## Calcul de la pression dans la résolution spectrale du problème de Stokes. (Computation of the pressure in the spectral approximation of the Stokes problem).(French)Zbl 0642.76037

The authors analyse different spectral methods to treat the Stokes problem of creeping flows in a square or cube (subjected to periodicity or Dirichlet or mixed boundary conditions) in a unifying manner by means of Brezzi’s and Raviarts’ theory of mixed approximation. For the methods of spectral Galerkin as well as spectral collocation, compatible discrete spaces have been theoretically constructed and optimal convergence results have been obtained for any kind of the boundary conditions as above. Numerical test computations are not included. It is worth mentioning that the theory of mixed approximation is actually not required at least in the simplest case of the periodicity condition. Concerning this, see the second author and A. Quarterioni, SIAM J. Numer. Anal. 19, 761-780 (1982; Zbl 0503.76035) or W. Borchers, “Eine Fourier-Spektralmethode für das Stokes-Resolventenproblem” (to appear in Numer. Math.), where even the additional error due to evaluating the Fourier coefficients by the trapezoidal rule has been estimated.
Reviewer: F.-K.Hebeker

### MSC:

 76D07 Stokes and related (Oseen, etc.) flows 35Q30 Navier-Stokes equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs

Zbl 0503.76035