Two-dimensional exponential fitting and applications to drift-diffusion models. (English) Zbl 0686.65088

Brezzi, Franco; Marini, Luisa Donatella; Pietra, Paola

SIAM J. Numer. Anal. 26, No. 6, 1342-1355 (1989).

The authors consider the numerical solution for u of the two-dimensional system \(\nabla \cdot (\nabla u+u\nabla \psi)=f\quad in\quad \Omega,\quad u=g\) on \(\Gamma_ 0\subset \partial \Omega;\quad \partial u/\partial n+u(\partial \psi /\partial n)=0\) on \(\Gamma_ 1=\partial \Omega \setminus \Gamma_ 0,\) when \(\psi\), f and g are given. \(\nabla \psi\) can be large and so the transformation \(u=\rho \exp (-\psi)\) is used.
They develop a number of methods for the discretization of the system, all of which involve the conservation of \(J=\nabla u+u\nabla \psi,\) and show that unique solutions exist. The results of the application of the ideas discussed to a particular problem are given.

Reviewer: Ll.G.Chambers

MSC:

65Z05	Applications to the sciences
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q99	Partial differential equations of mathematical physics and other areas of application
78A55	Technical applications of optics and electromagnetic theory

Keywords:

drift-diffusion models; exponential fitting; upwind; mixed finite element method; semiconductors