On the uniform convergence of Scharfetter-Gummel discretization in one dimension

A convergence analysis is given for the Scharfetter–Gummel discretization of proto-type one-dimensional continuity equations as arise in the drift-diffusion system modeling semiconductors. These are linear, second-order, boundary-value problems whose coefficient functions are $O(1)$ but can have derivatives that are $O({1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda })$, for a small positive parameter $\lambda $. For such problems, discretizations “of Scharfetter–Gummel type” are proved to be first-order accurate on general meshes uniformly in $\lambda $ in a strong global sense. These results improve upon previous analyses, where discrete convergence of $O(h + \lambda |\ln \lambda |)$ was believed to be the best possible.