On local algorithms for electrostatics

¹ Department of Mathematics, University of California, San Diego, La Jolla, California, USA
² Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, P.R. China
³ School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, P.R. China
⁴ MOE-LSC, CMA-Shanghai and Shanghai Center for Applied Mathematics, Shanghai Jiao Tong University, Shanghai, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 August 2025
Accepted: 14 March 2026

Abstract

We study finite-difference approximations of the Poisson-Boltzmann (PB) electrostatic energy functional of ionic concentrations and electric displacements constrained by Gauss’ law and the ionic mass conservation, and a class of local algorithms for minimizing the finite-difference discretized such energy functional. We prove that the discrete Boltzmann distributions characterize the finite-difference minimizer and obtain the uniform bounds and optimal error estimates in maximum norm for such a minimizer. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy by updating the concentrations and displacement one grid box at a time, keeping Gauss’ law and the mass conservation satisfied. A new local algorithm with a shift is constructed for minimizing the Poisson electrostatic energy (the part of the PB energy without ionic concentrations) with a variable dielectric coefficient. We prove the convergence of these local algorithms and present numerical tests to demonstrate the results of our analysis.