The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element

The conforming Scott-Vogelius pair for the stationary Stokes equation in 2D is a popular finite element which is inf-sup stable for any fixed regular triangulation. However, the inf-sup constant deteriorates if the "singular distance" (measured by some geometric mesh quantity $\Theta_{\min}>0$) of the finite element mesh to certain "singular" mesh configurations is small. In this paper we present a modification of the classical Scott-Vogelius element of arbitrary polynomial order $k\geq4$ for the velocity where a constraint on the pressure space is imposed if locally the singular distance is smaller than some control parameter $\eta>0$. We establish a lower bound on the inf-sup constant in terms of $\Theta_{\mathrm{\min}}+\eta>0$ independent of the maximal mesh width and the polynomial degree that does not deteriorate for small $\Theta_{\min} \ll 1$. The divergence of the discrete velocity is at most of size $\mathcal{O}(\eta)$ and very small in practical examples. In the limit $\eta\rightarrow0$ we recover and improve estimates for the classical Scott-Vogelius Stokes element.