On the parameter choice in grad-div stabilization for the Stokes equations

Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $mathcal O(1)$ . This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^{1}(Omega )$ error of the velocity and the $L^{2}(Omega )$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the $H^{1}(Omega )$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the $L^{2}(Omega )$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the $H^{1}(Omega )$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.