Méthodes de Galerkin non linéaires en discrétisation par éléments finis et pseudo-spectrale. Application à la mécanique des fluides

For computing the solution of partial differential equations in fluid mechanics and physics, some new numerical schemes, the nonlinear Galerkin methods, mo­ tivated by concepts of the dynamical systems theory were developped by Marion and Temam. In the case of a pseudo-spectral discretization, the first part is setting out theoretical motivations and principles of the new schemes with models of the turbulence. It is proved that the evolution of small scales can be frozen if the cut­ off number is near dissipation range : some criterions for the selection of small scales and some characteristic times are introduced and analyzed. The second part concerns the description of the classical mixed, conforming P1-4P1 finite element method for computing the solution of the Navier-Stokes equations. The numerical difficulties due to the incompressibility, the nonlinearity and time evolution are described. A code which permits to obtain some data bases for the driven cavity, has been completely developped. In the third part, small and large structures induced by the hierarchical basis are shown and studied (particularly in the large gradient region) in the perspective of implementing nonlinear Galerkin methods. Then a preconditioned conjugated gradient method for the Dirichlet problem is developped based on the multigrid structure of the hierarchical basis. After a study of the numerical stability of the schemes with a linear case, the description of algorithms which consist of determining separately the different structures are proposed in the fourth part of this thesis. Some numerical results are presented for Burgers and Navier-Stokes problems in dimension two.