Space-time variational saddle point formulations of Stokes and Navier–Stokes equations^∗

¹ Seminar for Applied Mathematics, ETH Zürich, CH 8092 Zürich, Switzerland
rafaela.guberovic@sam.math.ethz.c; christoph.schwab@sam.math.ethz.ch
² Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
R.P.Stevenson@uva.nl

Received: 5 October 2012
Revised: 1 August 2013

Abstract

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 H₁ and H'₂, both Hilbert spaces H₁ and H₂ 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 H₁ into H'₂, with a Fréchet derivative that, at any (u,p) ∈ H₁, 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.