## Preprint 2022-35

## Claudio I. Correa, Gabriel N. Gatica, Esteban Henriquez, Ricardo Ruiz-Baier, Manuel Solano:

### Banach spaces-based mixed finite element methods for the coupled Navier--Stokes and Poisson--Nernst--Planck equations

### Abstract:

In this paper we extend the Banach spaces-based fully mixed approach recently developed for the coupled Stokes and Poisson--Nernst--Planck equations, to cover the coupled Navier--Stokes and Poisson--Nernst--Planck equations. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernoulli-type stress tensor are added as further unknowns. Similarly, fully mixed formulations for the Poisson and Nernst--Planck sub-problems are achieved by considering, alongside the electrostatic potential and the concentration of ionized particles, the electric current field and total ionic fluxes as new mixed variables. As a consequence, two saddle-point problems, one of them non-linear, and both involving nonlinear source terms depending on the other unknowns, along with a perturbed saddle-point problem that is in turn further perturbed by a bilinear form depending on the remaining unknowns, constitute the resulting variational formulation of the whole coupled system. Fixed-point strategies are then employed to prove, under smallness assumptions on the data, the well-posedness of the continuous and associated Galerkin schemes, the latter for arbitrary finite element subspaces under suitable stability assumptions. The main theoretical tools utilized include the Babu\v ska--Brezzi and Banach--Ne\v{c}as--Babu\v{s}ka theories in Banach spaces, an abstract result for perturbed saddle-point problems (also in Banach spaces), and the classical Banach and Brouwer fixed-point theorems. Strang-type lemmas are then applied to establish a priori error estimates. Next, specific finite element subspaces (defined by Raviart--Thomas elements of order $k\geq 0$ and piecewise polynomials of degree $\le k$) are shown to satisfy the required hypotheses, and this yields specific convergence rates. Finally, several numerical results are reported, confirming the theoretical findings and illustrating the good performance of the method.