An HDG method for a convection-diffusion equation with non-linear boundary conditions.
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Date
2025
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Universidad de Concepción
Abstract
El principal objetivo de esta tesis es desarrollar un esquema de Galerkin Discontinuo Hibridizable (HDG) para una ecuación de convección-difusión con condiciones de contorno no lineales. La motivación proviene del proceso de ósmosis inversa aplicado a la desalinización de agua, que en su versión más completa involucra un sistema acoplado de ecuaciones de Navier Stokes y convección-difusión, con incógnitas correspondientes a la presión, la velocidad del f luido y la concentración de sal. En este trabajo nos enfocamos exclusivamente en la ecuación de convección-difusión, considerando una condición de borde no lineal sobre una parte de la frontera, donde la única incógnita es la concentración de sal.
En primer lugar, se analiza la existencia y unicidad de solución del problema a nivel continuo mediante una formulación variacional mixta en el contexto de espacios de Banach, utilizando un enfoque basado en la perturbación de un punto de silla. Para garantizar el buen planteamiento del problema, se adopta una estrategia de punto fijo de Banach, resolviendo una versión linealizada del sistema en la que aparece una condición de borde del tipo Robin, y aplicando el teorema de Banach-Nečas-Babuška junto con la teoría de Babuška –Brezzi.
Posteriormente, se propone un esquema HDG para aproximar la solución de la formulación variacional continua, cuya estructura también es no lineal. Se emplea nuevamente un esquema de punto fijo, y para establecer el buen planteamiento del esquema linealizado se demuestra primero la dependencia continua respecto a los datos, utilizando argumentos de energía y dualidad. La existencia y unicidad del punto fijo en el esquema discreto se obtiene de manera similar al caso continuo, aunque bajo hipótesis más restrictivas.
Finalmente, se realiza un análisis de error a priori, estudiando las proyecciones de los errores y obteniendo resultados de convergencia óptimos bajo suposiciones similares a las consideradas en el análisis discreto. Por último, se presentan ensayos numéricos que corroboran las cotas teóricas obtenidas.
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 motiva tion 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 perturba tion 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–Neˇcas–Babuˇska theorem along with the Babuˇska–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.
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 motiva tion 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 perturba tion 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–Neˇcas–Babuˇska theorem along with the Babuˇska–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.
Description
Tesis presentada para optar al título de Ingeniero/a Civil Matemático/a.
Keywords
Ecuaciones, Ecuaciones de Navier-Stokes, Espacios de Banach, Problemas de valores de contorno no lineales