Electro-osmotic flows in the vicinity of surface charge density transitions are studied using a two-dimensional model problem. The analysed configuration consists of an electrolyte solution in contact with a dielectric planar wall, on which the density transition is approximated by a finite jump. The flow field in the semi-infinite fluid domain is driven by an external electric field which is applied parallel to the charged wall, in the direction of the jump. The problem is analysed using both the thin-Debye-layer (TDL) formulation, which does not resolve the fine details of the Debye-layer structure, and an approximate electrokinetic model.