The paper is devoted to the convergence analysis of a well-known
cell-centered Finite Volume Method (FVM) for a
convection-diffusion problem in $\mathbb{R}^2$. This FVM is based on Voronoi
boxes and
exponential fitting. To prove the convergence of the FVM, we use
a new nonconforming Petrov-Galerkin Finite Element Method (FEM)
for which the system of linear equations coincides completely with
that of the FVM. Thus, by proving convergence properties of the
FEM we obtain similar ones for the FVM. For the error estimation
of the FEM well-known statements have to be modified.