Firedrake is a state of the art system for solving PDEs with finite element and spectral methods. The key idea of Firedrake is to write the PDE in a compact mathematical language; Firedrake then generates the low-level code for the assembly of vectors and matrices required to solve the PDE.

I will give a short course introducing Firedrake and showing how to solve some systems of interest with it, including

- solving Poisson with optimal complexity spectral methods and p-multigrid;
- mixed formulations for the Stokes and Navier-Stokes equations;
- bifurcation analysis of a compressible hyperelasticity problem with deflated continuation.

The slides for a longer version are available at https://pefarrell.org/files/icerm2024.pdf .