The spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations

We investigate the spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions. The theoretical results are derived for a general form of energy stable boundary conditions, and applied to a few commonly used ones. The analysis is done on a system that simultaneously covers the nonlinear incompressible Navier–Stokes, the Oseen and the Stokes equations. When the spectrum of the spatial operator is investigated, we restrict the analysis to the Oseen and Stokes equations. The continuous analysis carries over to the discrete setting by using the summation-by-parts framework.