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 ${\text{L}}^\infty ({\text{L}}^1 )$-error for the electron concentration and the ${\text{L}}^\infty ({\text{L}}^\infty )$-error of the electric field are both proven to be of order $\Delta x$. The error analysis is carried out first in the zero diffusion case in detail and then extended to the full unipolar model.