<p>The Poisson-Nernst-Planck equations play a crucial role in modeling ion mass conservation and electrostatic diffusion in various applications, such as biological ion channels, semiconductor devices, and nanopore systems. Due to the inherent strong coupling and nonlinearity, developing robust theoretical frameworks for discrete schemes and associated fast algorithms is challenging. While the finite element method and some improved methods are widely used to solve these equations, existing theoretical analyses are insufficient, particularly regarding the existence of solutions for discrete methods or nonlinear discrete systems, and the convergence of fast algorithms. This study aims to contribute to this area by proposing a framework that includes establishing the existence of solutions for an abstract discrete scheme, which encompasses schemes such as standard finite element and edge-averaged finite element methods. We establish the existence theory for the nonlinear discrete system by carefully constructing a suitable compact convex set for the fixed-point operator and then using the Brouwer’s fixed-point theorem. Additionally, we present a convergence theory for the Gummel iteration, a commonly used iteration for the Poisson-Nernst-Planck equations, associated with our abstract discrete scheme. We further explore the theoretical results for both standard and edge-averaged finite element schemes as applications of the framework. Numerical experiments are conducted to verify the theoretical results for the Gummel iteration of these two schemes.</p>