Banach spaces-based mixed finite element methods for the coupled stokes and poisson-nernst-planck equations.

Abstract

This work is divided in two main parts. In the first part we provide sufficient conditions for perturbed saddle-point formulations in Banach spaces and their associated Galerkin schemes to be well-posed. Our approach, which extends a similar procedure employed with Hilbert spaces, proceeds in two slightly different ways depending on whether the kernel of the adjoint operator induced by one of the bilinear forms is trivial or not. The applicability of the continuous solvability is illustrated with a mixed formulation for the decoupled Nernst-Planck equation. This part yielded the following work already published: C.I. Correa and G.N. Gatica, On the continuous and discrete well-posedness of perturbed saddle-point formulations in Banach spaces. Comput. Math. Appl. 117 (2022), 14–23. On the other hand, in the second part we employ a Banach spaces-based framework to intro duce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dy namics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incom pressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system. the well-posedness of the continuous formulation is a consequence of a fixed point strategy in combination with the Banach theorem, the Babuˇska–Brezzi theory, the solvability of abstract perturbed saddle point problem that will be developed in the first part of this thesis, and the Banach–Neˇcas–Babuˇska theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori vi vii error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ď k. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the balance-preserving properties and applicability of the proposed family of methods. This part yielded the following work, presently submitted: C.I. Correa, G.N. Gatica and R. Ruiz-Baier, New mixed finite element methods for the coupled Stokes and Poisson-Nernst-Planck equations in Banach spaces. Preprint 2022-26, Centro de Investigaci´on en Ingenier´ıa Matem´atica (CI2MA), Universidad de Concepci´on, (2022).