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

Guberovic, Rafaela; Schwab, Christoph; Stevenson, Rob

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 875-894.

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.

Classification : 35A15, 35Q30, 76D05, 76D07, 76M30
Mots clés : instationary Stokes and Navier−Stokes equations, space-time variational saddle point formulation, well-posed operator equation

[1] C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237-1271. | MR | Zbl

[2] C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437-455. | Numdam | MR | Zbl

[3] A. Cohen, W. Dahmen and R. Devore, Adaptive wavelet methods for elliptic operator equations - Convergence rates. Math. Comput. 70 (2001) 27-75. | MR | Zbl

[4] N.G. Chegini and R.P. Stevenson, Adaptive wavelets schemes for parabolic problems: Sparse matrices and numerical results. SIAM J. Numer. Anal. 49 (2011) 182-212. | MR | Zbl

[5] M. Dauge, 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 | Zbl

[6] R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Evolution problems I. Vol. 5. Springer-Verlag, Berlin (1992). | MR | Zbl

[7] 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). | MR | Zbl

[8] 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. | MR | Zbl

[9] 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. | MR | Zbl

[10] M.D. Gunzburger and A. Kunoth. Space-time adaptive wavelet methods for control problems constrained by parabolic evolution equations. SIAM J. Control. Optim. 49 (2011) 1150-1170. | MR | Zbl

[11] R.B. Kellogg and J.E. Osborn, A regularity result for the Stokes in a convex polygon. J. Funct. Anal. 21 (1976) 397-431. | MR | Zbl

[12] 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). | Zbl

[13] 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). | MR | Zbl

[14] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Paris (1967). | MR | Zbl

[15] R.A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349-357. | MR | Zbl

[16] J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213-231. | MR | Zbl

[17] V. Savcenco, Multirate Numerical Integration For Ordinary Differential Equations. Ph.D. thesis. Universiteit van Amsterdam (2008).

[18] Ch. Schwab and R.P. Stevenson, A space-time adaptive wavelet method for parabolic evolution problems. Math. Comput. 78 (2009) 1293-1318. | MR | Zbl

[19] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J. (1970). | MR | Zbl

[20] R.P. Stevenson, Adaptive wavelet methods for linear and nonlinear least squares problems. Technical report. KdVI, UvA Amsterdam. Submitted (2013). | MR

[21] R.P. Stevenson, 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 | Zbl

[22] R.P. Stevenson, Divergence-free wavelets on the hypercube: General boundary conditions. ESI preprint 2417. Erwin Schrödinger Institute, Vienna. Submitted (2013).

[23] 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). | MR | Zbl

[24] J. Wloka, Partielle Differentialgleichungen, Sobolevräume und Randwertaufgaben. Edited by B.G. Teubner, Stuttgart (1982). | MR | Zbl