A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements

We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u(h)(n) (the superscript n stands for the CG iteration number, the subscript It for the typical mesh size) and on heuristics to relate the error between u(h) and u(h)(n) to the residual vector.