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

TL;DR

This paper discusses two conjectures related to the stability and decomposition of discretizations of the 3D Stokes equations on Freudenthal meshes, supported by numerical evidence, aiming to improve pressure-robust methods.

Contribution

It proposes two new conjectures on the stability of Scott-Vogelius elements and divergence kernel decomposition in 3D, extending understanding beyond current literature.

Findings

Numerical evidence supports the conjectures.

Conjecture 1: Scott-Vogelius is inf-sup stable for k ≥ 4.

Conjecture 2: Stable divergence kernel decomposition exists for k ≥ 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.