Fast and robust solvers for large linear systems are a basic requirement for many real applications, often modeled by partial differential equations (PDEs). The resulting coefficient matrices usually have a hidden structure inherited by the continuous model and by the discretization method. The exploration of such a structure allows the derivation of extensive spectral information that is crucial for the definition and for the analysis of effective preconditioners for Krylov methods and for the design of multigrid methods.

As a case study we consider two specific differential problems [1,2,3]: we demonstrate how to analyse the spectral properties and then how to derive a fast and robust solver using a generalization of the local Fourier analysis. In particular, we consider a finite difference discretizzation of fractional diffusion equations in space and the discretizzation of elliptic PDEs by the isogeometric analysis.

Research partly carried out with M. Donatelli, C. Garoni, C. Manni, M. Mazza, H. Speleers.