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Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

Patrick E. Farrell, Lawrence Mitchell, L. Ridgway Scott

TL;DR

This work state two conjectures that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree $k \ge 4$ and that there exists a stable space decomposition of the kernel of the divergence for $k\ge 5$.

Abstract

In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust, i.e. the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions, but remain poorly understood in three dimensions. In this work we state two conjectures on this subject. The first is that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree $k \ge 4$; the best result available in the literature is for $k \ge 6$. The second is that there exists a stable space decomposition of the kernel of the divergence for $k \ge 5$. We present numerical evidence supporting our conjectures.

Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

TL;DR

This work state two conjectures that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree and that there exists a stable space decomposition of the kernel of the divergence for .

Abstract

In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust, i.e. the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions, but remain poorly understood in three dimensions. In this work we state two conjectures on this subject. The first is that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree ; the best result available in the literature is for . The second is that there exists a stable space decomposition of the kernel of the divergence for . We present numerical evidence supporting our conjectures.
Paper Structure (21 sections, 1 theorem, 46 equations, 3 figures, 6 tables)

This paper contains 21 sections, 1 theorem, 46 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Inf-sup stability of the Scott--Vogelius element
  3. Computing the inf-sup constant
  4. Using the power method
  5. Eigenvalue bounds
  6. Effect of round-off error
  7. Full algorithm
  8. 2D tests
  9. 3D tests
  10. Size of div V_h^k
  11. More general elements
  12. Space decompositions for the kernel of the divergence
  13. Background
  14. Space decompositions from de Rham complexes
  15. Second conjecture
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose that $\Vert v \Vert_V=\sqrt{a(v,v)}$ and $\Pi_h={{\nabla\cdot} \,} V_h$. Then where $\kappa$ is defined in eqn:bisbetaabndt.

Figures (3)

  • Figure 1: Unit cube for a structured mesh in three dimensions used for computational experiments. Given $N \in \mathbb{N}$, a $N \times N \times N$ mesh of cubes is generated, each of which is subdivided into 6 tetrahedra as shown. Vertex labels correspond to the vertices enumerated in \ref{['sec:freudenthal']}.
  • Figure 1: Solver diagram for \ref{['eq:nearly_singular']}.
  • Figure 2: Two types of regular meshes in two dimensions. The Type II mesh is also known as the Malkus split.

Theorems & Definitions (3)

  • Conjecture 1
  • Lemma 1
  • Conjecture 2