Résumé
This document contains lecture notes from the Ph.D. course given at Scuola
Superiore Meridionale by Daniele Di Pietro in February 2025. The goal of the
course is to provide an overview of polytopal methods, focusing on the Hybrid
High-Order (HHO) method. As a starting point, we study the Crouzeix-Raviart
method for a pure diffusion equation, with particular focus on its stability.
We then show that, switching to a fully discrete point of view, it is possible
to generalize it first to polyhedral meshes and then to arbitrary order,
leading to a method that belongs to the HHO family. A study of the stability
and consistency of this method reveals the need for a stabilization term, for
which we identify two key properties.