Primal-mixed finite element methods for the coupled Biot and Poisson–Nernst–Planck equations

We propose mixed finite element methods for the coupled Biot poroelasticity and Poisson–Nernst–Planck equations (modeling ion transport in deformable porous media). For the poroelasticity, we consider a primal-mixed, four-field formulation in terms of the solid displacement, the fluid pressure, the Darcy flux, and the total pressure. In turn, the Poisson–Nernst–Planck equations are formulated in terms of the electrostatic potential, the electric field, the ionized particle concentrations, their gradients, and the total ionic fluxes. The weak formulation, posed in Banach spaces, exhibits the structure of a perturbed block-diagonal operator consisting of perturbed and generalized saddle-point problems for the Biot equations, a generalized saddle-point system for the Poisson equations, and a perturbed twofold saddle-point problem for the Nernst–Planck equations. One of the main novelties here is the well-posedness analysis, hinging on the Banach fixed-point theorem along with small data assumptions, the Babuška–Brezzi theory in Banach spaces, and a slight variant of recent abstract results for perturbed saddle-point problems, again in Banach spaces. The associated Galerkin scheme is addressed similarly, employing the Banach fixed-point theorem to yield discrete well-posedness. A priori error estimates are derived, and simple numerical examples validate the theoretical error bounds, and illustrate the performance of the proposed schemes.