  For a model convection-diffusion problem, we obtain new error estimates for a
general upwinding finite element discretization based on bubble modification of
the test space. The key analysis tool is based on finding representations of
the optimal norms on the trial spaces at the continuous and discrete levels. We
analyze and compare the standard linear discretization, the saddle point least
square and upwinding Petrov-Galerkin methods. We conclude that the bubble
upwinding Petrov-Galerkin method is the most performant discretization for the
one dimensional model. Our results for the model convection-diffusion problem
can be extended for creating new and efficient discretizations for the
multidimensional cases.
