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

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.

References in corpus (3)

• An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number
• A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations
• Stability of Lagrange elements for the mixed Laplacian