2. More advanced issues in scientific computing

This chapter is the natural continuation of Section 1, where we have introduced the basic mathematical ingredients needed in the treatment of variational problems, as well as the convenient \(\texttt{FreeFem}\) environment for their implementation. The present section builds on the contents of Section 1 and jumps to slightly more specific tools, that are transverse to many application fields while being perhaps a little less widely discussed in the literature.

Contents:

• 2.1. Meshing images
• 2.2. Curvature
  • 2.2.1. Definition of curvature
  • 2.2.2. Calculation of the mean curvature from a mesh
• 2.3. Nonlinear problems
  • 2.3.1. The Newton-Raphson method
  • 2.3.2. An application example: a non linear Laplace equation
• 2.4. Variational inequalities
  • 2.4.1. A word about the modeling
  • 2.4.2. A glimpse of the theory
  • 2.4.3. A simple penalty method
  • 2.4.4. Use of a constrained optimization algorithm: \(\texttt{ipopt}\)
  • 2.4.5. An active set algorithm
• 2.5. Level Set method I
  • 2.5.1. Implicit representation
  • 2.5.2. Handling geometry with Level Set functions
  • 2.5.3. Application: fictitious domain methods
• 2.6. Level Set method II
  • 2.6.1. Initialization of a Level Set representation by the signed distance function
  • 2.6.2. Evolution of a domain by the Level Set method
  • 2.6.3. Miscellaneous operations on implicit domains
  • 2.6.4. Implementation of the Level Set Method
• 2.7. Going 3d
  • 2.7.1. Construction of a 3d mesh
  • 2.7.2. 3d visualization with \(\texttt{medit}\)
  • 2.7.3. Solution of the Finite Element problem