Existence of discrete DDFV-HA solutions on general unstructured grids

Prove the existence of discrete solutions for the fully coupled nonlinear semiconductor drift-diffusion system discretized by the DDFV-HA scheme on general unstructured grids.

Background

The DDFV-HA scheme is introduced for stationary semiconductor drift-diffusion equations and is designed to remain reliable on low-quality, non-Delaunay meshes. Numerical experiments indicate that the method produces stable results on highly distorted grids and outperforms the classical finite-volume Scharfetter–Gummel method in such settings.

Despite this numerical evidence, the paper does not establish a rigorous existence theorem for the discrete solution of the fully coupled nonlinear system comprising the electron and hole continuity equations and the Poisson equation when the DDFV-HA discretization is applied on general unstructured meshes. Establishing such a result would provide a mathematical foundation for the observed robustness of the method.

References

Nevertheless, a rigorous proof of the existence of discrete DDFV-HA solutions for the fully coupled nonlinear semiconductor system on general unstructured grids remains an open theoretical question.

— A discrete duality finite volume method with harmonic average for semiconductor drift-diffusion equations  (2608.12808 - Liu et al., 13 Aug 2026) in Section 5, subsection “Dependence on mesh quality”