Convergent finite element discretizations of the density gradient equation for quantum semiconductors

We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. In particular, we give a finite element description of the so-called nonlinear scheme introduced by Ancona. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, in particular, with respect to the influence of approximate vacuum boundary conditions.