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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.
Keywords: instationary Stokes and Navier−Stokes equations, space-time variational saddle point formulation, well-posed operator equation
@article{M2AN_2014__48_3_875_0,
author = {Guberovic, Rafaela and Schwab, Christoph and Stevenson, Rob},
title = {Space-time variational saddle point formulations of {Stokes} and {Navier-Stokes} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {875--894},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013124},
mrnumber = {3264338},
zbl = {1295.35354},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013124/}
}
TY - JOUR AU - Guberovic, Rafaela AU - Schwab, Christoph AU - Stevenson, Rob TI - Space-time variational saddle point formulations of Stokes and Navier-Stokes equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 875 EP - 894 VL - 48 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013124/ DO - 10.1051/m2an/2013124 LA - en ID - M2AN_2014__48_3_875_0 ER -
%0 Journal Article %A Guberovic, Rafaela %A Schwab, Christoph %A Stevenson, Rob %T Space-time variational saddle point formulations of Stokes and Navier-Stokes equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 875-894 %V 48 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013124/ %R 10.1051/m2an/2013124 %G en %F M2AN_2014__48_3_875_0
Guberovic, Rafaela; Schwab, Christoph; Stevenson, Rob. Space-time variational saddle point formulations of Stokes and Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 875-894. doi: 10.1051/m2an/2013124
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