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

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 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.

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