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

Abstract

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 \( \Theta \left( \textbf{z}\right) \ge 0\) for triangle vertices \(\textbf{z}\)) and imposes a local side condition on the pressure space associated to vertices \(\textbf{z}\) with \(\Theta \left( \textbf{z}\right) =0\). The method is inf-sup stable for any fixed regular triangulation and \(k\ge 4\). However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices \(0<\Theta \left( \textbf{z}\right) \ll 1\). In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.