Aspectos básicos de la modelización matemática y numérica de los medios continuos

Abstract

The main purpose of this paper is the construction and treatment of equations that are useful in Continuum Mechanichs from both, physical and mathematical points of view. In particular, we will be considering the Advection Difussión equation and the Elastisicy equation. We will work in Sobolev Spaces, which are the natural enviroment for PDEs to be developed. Through the theorems of Lax Milgram and Banach-Neças-Babuska, we will study under which conditions a configuration setting is well posed. Once the existence and uniqueness of the analytical solution is proven, we will approximate our problem using Galerkin and Petrov Galerkin methods. The existence and uniqueness of the approximate solution is studied with very similar results to the analytical ones, and its covergence to the exact solution will be a result of having both, consistency and stability properties. Next, the Elasticity and Advection-Diffusion equations will be modeled. We will study the conformal approximation of Elasticity problems with different boundary conditions and the conformal and nonconformal approximation of Advection-Difussion problems, also with different boundary conditions. Finally, we will briefly present some implementations of the solutions of these problems in FreeFem++.