Resumen de la Tesis:

The main objective of this thesis is to develop a Hybridizable Discontinuous Galerkin (HDG) scheme for a convection-diffusion equation with nonlinear boundary conditions. The motivation arises from the reverse osmosis process applied to water desalination, which, in its most complete form, involves a coupled system of Navier–Stokes and convection–diffusion equations, with unknowns corresponding to pressure, fluid velocity, and salt concentration. In this work, we focus exclusively on the convection–diffusion equation, considering a nonlinear boundary condition on part of the boundary, where the only unknown is the salt concentration. First, we analyze the existence and uniqueness of the continuous problem using a mixed variational formulation in the context of Banach spaces, based on a saddle-point perturbation approach. To ensure the well-posedness of the problem, a Banach fixed-point strategy is adopted, solving a linearized version of the system involving a Robin-type boundary condition, and applying the Banach-Nečas-Babuška theorem along with the Babuška–Brezzi theory. Then, an HDG scheme is proposed to approximate the solution of the continuous variational formulation, whose structure is also nonlinear. A fixed-point scheme is again employed, and to establish the well-posedness of the linearized scheme, we first prove continuous dependence on the data using energy and duality arguments. The existence and uniqueness of the fixed point in the discrete scheme are obtained similarly to the continuous case, but under more restrictive assumptions. Finally, an a priori error analysis is carried out, studying the projections of the errors and obtaining optimal convergence results under assumptions similar to those considered in the discrete analysis. Lastly, numerical experiments are presented that confirm the theoretical bounds obtained.