State of the art programs for the solution of the drift diffusion semiconductor equations are based on finite difference techniques, or on certain combinations of finite elements and finite differences. The extreme gradients which occur in semiconductor devices have motivated several attempts to exploit perturbation analysis in order to improve the numerical scheme. Selberherr, Markowich et. al. showed the singular perturbation nature of the drift diffusion equations. Their analytic approximations of diodes influenced some aspects of the Minimos and Bambi simulators. Asymptotic analyses of MOSFET were presented by Brews and by Ward. These solutions are mainly valid for long channels, and their accuracy is limited.
Efrat, I. and Israeli, M. (1991), "A HYBRID SOLUTION OF THE SEMICONDUCTOR DEVICE EQUATIONS", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 10 No. 4, pp. 215-229. https://doi.org/10.1108/eb051700
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