An Optimal Iterative Solver for Symmetric Indefinite Systems Stemming from Mixed Approximation

We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural energy norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a proof of concept MATLAB implementation of this algorithm, which we call EST_MINRES, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007).