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

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 Θ( 𝐳) ≥ 0 for triangle vertices 𝐳) and imposes a local side condition on the pressure space associated to vertices 𝐳 with Θ( 𝐳) =0. The method is inf-sup stable for any fixed regular triangulation and k≥ 4. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0<Θ( 𝐳) ≪ 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.