CALCOLO 38, 97 – 112 (2001) CALCOLO © Springer-Verlag 2001 Stopping criteria for iterative methods: applications to PDE’s M. Arioli 1 , E. Noulard 2 , A. Russo 1 1 Istituto di Analisi Numerica del C.N.R., Via Ferrata 1, 27100 Pavia, Italy e-mail: arioli@dragon.ian.pv.cnr.it; russo@dragon.ian.pv.cnr.it 2 Laboratoire d’Informatique et de Mathématiques Appliquées de l’ENSEEIHT, 2 rue Charles Camichel, 31071 Toulouse Cedex, France e-mail: noulard@enseeiht.fr Received: March 2000 / Accepted: October 2000 Abstract. We show that, when solving a linear system with an iterative method, it is necessary to measure the error in the space in which the residual lies. We present examples of linear systems which emanate from the finite element discretization of elliptic partial differential equations, and we show that, when we measure the residual in H -1 (), we obtain a true evaluation of the error in the solution, whereas the measure of the same residual with an algebraic norm can give misleading information about the convergence. We also state a theorem of functional compatibility that proves the existence of perturbations such that the approximate solution of a PDE is the exact solution of the same PDE perturbed. 1 Introduction Stationary physical phenomena are often driven by elliptic partial differential equations. The discretization of equations of this kind often leads to a real N × N linear system, A · x = b , which is normally solved by Krylov-based methods such as Conjugate Gradient ([8]) when A is symmetric positive definite or GMRES ([12]) in the general case. At each iteration step we compute an approximation x (n) ∈ R N of the solution of the linear system. It is necessary, at this point, to introduce a stopping criterion in order to test whether x (n) is accurate enough for our purposes. This work was supported by the “Istituto di Analisi Numerica – Consiglio Nazionale delle Ricerche” (Pavia, Italy) through the European programme HCM, contract no: ERBCHRXCT930420.