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

In this paper we present a novel exponentially fitted finite elementmethod with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices.The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkinfinite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded asan extension to two dimensions of the well-known Scharfetter-Gummel method.Error estimates for the approximate solution and its associated flux are given. These h-order error bounds depend on some first-order seminorms of the exact solution, the exact fluxand the coefficient function of the convection terms.A method is also proposed for the evaluation of terminal currentsand it is shown that the computed terminal currents are convergent andconservative.