1. Getting started

This chapter introduces the fundamental mathematical background for the Finite Element method. The first Section 1.1, Section 1.2 and Section 1.3 are mainly of mathematical nature, and contain rudiments about functional analysis, variational problems, and the theory behind the Finite Element Method. The next Section 1.4 and Section 1.5 are oriented towards practice: convenient numerical softwares are presented, notably the \(\texttt{FreeFem}\) environment and \(\texttt{medit}\). Eventually, Section 1.7, Section 1.8 and Section 1.9 are application examples that are interesting per se, and are carefully detailed, offering opportunities to showcase some of the building blocks of the methods developed in the subsequent chapters.

Contents:

• 1.1. Functional analysis
  • 1.1.1. Setting
  • 1.1.2. Differentiable functions, notations, and the Green’s formula
  • 1.1.3. Functional spaces
  • 1.1.4. The Poincaré’s inequality
• 1.2. Lax-Milgram framework
  • 1.2.1. From classical boundary-value problems to their variational formulations
  • 1.2.2. The abstract Lax-Milgram theory
  • 1.2.3. A worked example: homogeneous Neumann boundary conditions
• 1.3. The Finite Element method
  • 1.3.1. A short digression about meshes
  • 1.3.2. Finite Element spaces
  • 1.3.3. The discrete variational problem
  • 1.3.4. Convergence of the finite element method
  • 1.3.5. Towards implementation of the method
• 1.4. Laplace equation
  • 1.4.1. Motivation: modeling of thermal phenomena with the Laplace equation
  • 1.4.2. Getting started with \(\texttt{FreeFem}\): resolution of the Laplace equation
  • 1.4.3. Changing boundary conditions and considering inhomogeneous coefficients
  • 1.4.4. The unsteady Laplace equation
• 1.5. Visualization
  • 1.5.1. The \(\texttt{plot}\) command in \(\texttt{FreeFem}\)
  • 1.5.2. A simple and efficient visualization tool: \(\texttt{medit}\)
  • 1.5.3. Creating animated \(\texttt{.gif}\) files using \(\texttt{medit}\)
• 1.6. Meshes
  • 1.6.1. Handling meshes with \(\texttt{FreeFem}\)
  • 1.6.2. Adaptive mesh refinement
  • 1.6.3. Description of the \(\texttt{.mesh}\) and \(\texttt{.sol}\) formats
• 1.7. Eigenvalues
  • 1.7.1. Handling Finite Element matrices
  • 1.7.2. An eigenvalue problem
• 1.8. Penalization for Dirichlet conditions
• 1.9. Pure Neumann
  • 1.9.1. A bit of theory
  • 1.9.2. Two different numerical resolution methods
  • 1.9.3. Implementation of the pure Neumann problem