Introduction to the finite element method

This course is dedicated to the finite element method.
First, we present mechanical models whose variational form is based on the application of the principle of virtual work. A general mathematical framework is then established with the Lax-Milgram theorem.
We then introduce the finite element method, its principles and implementation.

First-year courses in analysis and numerical analysis.

1) Introduction
2) Variational formulation of elliptic boundary problems
2.1 Model problems: linear elasticity, special case of the elastic wire, "Laplacian-type" boundary problems, stationary Stokes problem, time-dependent problems
2.2 Variational problems: the Lax-Milgram theorem
2.3 Functional spaces: Sobolev spaces
2.4 Application of the variational method to some model problems
3) Numerical approximation of elliptic problems
3.1 General
3.2 The Galerkin method
3.3 The Legendre-Galerkin method: a spectral method
3.4 Introduction to the finite element method: Lagrange P1 elements, general definition
3.5 Properties of P1 finite elements
3.6 Implementation: meshing, calculation code (assembly), post-processing
3.7 Other examples of finite elements: simplicial Lagrange elements, Q_k, Hermite elements
4) Complements: unsteady problems, finite volumes and finite elements, Stokes problem

One course handout and one TD booklet