Preprint 2024-14
Gabriel N. Gatica, Cristian Inzunza, Ricardo Ruiz-Baier:
Primal-mixed finite element methods for the coupled Biot and Poisson-Nernst-Planck equations
Abstract:
We introduce and analyze conservative primal-mixed finite element methods for numerically solving 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 is posed in suitable Banach spaces, and it exhibits the structure of a perturbed block-diagonal operator consisting in turn of perturbed and generalized saddle-point problems for the Biot equations, a generalized saddle-point problem for the Poisson equations, and a perturbed twofold saddle-point problem for the Nernst--Planck equations. The well-posedness analysis hinges on the Banach fixed-point theorem along with small data assumptions, the Babuska-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 Brouwer and Banach theorems to yield existence and uniqueness of discrete solution. A priori error estimates are derived, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions are established. Finally, several numerical examples validating the theoretical error bounds, and illustrating the performance of the proposed family of finite element methods, are presented.
