Digital Object Identifier (DOI) 10.1007/s00211-003-0500-y Numer. Math. (2003) Numerische Mathematik A stopping criterion for the conjugate gradient algorithm in a finite element method framework M. Arioli Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK; e-mail: m.arioli@rl.ac.uk Received April 28, 2001 / Revised version received January 10, 2003 / Published online November 19, 2003 – c  Springer-Verlag 2003 Summary. The Conjugate Gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [13,19,20,22] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced in [3]. Moreover, we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the effec- tiveness of our stopping criterion. Mathematics Subject Classification (2000): 65F10, 65N30, 65F50 1 Introduction In this paper, we combine linear algebra techniques with finite element techniques to obtain a reliable stopping criterion for the conjugate gradient algorithm. The finite element method approximates the weak form of an self adjoint, coercive elliptic partial differential equation defined within a Hilbert space by a linear system of equations Ax = b where A ∈ R N ×N is symmetric and positive definite and b ∈ R N . The conjugate gradient method is a very effective iterative algorithm for solv- ing these linear systems. In particular, using the conjugate gradient algo- rithm, we will compute the information which is necessary to evaluate the