We present recent finite element numerical results on a model
convection-diffusion problem in the singular perturbed case when the convection
term dominates the problem. We compare the standard Galerkin discretization
using the linear element with a saddle point least square discretization that
uses quadratic test functions, trying to control and explain the non-physical
oscillations of the discrete solutions. We also relate the up-winding
Petrov-Galerkin method and the stream-line diffusion discretization method, by
emphasizing the resulting linear systems and by comparing appropriate error
norms. Some results can be extended to the multidimensional case in order to
come up with efficient approximations for more general singular perturbed
problems, including convection dominated models.Comment: 24 pages, 12 figure