Explaining oscillatory behavior in convection-diffusion discretization

For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions, and study difficulties in recovering standard approximation results for its solution. We justify the presence of non-physical oscillations and propose ways to eliminate oscillations. A new approach for error analysis that requires establishing optimal discrete infinity error as a first step is introduced and justified. We emphasize that the discretization of two dimensional convection dominated problems benefit from the efficient discretization of the corresponding one dimensional problem along each stream line. Our results are useful in building new and robust discretizations for multi-dimensional convection dominated problems.