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

TL;DR

This paper introduces a modified Scott-Vogelius finite element for the 2D Stokes problem that maintains mesh-robust stability even near nearly singular vertices, improving reliability in complex meshes.

Contribution

A simple parameter-dependent modification of the Scott-Vogelius element is proposed, ensuring inf-sup stability independent of nearly singular vertices in the mesh.

Findings

The modified element is inf-sup stable regardless of mesh singularities.

Numerical experiments confirm negligible impact on divergence-free velocity approximation.

The approach enhances robustness of finite element discretizations for the Stokes problem.

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( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $\Theta \left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<\Theta \left( \mathbf{z}\right) \ll 1$. 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.