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.
