Chapter 3: Krylov methods

In Chapter 1, we introduced Schwarz methods as iterative solvers. The aim of this chapter is to explain the benefit of using domain decomposition methods as preconditioners for Krylov methods such as CG (conjugate gradient), GMRES, or BiCGSTAB. In section 3.1, we present fixed-point methods. In section 3.2, we introduce the fundamental notion of the Krylov subspace. This yields two celebrated algorithms: the conjugate gradient (CG) method for symmetric positive definite problems in section 3.3 and the Generalized Minimal RESidual (GMRES) method for nonsymmetric linear systems in section 3.4. At the end of section 3.4, we explain why Krylov methods are always preferable to fixed-point iterations. In section 3.5, the case where the system is not invertible is dealt with. Finally, in section 3.6, we give FreeFem++ implementations and the use of Krylov methods in domain decomposition methods. It should be noted that the detailed and sometimes technical derivation of these methods is not necessary to be able to use them with success.