Numerical tours in scientific computing

This online book is a series of tutorials, intended as a comprehensive introduction to various aspects of scientific computing. It is mainly aimed to graduate students, although the most basic contents should already be available to undergraduates. It might also be useful to researchers that have little prior knowledge about this field and wish to get more familiar with some of its aspects. It is eventually for… me, a selfish excuse to learn many things!

These tours are intended to be pedagogic; the presentation is not minimal, and repetitions occur between different sections. They are oriented towards practical and numerical applications. (Sketches of) Proofs are given when they help intuition; further details for the curious reader are provided in appendices, or in hidden boxes.

These tours are nowhere set in stone; quite the contrary, they are meant to evolve continuously. Please do not hesitate to contact me for any mistake or inappropriate content… and also to suggest your own application that I will happily upload!

Before getting started, let us highlight a few useful references.

• The online \(\texttt{FreeFem}\) documentation is a very rich source of various physical problems addressed with \(\texttt{FreeFem}\).

• The online \(\texttt{FreeFem}\) tutorial by P. Jolivet is full of interesting mathematical models and advanced \(\texttt{FreeFem}\) syntax elements.

• The recent book [HLT24] is focused on PDE-constrained optimization problems, solved with \(\texttt{FreeFem}\). It can be freely downloaded here.

Additionally, other online tutorials are available about (more or less closely) related topics:

• The amazing Numerical Tours by G. Peyré offer an exhaustive overview of multiple issues in imaging and learning.

• The Numerical Tours in continuum mechanics by J. Bleyer revolve around various models in continuum mechanics.

• The NGSolve tutorials contain presentations of a full series of physical models, and their solution with the software \(\texttt{NGSolve}\).

Contents:

• 1. Generalities
  • 1.1. Functional analysis
  • 1.2. Lax-Milgram framework
  • 1.3. The Finite Element method
  • 1.4. Laplace equation
  • 1.5. Visualization
  • 1.6. Meshes
  • 1.7. Eigenvalues
  • 1.8. Penalization for Dirichlet conditions
  • 1.9. Pure Neumann
• 2. Advanced features
  • 2.1. Meshing images
  • 2.2. Curvature
  • 2.3. Nonlinear problems
  • 2.4. Variational inequalities
  • 2.5. Level Set method I
  • 2.6. Level Set method II
  • 2.7. Going 3d
• 3. Structure mechanics
  • 3.1. Linear elasticity
  • 3.2. Finite Elements for linear elasticity
  • 3.3. Hyperelasticity
  • 3.4. Contact
• 4. Fluid mechanics
  • 4.1. Theory
  • 4.2. Models
  • 4.3. Cavity
  • 4.4. Navier-Stokes
  • 4.5. Fluid Motion
  • 4.6. Fluid Motion with the Level Set method
• 5. Optimal design
  • 5.1. Density methods
  • 5.2. Geometric optimization
  • 5.3. The level set method
  • 5.4. Topological derivatives
  • 5.5. Shape optimization for fluids
• 6. Appendix
  • 6.1. Differential calculus
  • 6.2. Complements of functional analysis
  • 6.3. More advanced functional analysis
  • 6.4. Convexity
  • 6.5. Optimization theory
  • 6.6. PDE
• 7. Bibliography
• 8. Glossary