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The solution of matrices with $2\times 2$ block structure arises in numerousareas of computational mathematics, such as PDE discretizations based onmixed-finite element methods, constrained optimization problems, or theimplicit or steady state treatment of any system of PDEs with multipledependent variables. Often, these systems are solved iteratively using Krylovmethods and some form of block preconditioner. Under the assumption that onediagonal block is inverted exactly, this paper proves a direct equivalencebetween convergence of $2\times2$ block preconditioned Krylov or fixed-pointiterations to a given tolerance, with convergence of the underlyingpreconditioned Schur-complement problem. In particular, results indicate thatan effective Schur-complement preconditioner is a necessary and sufficientcondition for rapid convergence of $2\times 2$ block-preconditioned GMRES, forarbitrary relative-residual stopping tolerances. A number of corollaries andrelated results give new insight into block preconditioning, such as the factthat approximate block-LDU or symmetric block-triangular preconditioners offerminimal reduction in iteration over block-triangular preconditioners, despitethe additional computational cost. Theoretical results are verified numericallyon a nonsymmetric steady linearized Navier-Stokes discretization, which alsodemonstrate that theory based on the assumption of an exact inverse of onediagonal block extends well to the more practical setting of inexact inverses.

On fixed-point, Krylov, and $2\times 2$ block preconditioners for nonsymmetric problems