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

Rafaela Guberovic; Christoph Schwab; Rob Stevenson

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 875-894
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Guberovic, Rafaela, Schwab, Christoph, and Stevenson, Rob. "Space-time variational saddle point formulations of Stokes and Navier–Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 875-894. <http://eudml.org/doc/273330>.

@article{Guberovic2014,
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.},
author = {Guberovic, Rafaela, Schwab, Christoph, Stevenson, Rob},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {instationary Stokes and Navier−Stokes equations; space-time variational saddle point formulation; well-posed operator equation; Stokes; Navier-Stokes; space-time variational saddle-point formulation; well-posedness},
language = {eng},
number = {3},
pages = {875-894},
publisher = {EDP-Sciences},
title = {Space-time variational saddle point formulations of Stokes and Navier–Stokes equations},
url = {http://eudml.org/doc/273330},
volume = {48},
year = {2014},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 875
EP - 894
AB - 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.
LA - eng
KW - instationary Stokes and Navier−Stokes equations; space-time variational saddle point formulation; well-posed operator equation; Stokes; Navier-Stokes; space-time variational saddle-point formulation; well-posedness
UR - http://eudml.org/doc/273330
ER -

References

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