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 Θ ( 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 Θ ( 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<Θ ( z) 1 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.
No takes yet. Share an insight, caveat, or question.
Gräßle et al. (2024) studied this question.