Uniform Convergence of an Exponentially Fitted Scheme for the Quantum Drift Diffusion Model

We analyze an exponentially fitted finite element scheme for the unipolar quantum drift diffusion model in one-dimensional space. The existence of discrete solutions is shown under very mild assumptions, and convergence of a subsequence is proved by compactness arguments. The scheme is constructed in such a way that it reduces in the semiclassical limit to the well-known Scharfetter--Gummel discretization for the classical drift diffusion model. We derive uniform error bounds which allow for the semiclassical limit on the discrete level. Numerical tests underlining the analytical results are presented.