  The Scott-Vogelius finite element pair for the numerical discretization of
the stationary Stokes equation in 2D is a popular element which is based on a
continuous velocity approximation of polynomial order $k$ and a discontinuous
pressure approximation of order $k-1$. It employs a "singular distance"
(measured by some geometric mesh quantity $ \Theta \left( \mathbf{z}\right)
\geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition
on the pressure space associated to vertices $\mathbf{z}$ with $\Theta \left(
\mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular
triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the
triangulation contains nearly singular vertices $0<\Theta \left(
\mathbf{z}\right) \ll 1$.
  In this paper, we introduce a very simple parameter-dependent modification of
the Scott-Vogelius element such that the inf-sup constant is independent of
nearly-singular vertices. We will show by analysis and also by numerical
experiments that the effect on the divergence-free condition for the discrete
velocity is negligibly small.
