Lederer, J. Schöberl, C. Merdon:
"Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods";
arXiv.org e-Print archive,
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 Navier--Stokes equations. The novel error estimators only take the curl of the right-hand 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 Taylor--Hood and mini finite element methods.
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