A finite element method with energy-adaptive grids for the coupled Schrodinger-Poisson-drift-diffusion model

In this paper, we propose a finite element method for solving the coupled model of classical drift-diffusion equations and Schrodinger-Poisson equations in simulating a resonant tunneling diode (RTD). The coupling coefficient Theta and the quantum electron density n(qu) are defined by weighted integrals of vertical bar psi(p)vertical bar(2) where psi(p) is the wave function of Schrodinger equation for each momentum p is an element of R. We first prove a growth rate of vertical bar psi(p)vertical bar as vertical bar p vertical bar -> +infinity. This enables us to truncate the momentum space R to a bounded interval [-P, P], which corresponds to an energy interval [0, E-P]. We propose an energy-adaptive algorithm based on a posteriori error estimate for computing Theta and n(qu) approximately on [0, E-P]. The algorithm reduces the number of nodes greatly by capturing resonance peaks accurately and adjusting the partition of [0, E-P] adaptively. Numerical experiment for an RTD shows that, based on energy-adaptive grids, the Gummel method for solving the coupled model is robust and can converge in several iterative steps.