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Browsing Tesis Doctorado by Subject "AGUA limpia y saneamiento"
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Item Métodos de galerkin discontinuos para problemas de interfaz: aplicación a procesos de desalinización del agua.(Universidad de Concepción, 2024) Bermúdez Montiel, Isaac; Solano Palma, Manuel; Camaño Valenzuela, Jessika PamelaThe aim of this thesis is to develop continuous and discontinuous Galerkin-type discretizations applied to interface problems modelled by systems of partial differential equations (PDEs). The com plexity of these problems lies in the unknowns and their associated source terms arising from coupling interfaces, as well as the non-linearity of the equations. First, we consider a coupled Navier–Stokes/transport system inspired by the modeling of 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 and 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 struc ture. We analyze the continuous and discrete solvability of this problem by linearizing the perturbation term 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 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. Once we have established the theoretical foundations to ensure the solvability of the variational scheme, we take advantage of them to numerically address different approaches closely related to reverse osmosis processes by considering coupled Brinkman–Forchheimer/transport equations in addition to Navier–Stokes/transport equations. The cases of a single channel and coupled feed/permeate channels are covered. Thus, through diverse numerical simulations and a variety of configurations, we illustrate the capability of the method to accurately capture the behavior of saline water when passing through membrane-based reverse osmosis desalination channels. Onthe other hand, we present and analyze a hybridizable discontinuous Galerkin (HDG) method for coupling Stokes and Darcy equations, whose domains are discretized by two independent triangulations. This causes non-conformity at the intersection of the subdomains or originates a gap (unmeshed region) between them. In order to properly couple the two different discretizations and obtain a high order scheme, we propose suitable transmission conditions based on mass conservation, equilibrium of normal forces, and the Beavers–Joseph–Saffman law. Since the meshes do not necessarily coincide at the interface, we use the Transfer Path Method (TPM) to tie them. We establish the well-posedness of the method and provide error estimates where the influences of the non-conformity and the gap are explicit in the constants. Finally, numerical experiments that illustrate the performance of the method are shown.