Stopping criteria for iterative methods: applications to PDE's

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(Ω), 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