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VLSI Design
Volume 10 (2000), Issue 4, Pages 485-529

Discretization of Anisotropic Convection-diffusion Equations, Convective M-matrices and their Iterative Solution

Received 16 December 1998; Accepted 14 December 1999

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We derive the constant-j box method discretization for the convection-diffusion equation, j=f, with j=αu+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.