SPACE-TIME VARIATIONAL SADDLE POINT FORMULATIONS OF STOKES AND NAVIER-STOKES EQUATIONS

The instationary Stokes and Navier-Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities u and pressure p. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between H-1 and H-2', both Hilbert spaces H-1 and H-2 being Cartesian products of (intersections of) Bochner spaces, or duals of those. Based on these results, the operator that corresponds to the Navier-Stokes equations is shown to map H-1 into H-2', with a Frechet derivative that, at any (u, p) is an element of H-1, is boundedly invertible. These results are essential for the numerical solution of the combined pair of velocities and pressure as function of simultaneously space and time. Such a numerical approach allows for the application of (adaptive) approximation from tensor products of spatial and temporal trial spaces, with which the instationary problem can be solved at a computational complexity that is of the order as for a corresponding stationary problem.