Discretization of Three Dimensional Drift-Diffusion Equations by Numerically Stable Finite Elements

Abstract

Many of the commonly employed discretizations of the drift-diffusion current continuity equations (including the Scharfetter-Gummel perpendicular bisector box-method discretization) can be expressed in terms of the Slotboom variables ν and ω. The exponential upwinding techniques, which are usually included, bring the drift-diffusion current continuity equations in terms of the Slotboom variables in self-adjoint form.

The piecewise linear finite element approximation which solves Galerkin’s equations is shown to be maximum stable for mixed Neumann-Dirichlet boundary conditions in two and three dimensions, even if no discrete extrema principles prevail. For the Delaunay triangulation in two dimensions, and the Delaunay tetrahedryzation in three dimensions, the box method discretization based on perpendicular bisectors yields a discretized system with extrema principles for the Slotboom variables. Box methods can be considered as Petrov-Galerkin methods for a piecewise linear approximation with piecewise constant test-functions.

Petrov-Galerkin finite element error analysis requires demonstration of an appropriate inf-sup condition. Complications arise in demonstrating this condition because piecewise constant test-functions are not of H ¹ regularity. These complications can be circumvented by approximating the piecewise constant test functions by continuous piecewise polynomial test functions. The box-method for a piecewise linear approximation function need not generate a system of linear equations which is identical to Galerkin’s equations. Nevertheless, under rather general conditions, the piecewise linear approximation that is obtained by the box-method realizes the same order of accuracy as the solution to Galerkin equations.