Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Benedikt Gräßle, Nis-Erik Bohne, Stefan A. Sauter

TL;DR

This work introduces the pressure-wired Stokes element, a simple parameter-driven modification of the Scott-Vogelius discretization that attains mesh-robust inf-sup stability independent of nearly-singular vertices by constraining the pressure space at $\eta$-critical vertices. The key insight is a lower bound on the inf-sup constant proportional to $\Theta_{\min}+\eta$, and the construction of a right-inverse for divergence that preserves stability while allowing a controlled, localized divergence that vanishes away from the $\eta$-critical region under mild conditions. Theoretical results are complemented by numerical experiments showing optimal convergence rates and negligible discrete divergence for practical choices of $\eta$, highlighting the method's robustness without mesh modification. The approach offers a simple, effective remedy to the sensitivity of the Scott-Vogelius element to nearly singular mesh configurations, with potential for reliable velocity-pressure approximations in incompressible flow simulations.

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 $ Θ\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 $Θ\left( \mathbf{z}\right) =0$. 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 $0<Θ\left( \mathbf{z}\right) \ll 1$. 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.

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

TL;DR

This work introduces the pressure-wired Stokes element, a simple parameter-driven modification of the Scott-Vogelius discretization that attains mesh-robust inf-sup stability independent of nearly-singular vertices by constraining the pressure space at -critical vertices. The key insight is a lower bound on the inf-sup constant proportional to , and the construction of a right-inverse for divergence that preserves stability while allowing a controlled, localized divergence that vanishes away from the -critical region under mild conditions. Theoretical results are complemented by numerical experiments showing optimal convergence rates and negligible discrete divergence for practical choices of , highlighting the method's robustness without mesh modification. The approach offers a simple, effective remedy to the sensitivity of the Scott-Vogelius element to nearly singular mesh configurations, with potential for reliable velocity-pressure approximations in incompressible flow simulations.

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 and a discontinuous pressure approximation of order . It employs a "singular distance" (measured by some geometric mesh quantity for triangle vertices ) and imposes a local side condition on the pressure space associated to vertices with . The method is inf-sup stable for any fixed regular triangulation and . However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices . 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.
Paper Structure (8 sections, 12 theorems, 80 equations, 6 figures)

This paper contains 8 sections, 12 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Stokes problem and its numerical discretization
  3. The pressure-wired Stokes element
  4. Inf-sup stability of the pressure-wired Stokes element
  5. Divergence estimate of the discrete velocity
  6. Analysis for $\eta$-critical vertices
  7. Proof of Lemma \ref{['lem:div_est']}
  8. Numerical experiments

Key Result

Lemma 1

There exists a constant $\eta_{0}>0$ only depending on the shape-regularity of the mesh and the minimal outer angle $\alpha_{\mathcal{T}}$ of $\Omega$ such that $\Theta(\mathbf{z})\leq \eta_{0}$ for $\mathbf{z}\in\mathcal{V}(\mathcal{T})$ implies

Figures (6)

  • Figure 1: Reference triangle (left) and physical triangle (right)
  • Figure 2: Singular configuration (top) and generic configuration (bottom) of a vertex patch for an interior vertex $\mathbf{z}\in \mathcal{V}_{\Omega }(\mathcal{T})$ with $N_{\mathbf{z}}=4$ (resp. boundery vertex $\mathbf{z}\in \mathcal{V}_{\partial \Omega }(\mathcal{T})$ with $N_{\mathbf{z}}=1,2,3$) triangles
  • Figure 3: Setting of Lemma \ref{['lem:A_z_explicit']}
  • Figure 4: Perturbation $\mathcal{T}_\varepsilon$ (blue) of the criss-cross triangulation $\mathcal{T}_0$ (gray) of the square
  • Figure 5: Convergence history of the total error (left) and of the divergence (right) with $\mathbf{z}_\varepsilon$ treated as $\eta$-critical $\mathbf{z}_\varepsilon\in\mathcal{C}_\mathcal{T}(\eta)$ or non-singular $\mathbf{z}_\varepsilon\not\in\mathcal{C}_\mathcal{T}(\eta)$ vertex for the $h$-method
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1: SauterCR_prob
  • Definition 3
  • Proposition 1: Scott-Vogelius
  • Theorem 2: inf-sup stability
  • Lemma 2
  • Lemma 3: AP_Locking
  • proof
  • ...and 19 more