A Finite Element Approximation Theory for the Drift Diffusion Semiconductor Model

Two-sided estimates are derived for the approximation of solutions to the drift-diffusion steady-state semiconductor device system which are identified with fixed points of Gummel’s solution map. The approximations are defined in terms of fixed points of numerical finite element discretization maps. By use of a calculus developed by Krasnosel’skii and his coworkers, it is possible both to locate approximations near fixed points in an “a priori” manner, as well as fixed points near approximations in an “a posteriors” manner. These results thus establish a nonlinear approximation theory, in the energy norm, with rate keyed to what is possible in a standard linear theory. This analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation.