We give an interpretation of the Scharfetter-Gummel (SG) scheme in the theory of semiconductor devices. The key fact is that the SG scheme is based on a harmonic relation between the Green function and the Green matrix for a two-point boundary value problem. An a consequence of our interpretation, we obtain error estimates in $L^r$, $r\in[2,\infty]$, where the coefficient functions may be discontinuous.
Norikazu Saito. "An interpretation of the Scharfetter-Gummel finite difference scheme." Proc. Japan Acad. Ser. A Math. Sci. 82 (10) 187 - 191, JANUARY 2007. https://doi.org/10.3792/pjaa.82.187