Using a technique for constructing analytic expressions for discrete solutions to the convection-diffusion equation, we examine and characterise the effects of upwinding strategies on solution quality. In particular, for grid-aligned flow and discretisation based on bilinear finite elements with streamline upwinding, we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when boundary layers are present. In addition, we show that the same analytic techniques provide insight into other discretisations, such as a finite difference method that incorporates streamline diffusion, and the isotropic artificial diffusion method. (Also cross-referenced as UMIACS-TR-2000-50)