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

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.