Linear Reduction and Homotopy Control for Steady Drift-Diffusion Systems in Narrow Convex Domains

TL;DR

This paper introduces a homotopy-based method for solving steady drift-diffusion equations in narrow convex domains, validating a predictor/corrector scheme without relying on Boltzmann statistics or Einstein relations.

Contribution

It develops a new theoretical framework for homotopy curves applicable to drift-diffusion systems in narrow domains, extending previous results and validating computational methods.

Findings

Existence of homotopy curves for drift-diffusion equations in narrow domains

Validation of predictor/corrector computational scheme

No assumptions on domain diameter, only narrowness

Abstract

This article develops and applies results, originally introduced in earlier work, for the existence of homotopy curves, terminating at a desired solution. We describe the principal hypotheses and results in section two; right inverse approximation is at the core of the theory. We apply this theory in section three to the basic drift-diffusion equations. The carrier densities are not assumed to satisfy Boltzmann statistics and the Einstein relations are not assumed. By proving the existence of the homotopy curve, we validate the underlying computational framework of a predictor/corrector scheme, where the corrector utilizes an approximate Newton method. The analysis depends on the assumption of domains of narrow width. However, no assumption is made regarding the domain diameter.