Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T17:14:42.451Z Has data issue: false hasContentIssue false
  • English
  • Français

Space-time variational saddle point formulations of Stokes and Navier–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. rafaela.guberovic@sam.math.ethz.c; christoph.schwab@sam.math.ethz.ch
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, CH 8092 Zürich, Switzerland. rafaela.guberovic@sam.math.ethz.c; christoph.schwab@sam.math.ethz.ch
Rob Stevenson
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands; R.P.Stevenson@uva.nl
Get access

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernardi, C., Canuto, C. and Maday, Y., Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237–1271. Google Scholar
Bernardi, C. and Verfürth, R., A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437–455. Google Scholar
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet methods for elliptic operator equations – Convergence rates. Math. Comput. 70 (2001) 27–75. Google Scholar
Chegini, N.G. and Stevenson, R.P., Adaptive wavelets schemes for parabolic problems: Sparse matrices and numerical results. SIAM J. Numer. Anal. 49 (2011) 182–212. Google Scholar
Dauge, M., Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations. SIAM J. Math. Anal. 20 (1989) 74–97. Google Scholar
R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Evolution problems I. Vol. 5. Springer-Verlag, Berlin (1992).
G. de Rham, Differentiable manifolds. Forms, currents, harmonic forms. Vol. 266 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Translated from the French by F.R. Smith, With an introduction by S.S. Chern. Springer-Verlag, Berlin (1984).
R.E. Ewing and R.D. Lazarov, Approximation of parabolic problems on grids locally refined in time and space, in vol. 14 of Proc. of the Third ARO Workshop on Adaptive Methods for Partial Differential Equations. Troy, NY 1992 (1994) 199–211.
I. Faille, F. Nataf, F. Willien and S. Wolf, Two local time stepping schemes for parabolic problems. In vol. 29, Multiresolution and adaptive methods for convection-dominated problems. ESAIM Proc. EDP Sciences, Les Ulis (2009) 58–72.
Gunzburger, M.D. and Kunoth, A.. Space-time adaptive wavelet methods for control problems constrained by parabolic evolution equations. SIAM J. Control. Optim. 49 (2011) 1150–1170. Google Scholar
Kellogg, R.B. and Osborn, J.E., A regularity result for the Stokes in a convex polygon. J. Funct. Anal. 21 (1976) 397–431. Google Scholar
S.G. Kreĭn, Yu.Ī. Petunīn and E.M. Semënov, Interpolation of linear operators. In vol. 54 of Translations of Mathematical Monographs. Translated from the Russian by J. Szűcs. American Mathematical Society, Providence, R.I. (1982).
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. In vol. I. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York (1972).
J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Paris (1967).
Nicolaides, R.A., Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349–357. Google Scholar
Pousin, J. and Rappaz, J., Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213–231. Google Scholar
V. Savcenco, Multirate Numerical Integration For Ordinary Differential Equations. Ph.D. thesis. Universiteit van Amsterdam (2008).
Schwab, Ch. and Stevenson, R.P., A space-time adaptive wavelet method for parabolic evolution problems. Math. Comput. 78 (2009) 1293–1318. Google Scholar
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J. (1970).
R.P. Stevenson, Adaptive wavelet methods for linear and nonlinear least squares problems. Technical report. KdVI, UvA Amsterdam. Submitted (2013).
Stevenson, R.P., Divergence-free wavelets on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations. Math. Comput. 80 (2011) 1499–1523. Google Scholar
R.P. Stevenson, Divergence-free wavelets on the hypercube: General boundary conditions. ESI preprint 2417. Erwin Schrödinger Institute, Vienna. Submitted (2013).
R. Temam, Navier-Stokes equations. Theory and numerical analysis, with an appendix by F. Thomasset. In vol. 2 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam, revised edition (1979).
J. Wloka, Partielle Differentialgleichungen, Sobolevräume und Randwertaufgaben. Edited by B.G. Teubner, Stuttgart (1982).