A model of the time-dependent convection-diffusion problem whose solution may exhibit interior and boundary layers is considered. The standard streamline diffusion scheme, with piecewise linear elements on a uniform mesh for such a problem is convergent only at points that are not close to any layer. The main of the paper is to replace the uniform mesh by a special piecewise uniform mesh that it chosen a priori and resolves the part of any outflow boundary layer. The resulting method is convergent, independently of the diffusion parameter, with a pointwise accuracy almost of order 1.25 away from layers and almost of the order 0.75 inside the boundary layer.