Discretization of anisotropic convection-diffusion equations, convective M-matrices and their iterative solution

We derive the constant-j box method discretization for the convection-diffusion equation, del j=f, with j= -alpha del u + beta u. In two dimensions, alpha is a 2 x 2 symmetric, positive definite tensor field and beta 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 alpha and beta, 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 (beta=0) problem under certain conditions.