The cell-vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics: more recently, one of us has proposed an extension for viscous flows. The purpose of the present paper is two-fold: first we have applied this scheme to a well-known convection-diffusion model problem, involving flow round a 180 ∘ 180^\circ bend, which highlights some of the issues concerning the application of the boundary conditions in such cell-based schemes. The results are remarkably good when the boundary conditions are applied in an appropriate manner. In our efforts to explain the high quality of the results we were led to a detailed analysis of the corresponding one-dimensional problem. Our second purpose is thus to gather together various approaches to the analysis of this problem and to draw attention to the supra-convergence phenomena enjoyed by the proposed methods.