An optimal iterative solver for symmetric indefinite linear systems with PDE origins: Balanced black-box stopping tests

This work discusses the design of efficient algorithms for solving symmetric indefinite linear systems arising from FEM approximation of PDEs. The distinctive feature of the preconditioned MINRES solver that is used here is the incorporation of error control in the ‘natural norm’ in combination with an effective a posteriori estimator for the PDE approximation error. This leads to a robust and optimal blackbox stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error