(2024/2025) Numerical Methods for Partial Differential Equations

These notes are an unofficial resource and shouldn’t replace the course material or any other book on numerical methods for partial differential equations. It is not made for commercial purposes. I’ve made the following notes to help me improve my knowledge and maybe it can be helpful for everyone.

As I have highlighted, a student should choose the teacher’s material or a book on the topic. These notes can only be a helpful material.

The notes are taken from the books required for the course:

• Course slides.

You can view/download the PDF here. In the notes folder, you can also see the source code.

In the CHANGELOG file you can see the changes made to each version of the PDF file. The versioning can be helpful if you want to understand if there are any new features/fixes in the file.

For any issue, use the appropriate section.

Course Syllabus

According to the official course syllabus:

• Variational formulation of PDEs.
  • Elliptic and parabolic PDEs (Poisson, advection-diffusion-reaction, and heat equations);
  • Boundary and initial conditions;
  • Strong and weak formulations of the PDEs;
  • Lax-Milgram lemma.
• Finite Element method for elliptic PDEs in 1D/2D/3D.
  • Galerkin method, consistency, stability and convergence.
  • The finite element method in 1D/2D/3D;
  • Continuous Lagrangian basis functions;
  • Finite elements and meshes;
  • Algebraic properties of the fully discrete problem;
  • Accuracy and computational costs;
  • Error estimates and error analysis.
  • Finite element approximation of advection-reaction-diffusion equations and vectorial problems.
• Numerical approximation of time-dependent problems.
  • Heat equation, semi-discrete problem and Galerkin method.
  • Time discretization, explicit and implicit methods, theta-method, accuracy and stability properties.
• Advanced topics.
  • Finite element approximation of saddle-point problems, Stokes equations.
  • Finite Element approximation of nonlinear PDEs, Newton method.
  • Multiphysics problems and PDEs coupled through interfaces.
  • Domain decomposition methods and introduction to parallel computing and preconditioners.