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Analysis of a finite element method for the Stokes--Poisson--Boltzmann equations

Abeer F. AlSohaim, Ricardo Ruiz-Baier, Segundo Villa-Fuentes

TL;DR

The paper develops a finite element method for the coupled Stokes–Poisson–Boltzmann equations modeling electrokinetic flows, with a novel reformulation of the potential–drag coupling as a weighted advection term to facilitate analysis. A Banach fixed-point argument, together with Babuška–Brezzi theory and Minty–Browder, yields existence and uniqueness of a weak solution under a small-data regime, and a Taylor–Hood FE discretisation provides Céa-type error estimates. The discrete scheme achieves optimal convergence rates, and numerical experiments on manufactured solutions, micro-annuli, and nanopore geometries confirm robustness and applicability to electro-osmotic flows in microchannels. The results offer a rigorous, practically applicable framework for simulating electrokinetic phenomena in microfluidic devices, contributing to design and analysis of nanoscale channels and nanopore-based sensing.

Abstract

We define a finite element method for the coupling of Stokes and nonlinear Poisson--Boltzmann equations. The novelty in the formulation 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 Babuška--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 Céa 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.

Analysis of a finite element method for the Stokes--Poisson--Boltzmann equations

TL;DR

The paper develops a finite element method for the coupled Stokes–Poisson–Boltzmann equations modeling electrokinetic flows, with a novel reformulation of the potential–drag coupling as a weighted advection term to facilitate analysis. A Banach fixed-point argument, together with Babuška–Brezzi theory and Minty–Browder, yields existence and uniqueness of a weak solution under a small-data regime, and a Taylor–Hood FE discretisation provides Céa-type error estimates. The discrete scheme achieves optimal convergence rates, and numerical experiments on manufactured solutions, micro-annuli, and nanopore geometries confirm robustness and applicability to electro-osmotic flows in microchannels. The results offer a rigorous, practically applicable framework for simulating electrokinetic phenomena in microfluidic devices, contributing to design and analysis of nanoscale channels and nanopore-based sensing.

Abstract

We define a finite element method for the coupling of Stokes and nonlinear Poisson--Boltzmann equations. The novelty in the formulation 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 Babuška--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 Céa 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.

Paper Structure

This paper contains 10 sections, 5 theorems, 38 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scope.
  3. Outline.
  4. The Stokes--Poisson--Boltzmann equations.
  5. Existence and uniqueness of weak solution
  6. Weak formulation.
  7. Well-posedness analysis.
  8. Finite element discretisation
  9. Numerical results
  10. Acknowledgements.

Key Result

Lemma 2.1

Assume that $\boldsymbol{f} \in \mathbf{L}^2(\Omega)$ and $g \in L^2(\Omega)$ satisfy Then, the operator $\mathbf{T}$ is well-defined and $\mathbf{T}(\mathbf{W})\subseteq \mathbf{W}$.

Figures (3)

  • Figure 4.1: Approximate solutions for the convergence test on the unit square domain, using the lowest-order Taylor--Hood elements.
  • Figure 4.2: Approximate velocity magnitude and streamlines, pressure profile, and potential for the flow-potential test using Taylor--Hood elements.
  • Figure 4.3: Flow patterns of electrically charged fluid in a nanosensor. Approximate line integral contour of velocity, pressure, and potential.

Theorems & Definitions (8)

  • Lemma 2.1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4