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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {875-894},
doi = {10.1051/m2an/2013124},
mrnumber = {3264338},
zbl = {1295.35354},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 875-894. doi : 10.1051/m2an/2013124. http://gdmltest.u-ga.fr/item/M2AN_2014__48_3_875_0/
[1] , and , Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237-1271. | MR 972452 | Zbl 0666.76055
[2] and , A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437-455. | Numdam | MR 2075754 | Zbl 1079.76042
[3] , and , Adaptive wavelet methods for elliptic operator equations - Convergence rates. Math. Comput. 70 (2001) 27-75. | MR 1803124 | Zbl 0980.65130
[4] and , Adaptive wavelets schemes for parabolic problems: Sparse matrices and numerical results. SIAM J. Numer. Anal. 49 (2011) 182-212. | MR 2783222 | Zbl 1225.65094
[5] , 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. | MR 977489 | Zbl 0681.35071
[6] and , Mathematical analysis and numerical methods for science and technology. Evolution problems I. Vol. 5. Springer-Verlag, Berlin (1992). | MR 1156075 | Zbl 0755.35001
[7] , 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). | MR 760450 | Zbl 0534.58003
[8] and , 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. | MR 1273825 | Zbl 0811.65080
[9] , , and , 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. | MR 2768221 | Zbl 1181.65119
[10] and . Space-time adaptive wavelet methods for control problems constrained by parabolic evolution equations. SIAM J. Control. Optim. 49 (2011) 1150-1170. | MR 2806579 | Zbl 1232.65099
[11] and , A regularity result for the Stokes in a convex polygon. J. Funct. Anal. 21 (1976) 397-431. | MR 404849 | Zbl 0317.35037
[12] , and , 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). | Zbl 0493.46058
[13] and , 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). | MR 350177 | Zbl 0223.35039
[14] , Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Paris (1967). | MR 227584 | Zbl 1225.35003
[15] , Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349-357. | MR 650055 | Zbl 0485.65049
[16] and , Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213-231. | MR 1310318 | Zbl 0822.65034
[17] , Multirate Numerical Integration For Ordinary Differential Equations. Ph.D. thesis. Universiteit van Amsterdam (2008).
[18] and , A space-time adaptive wavelet method for parabolic evolution problems. Math. Comput. 78 (2009) 1293-1318. | MR 2501051 | Zbl 1198.65249
[19] , Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J. (1970). | MR 290095 | Zbl 0207.13501
[20] , Adaptive wavelet methods for linear and nonlinear least squares problems. Technical report. KdVI, UvA Amsterdam. Submitted (2013). | MR 3179584 | Zbl pre06339560
[21] , Divergence-free wavelets on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations. Math. Comput. 80 (2011) 1499-1523. | MR 2785466 | Zbl 1220.35134
[22] , Divergence-free wavelets on the hypercube: General boundary conditions. ESI preprint 2417. Erwin Schrödinger Institute, Vienna. Submitted (2013).
[23] , 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). | MR 603444 | Zbl 0426.35003
[24] , Partielle Differentialgleichungen, Sobolevräume und Randwertaufgaben. Edited by B.G. Teubner, Stuttgart (1982). | MR 652934 | Zbl 0482.35001