We derive the constant-j box method discretization for the convection-diffusion
equation, ∇j=f, with j=−α∇u+βu
. In two dimensions, α is a 2 × 2 symmetric, positive
definite tensor field and β is a two-dimensional vector field. This derivation generalizes
the well-known Scharfetter-Gummel discretization of the continuity equations in
semiconductor device simulation. We define the anisotropic Delaunay condition and
show that under this condition and appropriate evaluations of α and β, the stiffness
matrix, M, of the discretization is a convective M-matrix. We then examine classical
iterative splittings of M and show that convection (even convection dominance) does
not degrade the rate of convergence of such iterations relative to the purely diffusive (β=0) problem under certain conditions.