Title:
Pressure-robust mixed finite element methods and refined a posteriori error control for the Stokes problem


Abstract:

Classical inf-sup stable mixed finite element methods for the steady
incompressible Stokes equations that relax the divergence constraint lead to
a pressure-dependent contribution in the velocity error, which is proportional
to the inverse of the viscosity.

A recently proposed modification of the testfunctions only in the right hand
side leads to a discretization that is pressure-robust, i.e., its velocity
error converges with optimal order and is independent of the pressure and the
smallness of the viscosity. This talk reports on recent developments of this
approach.

The main part concerns a refined a posteriori velocity error control that
reflects the pressure-robustness. Here, the main difficulty lies in the volume
contribution of the standard residual-based approach that includes the
\(L^2\)-norm of the right-hand side. However, the velocity is only steered
by the divergence-free part of this source term and 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 only takes
the $\mathrm{curl}$ of the right-hand side into account. The pressure-robustness and efficiency of the error estimator is confirmed by some numerical examples.