Résumé:
In this thesis, some novel discrete formulations for stabilizing the mixed nite element method Q1-Q0 (bilinear velocity and constant pressure approximations) are introduced and discussed for the generalized Stokes problem. These are based on stabilizing discontinuous pressure approximations via local jump operators. The developing idea consists in a reduction of terms in the local jump formulation, introduced earlier, in such a way that stability and convergence properties are preserved. Moreover, some iterative methods of conjugate gradient type are discussed and their algorithms are presented. These are used for the iterative solution of the algebraic systems of linear equations which arise from the spatial discretization of the continuous problem. The computer implementation aspects and numerical evaluation of the stabilized discrete formulations are also considered. For illustrating the numerical performance of the proposed approaches and comparing the three versions of the local jump methods against the global jump setting, some obtained results for two test generalized Stokes problems are presented. Numerical tests con rm the stability and accuracy characteristics of the resulting approximations. Likewise, the numerical reliability of the discussed iterative solvers is assessed.
