A Modified Implementation of MINRES to Monitor Residual Subvector Norms for Block Systems

Abstract

Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows subvectors of the residual to be monitored individually. Compared to the implementation from the textbook of [H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014], our method requires one extra vector of storage and no additional applications of the preconditioner. Numerical experiments are included.