Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem.

*(English)*Zbl 0666.76055The Stokes flow problem and its approximations are considered in an abstract framework making use of a Galerkin method and a collocation method formulated in Chebyshev weighted Sobolev function spaces and their approximations. The Stokes problem has been transformed into a variational formulation in the continuous case and the finite-dimensional approximation and then a weighted variational formulation of the problem has been given in order to analyse a spectral Chebyshev approximation of the Stokes problem. The inf-sup conditions have been determined which are necessary and sufficient for the existence of a unique variational solution to the Stokes problem under consideration. A Galerkin spectral method is employed to discretize the problem and the spectral collocation method is analysed for the Stokes problem.

Reviewer: Z.Dżygadło

##### 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 |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |