The Mathematical Study and Approximation of Semiconductor
By: Joseph W. Jerome
Emphasis in this paper is upon the development of the
mathematical properties of the drift-diffusion
semiconductor device model, especially steady-state properties, where
the system is incorporated into a fixed point mapping framework. An essential
feature of the paper is the description of the Krasnosel'skii calculus,
the appropriate extension to nonlinear equations and systems of the
famous inf-sup saddle point
approximation theory for linear equations. The introductory
section describes certain scaling effects and the association with
numerical methods constructed via exponential fitting. The fixed point and
numerical fixed maps are introduced in the next section, while the
finite element approximation theory is described in the third section.
The fourth section deals with the evolution system and modular pre-algorithms
defined by Newton's method, combined with a fully implicit time
semidiscretization. More general moment models are introduced in
the fifth section, and some historical comments appear in the final section.
Specific algorithms and implementation issues are
not discussed. The principal section headings are as follows.
1. The Drift-Diffusion Model, Scaling, and Exponential Fitting.
2. Steady-State: The Fixed Point and Numerical Fixed Point Maps.
3. A Nonlinear Finite Element Convergence Theory.
4. The Evolution System and Newton's Method.
5. More General Moment Models: A Review.
6. Epilogue: Historical Perspective.
This paper appeared in: Large Scale Matrix Problems and the Numerical
Solution of Partial Differential Equations (Gilbert and Kershaw, eds.),
Oxford University Press (1994), pp. 157--204.
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