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A mixed finite element method for a reverse osmosis model.
(Universidad de Concepción, 2025) Burgos Villanueva, Víctor Manuel; Oyarzúa Vargas, Ricardo Elvis
We develop and analyze a numerical method to approximate the solution of a partial differential equation arising from a phenomenological model of water desalination through reverse osmosis within a channel module. The problem involves a coupled nonlinear system that accounts for the steady state of mass transport phenomena via a convection-diffusion equation and linear momentum balance through the Navier-Stokes equation. To address this problem, we introduce a mixed variational formulation based on Banach spaces for both phenomena, utilizing appropriate Lebesgue spaces to define the nonlinear terms and introducing a Lagrange multiplier that couples both phenomena at the boundary. We establish the existence and uniqueness of the solution under smallness assumptions on the physical parameters. We consider conforming subspaces, demonstrate the well-posedness of the discrete formulation, and derive the respective a priori error estimates. Finally, the model is verified against analytical solutions and compared with a related literature study under realistic conditions.