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 k - 1. It employs a “singular distance” (measured by some geometric mesh quantity (z) 0 Θ z ≥ 0 for triangle vertices z z) and imposes a local side condition on the pressure space associated to vertices z z with (z) =0 Θ z = 0. The method is inf-sup stable for any fixed regular triangulation and k 4 k ≥ 4. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0 0 Θ z ≪ 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.