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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.

MSC:

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

Citations:

Zbl 0338.90047
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References:

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