Optimal convergence analysis of arbitrary Lagrangian–Eulerian finite element methods in energy norm for Poisson–Nernst–Planck moving boundary problems

Xiaomeng Lin, Mingyan He and Pengtao Sun

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 244, issue C, 162-180

Abstract: In this paper, an arbitrary Lagrangian–Eulerian (ALE)-based finite element method (FEM) is developed and analyzed for a class of Poisson–Nernst–Planck (PNP) moving boundary problems, where the optimal convergence rate in energy norm and suboptimal convergence rate in L2 norm are obtained for a fully discrete, linearized, ALE-based finite element scheme. One key analytical technique is the introduction of a novel H1-projection that is associated with a coupling between the electric potential and ionic concentrations over a moving/deforming domain. Numerical experiments are carried out to validate all derived theoretical results. As a starting point, the developed ALE-finite element scheme and its analytical techniques can be extended to moving interface problems of PNP system and more beyond, of PNP–Navier–Stokes coupling system occurring in ion channels and their surrounding cellular environment that will be studied in our future work.

Keywords: Arbitrary Lagrangian–Eulerian-finite element method (ALE-FEM); Poisson–Nernst–Planck (PNP) moving boundary problems; H1-projection; Linearized backward Euler scheme; Optimal error estimate; Energy norm (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:244:y:2026:i:c:p:162-180

DOI: 10.1016/j.matcom.2025.12.020

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