Analysis and finite element approximation of Schrödinger–Poisson system with open Schrödinger boundary conditions

Uranchimeg Dorligjav; Minji Seo; Eunjung Lee; Uranchimeg Dorligjav; Minji Seo; Eunjung Lee

doi:10.3934/math.20261131

AIMS Mathematics

2026, Volume 11, Issue 9: 28409-28435. doi: 10.3934/math.20261131

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Research article

Analysis and finite element approximation of Schrödinger–Poisson system with open Schrödinger boundary conditions

School of Mathematics and Computing, Yonsei University, Seoul, 03722, South Korea

Received: 12 June 2026 Revised: 28 August 2026 Accepted: 02 September 2026 Published: 04 September 2026
MSC : 65N30, 65N12

We consider a stationary Schrödinger–Poisson system with open boundary conditions arising in quantum transport modeling. Despite its widespread use in device simulation, the mathematical analysis of such coupled systems remains challenging due to the noncoercivity of the Schrödinger operator and the nonlinear interaction through the electron density and quasi-Fermi level. In this work, we present, to our knowledge, the first comprehensive analysis and finite element framework for the self-consistent system with open boundary conditions. The approach is based on a quasi-Fermi level reparameterization that reveals a monotone structure in the nonlinear Poisson equation and enables a stable decoupling. The resulting formulation combines a Fredholm-based treatment of the Schrödinger equation with a Newton linearization for the Poisson problem, naturally leading to a Gummel-type iteration. We develop finite element discretizations of the subproblems, prove quasi-optimal convergence for the Schrödinger problem and optimal convergence for the Poisson problem, and, using the reparameterization and compactness properties of the electron density, show that, under suitable positivity and boundedness conditions and assuming uniqueness of the fully coupled solution, the finite element iterates converge to that solution. The results provide a consistent and stable theoretical framework for finite element simulations of the Schrödinger–Poisson system and a rigorous basis for numerical methods commonly employed in quantum transport simulations.
- quantum transport,
- Schrödinger open boundary conditions,
- finite element method,
- Gummel's iteration,
- error analysis
Citation: Uranchimeg Dorligjav, Minji Seo, Eunjung Lee. Analysis and finite element approximation of Schrödinger–Poisson system with open Schrödinger boundary conditions[J]. AIMS Mathematics, 2026, 11(9): 28409-28435. doi: 10.3934/math.20261131

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Abstract

We consider a stationary Schrödinger–Poisson system with open boundary conditions arising in quantum transport modeling. Despite its widespread use in device simulation, the mathematical analysis of such coupled systems remains challenging due to the noncoercivity of the Schrödinger operator and the nonlinear interaction through the electron density and quasi-Fermi level. In this work, we present, to our knowledge, the first comprehensive analysis and finite element framework for the self-consistent system with open boundary conditions. The approach is based on a quasi-Fermi level reparameterization that reveals a monotone structure in the nonlinear Poisson equation and enables a stable decoupling. The resulting formulation combines a Fredholm-based treatment of the Schrödinger equation with a Newton linearization for the Poisson problem, naturally leading to a Gummel-type iteration. We develop finite element discretizations of the subproblems, prove quasi-optimal convergence for the Schrödinger problem and optimal convergence for the Poisson problem, and, using the reparameterization and compactness properties of the electron density, show that, under suitable positivity and boundedness conditions and assuming uniqueness of the fully coupled solution, the finite element iterates converge to that solution. The results provide a consistent and stable theoretical framework for finite element simulations of the Schrödinger–Poisson system and a rigorous basis for numerical methods commonly employed in quantum transport simulations.

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