Introduction to the finite element method and application to the Stokes problem

Kurzfassung

This work presents a rigorous introduction to the Finite Element Method and its application to the two--dimensional stationary Stokes problem. The main objective is to establish a solid mathematical basis that allows the understanding of the numerical formulation and discretization of the problem, starting from its partial differential equation form, its variational formulation, and its discretization through the Galerkin method. The study is developed in the context of a research project linked to the German Aerospace Center, related to the structural health monitoring of wind turbine blades. In particular, it arises from the need to analyze the relationship between the aerodynamic pressure distribution and the angle of attack, with the aim of a future formulation of an inverse problem oriented to the identification of structural damage. First, the functional foundations required for the variational formulation of partial differential equation problems are introduced, including Sobolev spaces. Afterwards, the formal definition of a finite element and the construction of nodal bases and interpolation operators, both local and global, are presented. On this basis, the discretization of the Stokes problem with mixed boundary conditions is developed using the Galerkin method. Finally, numerical results corresponding to the simulation of a two--dimensional viscous flow around an aerodynamic profile on conforming triangular meshes are included. These results illustrate the behavior of the velocity and pressure fields. This work does not aim to completely solve the inverse problem that motivates the application context, but to provide the theoretical framework necessary for its future development, as well as to serve as a bridge between rigorous mathematical analysis and numerical simulation in fluid mechanics.