The solution of matrices with a $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, this paper proves a direct equivalence between convergence of $2\times2$ block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of $2\times 2$ block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. Theoretical results are verified numerically on a nonsymmetric steady linearized Navier--Stokes discretization, which also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses.


  1. Krylov
  2. GMRES
  3. block preconditioning

MSC codes

  1. 65F08

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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 871 - 900
ISSN (online): 1095-7162


Submitted: 8 November 2019
Accepted: 25 March 2020
Published online: 11 June 2020


  1. Krylov
  2. GMRES
  3. block preconditioning

MSC codes

  1. 65F08



Funding Information

Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : RGPIN-05606-2015, RGPAS-478018-2015
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-NA0002376

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