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

by R. Guberovic and Ch. Schwab and R. Stevenson

(Report number 2011-66)

Abstract
The instationary Stokes and Navier-Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities ${\bf u}$ 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 Fréchet derivative that, at any $({\bf u},p) \in H_1$, 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.

Keywords: Instationary Stokes and Navier-Stokes equations, space-time variational saddle point formulation, well-posed operator equation

@Techreport{GSS11_115,
  author = {R. Guberovic and Ch. Schwab and R. Stevenson},
  title = {Space-time variational saddle point formulations of Stokes and Navier-Stokes equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-66},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-66.pdf },
  year = {2011}
}

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