Two-Dimensional Exponential Fitting and Applications to Drift-Diffusion Models

A class of methods is introduced that generalizes the so-called exponential fitting method to two-dimensional problems and, more specifically, to equations of the type div (0.3em{-}∇ u + u0.3em{-}∇ ψ ) = f, with ψ given and |0.3em{-}∇ ψ | big in a part of the domain. The basic idea is the following: (i) write the equation as div (e- ψ 0.3em{-}∇ ρ ) = f through the change of variables u = ρ e- ψ; (ii) discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient e- ψ (mixed, hybrid, etc.); (iii) write the discrete scheme in terms of the nodal values of u = e- ψ ρ. This produces an M-matrix.