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Galerkin methods with finite elements are very widely used for parabolic systems like
Navire-Stokes. The conventional Galerkin Finite Elements (GFEM) are as estable for these
systems.  However, for predominantly hyperbolic systems (Euler Equations, Vlasov Equations),
GFEM loses its stability.  As a result, some form of stabilization is necessary like
a streamline upwinding has to be added.  There are also other ways to stabilize the GFEM
for hyperbolic systemsby using the so-called Petrov-Glaerkin techniques in which you
have the rial functions different from the test functions.  SOme error estimates are
available for different methods (Claes Johnson (1986), T.J. Hughes (1982)) and so on.

	Now for non-linear systems; It is well known that Navier-Stokes equations can
be primary source of chaos under appropriate parameter conditions (the Lorenz equations).
It is possible to use GFEM for such systems as well. However, it should be noted that
chaotic behavior is a characteristic of the "system of equations that one is solving" and
will manifest independent of the numerical technique.  Appropriate manipulation of the
same technique (GFEM for instanace) should not be a problem to study this behavior.

	It has also been observed that systems of equations can change their behavior
inside the domain.  FOr example parabolic systems can be come hyperbolic (boundary layer
flow, non-newtonian polymeric fluids).  In cases like these, GFEM (assuming that stabilizing
properties are already incoporated) can be used.  
	Hope that the above lines answer at least some of your questions.

S.Shankar
Tech CAD
INTEL Corporation