Error estimates for a finite element method for the drift-diffusion semiconductor device equations

In this paper, optimal error estimates are obtained for a method for numerically solving the so-called unipolar model (a one-dimensional simplified version of the drift-diffusion semi-conductor device equations). The numerical method combines a mixed finite element method using a continuous piecewise-linear approximation of the electric field with an explicit upwinding finite element method using a piecewise-constant approximation of the electron concentration. For initial and boundary data ensuring that the electron concentration is smooth, the L^∞(L¹)-error for the electron concentration and the L^∞(L^∞)-error of the electric field are both proven to be of order Δx. The error analysis is carried out first in the zero diffusion case in detail and then extended to the full unipolar model.