Two Conjectures on the Stokes Complex in Three Dimensions on Freudenthal Meshes
Patrick E. Farrell, Lawrence Mitchell, Larkin Ridgway Scott · SIAM Journal on Scientific Computing · 2024
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 [Formula: see text]; the best result available in the literature is for [Formula: see text]. The second is that there exists a stable space decomposition of the kernel of the divergence for [Formula: see text]. We present numerical evidence supporting our conjectures.