Singular perturbations of saddle point problems and preconditioning of the Stokes problem. (Perturbations singulières des problèmes de point selle et préconditionnement du problème de Stokes.) (French. English summary) Zbl 0822.65090

Time discretization of Navier-Stokes equations (for pressure and a velocity field) yields singularly perturbed elliptic operators. For a generalized version of the problem (with general linear, singularly perturbed operators) a uniquely determined decomposition of the space containing the solutions separates “velocity” and “pressure”. Using this, the convergence (in an appropriate sense) of the perturbed solution to the solution of the unperturbed equation is shown in the framework of saddle point problems, which had been considered in the beginning e.g. by F. Brezzi [Revue Franc. Automat. Inform. Rech. Oper. 8, R-2, 129- 151 (1974; Zbl 0338.90047)].
The results allow the construction of preconditioners for the (generalized version of the) Uzawa operator, especially for the application of finite element approximations or spectral approximations.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35Q30 Navier-Stokes equations


Zbl 0338.90047
Full Text: DOI EuDML


[1] M. AZAIEZ, 1990, Calcul de la pression dans le problème de Stokes pour les fluides visqueux incompressibles par une méthode spectrale de collocation, Thèse, Université de Paris-Sud.
[2] M. AZAIEZ, Preconditionning of Stokes problem, en préparation.
[3] M. AZAIEZ, A. FIKRI, G. LABROSSE, 1992, Direct simulation of 3-D driven cavity flow by collocation spectral method, ICOSAHOM 92, Montpellier.
[4] C. BERNARDI, Y. MADAY, 1992, Approximations spectrales des problèmes aux limites elliptiques, Mathématiques et Applications 10, Springer-Verlag. Zbl0773.47032 MR1208043 · Zbl 0773.47032
[5] H. BREZIS, 1983, Analyse fonctionnelle, théorie et applications, Masson, Paris. Zbl0511.46001 MR697382 · Zbl 0511.46001
[6] [6] F. BREZZI, 1974, On the existence uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers, R.A.I.R.O., R. 2, 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047
[7] C. CANUTO, A. QUARTERONI, 1982, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38, 67-86. Zbl0567.41008 MR637287 · Zbl 0567.41008
[8] A. CHORIN, 1968, Numerical simulation of the Navier-Stokes equations, Math. Comp., 22, 745-762. Zbl0198.50103 MR242392 · Zbl 0198.50103
[9] J. CAHOUET, J.-P. CHABARD, 1988, Some fast 3-D finite element solvers for generalized Stokes problem, Int. J. Num. Meth. in Fluids, 8, 269-295. Zbl0665.76038 MR953141 · Zbl 0665.76038
[10] V. GIRAULT, P.-A. RAVIART, 1986, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, 5, Springer-Verlag. Zbl0585.65077 MR851383 · Zbl 0585.65077
[11] D. HUET, 1964, Perturbations singulières, C. R. Acad. Sci. Paris, 258, 6320-6322. Zbl0137.29902 MR167711 · Zbl 0137.29902
[12] G. LABADIE, P. LASBLEIZ, 1983, Quelques méthodes de résolution du problème de Stokes en éléments finis, E.D.F. rapport HE41/83.01
[13] J.-L. LIONS, 1973, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, 323, Springer-Verlag. Zbl0268.49001 MR600331 · Zbl 0268.49001
[14] J.-L. LIONS, E. MAGENES, 1968, Problèmes aux limites non homogènes et applications, Dunod, Paris. Zbl0165.10801 · Zbl 0165.10801
[15] R. TEMAM, 1977, Navier-Stokes Equations, Studies in Mathematics and its Applications, 2, North-Holland. Zbl0383.35057 MR609732 · Zbl 0383.35057
[16] [16] R. TEMAM, 1968, Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France, 98, 115-152. Zbl0181.18903 MR237972 · Zbl 0181.18903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.