GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Iterative solution methods
Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term
SIAM Journal on Scientific Computing
Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow
ACM Transactions on Mathematical Software (TOMS)
An Augmented Lagrangian-Based Approach to the Oseen Problem
SIAM Journal on Scientific Computing
An Optimal Iterative Solver for Symmetric Indefinite Systems Stemming from Mixed Approximation
ACM Transactions on Mathematical Software (TOMS)
Preconditioning Stochastic Galerkin Saddle Point Systems
SIAM Journal on Matrix Analysis and Applications
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
Journal of Computational Physics
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The convergence of numerical approximations to the solutions of differential equations is a key aspect of numerical analysis and scientific computing. Iterative solution methods for the systems of linear(ized) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often paid to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a 'right' norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer.