Necessary L2-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems

We derive necessary conditions for the uniform convergence (with respect to the perturbation parameter) in L2 of schemes for singularly perturbed convection-diffusion problems. Applying these conditions, we show that the convergence order of the streamline diffusion finite element method with piecewise linear trial functions cannot be better than 1/2, uniformly in the singular perturbation parameter.