A discontinuous Poisson--Boltzmann equation with interfacial transfer: homogenisation and residual error estimate

TL;DR

This paper develops a homogenisation approach for a nonlinear Poisson--Boltzmann equation with discontinuous solutions due to interfacial transfer, providing rigorous residual error estimates for multiscale electrostatic modeling.

Contribution

It introduces a homogenisation framework for a discontinuous Poisson--Boltzmann equation with transmission conditions, including a first-order residual error estimate.

Findings

Homogenised model captures multiscale electrostatics with interface jumps.

Derived residual error estimate quantifies approximation accuracy.

Framework applicable to periodic multiphase media with dilute particles.

Abstract

A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue is that the interfacial transfer allows jumps and thus discontinuous solutions of the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem subject to inhomogeneous transmission interface conditions. Moreover, we establish a rigorous residual error estimate up to the first order correction.