# Efficient preconditioning strategies in Isogeometric analysis

**Efficient preconditioning strategies in Isogeometric analysis**

Isogeometric analysis is a high-order numerical method used to solve Partial Differential Equations (PDEs). In this talk I will focus on the k-method, that consists of using splines of degree p and regularity p-1, i.e. the maximal regularity allowed. The cost of the k-method is essentially concentrated into two processes: the formation of the linear system and its numerical solution. I will focus on the second computational challenge, i.e. the solution of the system, and I will present some ecient and robust preconditioning methods suited to solve dierent kind of PDEs, also in the multipatch setting. The basis of all the strategies is the Fast Diagonalization method, a fast solver for Sylvester-like equations. Starting from the Poisson problem, I will then consider Stokes system and, nally, I will give a special attention to space-time solving strategies for the heat equation.