IMPLEMENTATION AND NUMERICAL-ANALYSIS OF THE NONLINEAR GALERKIN METHODS WITH FINITE-ELEMENTS DISCRETIZATION

Some innovative methods in numerical analysis, called nonlinear Galerkin methods and stemmed from results of the dynamical systems theory, have been recently proposed by M. Marion and R. Temam in the context of numerical simulation of turbulent flow. These methods have been implemented with success in the case of pseudo-spectral discretization. Now we propose to adapt ideas of these algorithms in the case of a finite element discretization which is more relevant in fluid dynamics. We here review theoretical and numerical backgrounds of these schemes. Then on a computational side, we recall the definition of the hierarchical basis and analyze the structures associated to this basis. Finally we present the methods, we report on numerical experiments on two-dimensional Burgers' and Navier-Stokes problems and discuss their consistency with the approximations we make. Simulations were performed on the Cray 2 of the Centre de Calcul Vectoriel pour la Recherche in Palaiseau.