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

Guberovic, Rafaela; Schwab, Christoph; Stevenson, Rob

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 875-894.

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.

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