  This paper introduces a discretization-accurate stopping criterion of
symmetric iterative methods for solving systems of algebraic equations
resulting from the finite element approximation. The stopping criterion
consists of the evaluations of the discretization and the algebraic error
estimators, that are based on the respective duality error estimator and the
difference of two consecutive iterates. Iterations are terminated when the
algebraic estimator is of the same magnitude as the discretization estimator.
Numerical results for multigrid $V(1,1)$-cycle and symmetric Gauss-Seidel
iterative methods are presented for the linear finite element approximation to
the Poisson equations. A large reduction in computational cost is observed
compared to the standard residual-based stopping criterion.
