Discontinuous finite elements for coupled problems

Abstract

El objetivo de este trabajo es estudiar las propiedades de formulaciones de elementos finitos para problemas acoplados Darcy-Stokes basadas en HDG.Las tareas a elegir entre los estudiantes y los supervisores son: colaborar en la propuesta y análisis de nuevas formulaciones para el problema acoplado, demostrando teoremas de convergencia y estabilidad, olaborar en la adaptación del código de elementos finitos HDG disponible para Darcy, para el tratamiento del problema acoplado, estudiar mediante ejemplos numéricos las propiedades de convergencia y estabilidad

The ltration of uids through porous media is a challenging problem with many relevant applications in ltration problems, such as the ltration of blood through arterial vessel walls or the ltration of water through sand. To model this problem we consider di erent systems of partial di erential equations on each domain: in the free domain we will use Stokes equations, while the uid in the porous domain is modelled using Darcy equations. In the free domain the

uid is discretized by the Continuous Galerkin method, but in the porous domain we will use de Hybridizable Discontinuous Galerkin method. The project is focused in the Hybridizable Discontinuous Galerkin method, which is a new method that combines the advantatges of the Discrete Galerkin methods with the computational e ciency of the Continuous Galerkin methods. The main advantatges of this method are: Reduced number of degrees of freedom. With the hybridization process we can reduce the number of degrees of freedom at the boundaries of each element. Optimal convergence. It converges with order k + 1 in the L2 norm, where k is the degree of the polynomials used to approximate the solution. This convergence is for the solution and the derivative of the solution. Superconvergence and local postprocessing. The method alloes us to use an element-by-element postprocessing to obtain a new and better approximation with order k + 2, for k 1. The structure of the project is as follows. At the rst Chapter we set the problem: the equations that we will use and the boundary conditions. Next Chapter is dedicated to the Hybridizabe Galerkin method and to the analysis of the error of this method, for this Chapter we referenciate to [1], [3], [4], [7], [5]. We continue with a Chapter dedicated to the coupling, where we remark the article [6]. And nally we apply our problem to the ltration of water through sand.