Space-time variational saddle point formulations of Stokes and Navier–Stokes equations∗. (24th April 2014) - Guberovic, Rafaela; Schwab, Christoph; Stevenson, Rob
Abstract : The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocitiesu 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 Fréchet derivative that, at any (u, p ) ∈ 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.