Analysis of a finite element method for the Stokes--Poisson--Boltzmann equations
Date Issued
2026
Author(s)
Abeer F. AlSohaim
Monash University
Ruiz Baier Ricardo
Monash University, Society for Industrial and Applied Mathematics, Universidad de Concepcion, University of Oxford
Villa-fuentes Segundo
Monash University, Universidad de Los Lagos, Universidad del Bío-Bío, University of Bío-Bío
DOI
https://doi.org/10.48550/arXiv.2502.06455
Abstract
We define a finite element method for the coupling of Stokes and
nonlinear Poisson–Boltzmann equations. The novelty in the formula-
tion is that the coupling from the electric potential to the drag in the
momentum balance is rewritten as a weighted advection term. Using
Banach’s contraction principle, the Babuˇska–Brezzi theory, and the
Minty–Browder theorem, we show that the governing equations have
a unique weak solution. We also show that the discrete problem is
well-posed, establish C´ea estimates, and derive convergence rates. We
exemplify the properties of the proposed scheme via some numerical
experiments showcasing convergence and applicability in the study of
electro-osmotic flows in micro-channels.
nonlinear Poisson–Boltzmann equations. The novelty in the formula-
tion is that the coupling from the electric potential to the drag in the
momentum balance is rewritten as a weighted advection term. Using
Banach’s contraction principle, the Babuˇska–Brezzi theory, and the
Minty–Browder theorem, we show that the governing equations have
a unique weak solution. We also show that the discrete problem is
well-posed, establish C´ea estimates, and derive convergence rates. We
exemplify the properties of the proposed scheme via some numerical
experiments showcasing convergence and applicability in the study of
electro-osmotic flows in micro-channels.
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