A conforming mixed finite element method for a coupled Navier–Stokes/transport system modeling reverse osmosis processes

We consider the coupled Navier–Stokes/transport equations with nonlinear transmission conditions, which constitute one of the most common models utilized to simulate a reverse osmosis effect in water desalination processes when considering feed and permeate channels coupled through a semi-permeate membrane. The variational formulation consists of a set of equations where the velocities, the concentrations, along with tensors and vector fields introduced as auxiliary unknowns and two Lagrange multipliers are the main unknowns of the system. The latter are introduced to deal with the trace of functions that do not have enough regularity to be restricted to the boundary. In addition, the pressures can be recovered afterwards by a postprocessing formula. As a consequence, we obtain a nonlinear Banach spaces-based mixed formulation, which has a perturbed saddle point structure. We analyze the continuous and discrete solvability of this problem by linearizing the perturbation and applying the classical Banach fixed point theorem along with the Banach–Nečas–Babuška result. Regarding the discrete scheme, feasible choices of finite element subspaces that can be used include Raviart–Thomas spaces for the auxiliary tensor and vector unknowns, piecewise polynomials for the velocities and the concentrations, and continuous polynomial space of lowest order for the traces, yielding stable discrete schemes. An optimal a priori error estimate is derived, and numerical results illustrating both, the performance of the scheme confirming the theoretical rates of convergence, and its applicability, are reported.