how to solve nonlinear systems ?

[uunet!tequila.entpe.fr!cornet]

  I'm studying fluid dynamics and I work with Navier-Stokes equations.
My purpose is actually to find the steady-state solution --- stable or
not. In the case of unstable solution, I'm obliged to solve the direct
formulation of steady-state system, with a fully-implicit quadratic part.

  I use Finite Element Method on a square domain with triangular elements
and 2nd order Lagrange functions for velocity, 1st order affine functions
for temperature and pressure.

  I obtain an important number of nonlinear equations (due to quadratic
part of steady-state formulation), and I'm searching about an efficient
numerical method to solve this system.

  I've tried Newton method, continuation method, ... but they are not very
efficient when the number of equations inceases. Moreover, it appears that
a precise initial condition is the key of convergence...

  Is there anybody who can help me for
 - finding a good initial condition,
 - suggest me an efficient numerical procedure for this problem.

  Thank's...
                                                             Jean-Michel CORNET
                                  Ecole Nationale des Travaux Publics de l'Etat
                              Departement Genie Civil et Batiment URA CNRS 1652
                                          Laboratoire des Sciences de l'Habitat
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