The pressure-wired Stokes element: a mesh-robust version of the Scott-Vogelius element
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The conforming Scott-Vogelius pair for the stationary Stokes equation in 2D is a popular finite element which is inf-sup stable for any fixed regular triangulation. However, the inf-sup constant deteriorates if the "singular distance" (measured by some geometric mesh quantity Θ_min>0) of the finite element mesh to certain "singular" mesh configurations is small. In this paper we present a modification of the classical Scott-Vogelius element of arbitrary polynomial order k≥4 for the velocity where a constraint on the pressure space is imposed if locally the singular distance is smaller than some control parameter η>0. We establish a lower bound on the inf-sup constant in terms of Θ_min+η>0 independent of the maximal mesh width and the polynomial degree that does not deteriorate for small Θ_min≪ 1. The divergence of the discrete velocity is at most of size 𝒪(η) and very small in practical examples. In the limit η→0 we recover and improve estimates for the classical Scott-Vogelius Stokes element.
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