Mixed FEM for coupled unsteady fluid flow problems with $p$-type Brinkman-Forchheimer framework and its application for reverse-osmosis desalination

This work analyzes a fully discrete mixed finite element method in a Banach space framework for solving nonstationary coupled fluid flow problems modeled by the Brinkman-Forchheimer equations, with applications to reverse osmosis. The model couples unsteady $p$-type convective Brinkman-Forchheimer and transport equations with nonlinear boundary conditions across a semi-permeable membrane. A mixed formulation is used for the fluid equation (pseudostress-velocity) and for the transport equation (concentration, its gradient, and a Lagrange multiplier from the membrane condition). The continuous problem is reformulated in Banach spaces as a fixed-point problem, enabling a well-posedness analysis via differential-algebraic system theory. Spatial discretization employs lowest-order Raviart-Thomas elements for fluxes and piecewise constants for primal variables, while linear elements are used for the Lagrange multiplier. A fully discrete Galerkin scheme with backward Euler time-stepping is proposed. Its well-posedness and stability are proven using a fixed-point argument, and optimal convergence rates are established. Numerical results confirm the theoretical error estimates and demonstrate the method's effectiveness.