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Space-time variational saddle point formulations of Stokes andNavier–Stokes equations∗

Published online by Cambridge University Press:  24 April 2014

Rafaela Guberovic ,
Christoph Schwab  and
Rob Stevenson
Rafaela Guberovic
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, CH 8092 Zürich, Switzerland. [email protected]; [email protected]
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, CH 8092 Zürich, Switzerland. [email protected]; [email protected]
Rob Stevenson
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands; [email protected]
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Abstract

The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-timevariational saddle point formulation, so involving both velocities u and pressure p. For the instationaryStokes problem, it is shown that the corresponding operator is a boundedlyinvertible linear mapping between H1 and H'2, both Hilbertspaces H1 and H2 beingCartesian products of (intersections of) Bochner spaces, or duals of those. Based on theseresults, the operator that corresponds to the Navier−Stokes equations is shown to mapH1 into H'2, with a Fréchetderivative that, at any (u,p) ∈H1, is boundedly invertible. These resultsare essential for the numerical solution of the combined pair of velocities and pressureas function of simultaneously space and time. Such a numerical approach allows for theapplication of (adaptive) approximation from tensor products of spatial and temporal trialspaces, with which the instationary problem can be solved at a computational complexitythat is of the order as for a corresponding stationary problem.

Keywords

Instationary Stokes and Navier−Stokes equationsspace-time variational saddle point formulationwell-posed operator equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 875 - 894
Copyright
© EDP Sciences, SMAI, 2014

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