A brief introduction to pseudo-spectral methods: application to   diffusion problems
  The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type discretizations, even if some indications on how to handle non-periodic problems via Tchebyshev and Legendre approaches are made as well. The applications presented here are diffusion-type problems in accordance with the topics of the PhD school. 
Denys Dutykh
math.NA