Preprint 2026-33
Eider Aldana, Ricardo Oyarzúa:
A Lagrange multiplier-based finite element method for a reverse osmosis model
Abstract:
In this work, we propose a Lagrange multiplier approach for a reverse osmosis model describing the flow of a saline solution through a channel bounded by semipermeable membranes. The model couples the incompressible Navier–Stokes equations with a convection– diffusion equation for the solute concentration, both through the convective term and through the membrane conditions: the normal velocity depends linearly on the concentration, the tangential velocity vanishes, and the concentration satisfies a Robin-type condition with a non-negativity constraint. In addition, Dirichlet conditions are prescribed at the inlet and directional do-nothing conditions at the outlet. We consider a velocity–pressure formulation for the fluid and a primal formulation for the concentration, imposing the Dirichlet-type conditions weakly through Lagrange multipliers. Then, by means of a fixed-point strategy and the Banach fixed-point theorem, we establish, under suitable smallness assumptions on the data, the well-posedness of the continuous problem and show that the concentration remains non-negative. Similarly, we prove the well-posedness of the associated Galerkin scheme for generic finite-dimensional subspaces satisfying suitable hypotheses, and derive a Strang-type a priori error estimate. In particular, Taylor–Hood elements for the velocity and pressure, continuous piecewise polynomials for the concentration, and piecewise polynomials on independent boundary partitions for the multipliers satisfy these hypotheses and yield optimal rates of convergence. Finally, numerical experiments confirm the theoretical results and illustrate the performance of the method in a realistic membrane channel.
