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

TL;DR

This paper establishes a theoretical link between the convergence of block preconditioned Krylov methods and the underlying Schur-complement problem for nonsymmetric systems, providing new insights into effective preconditioning strategies.

Contribution

It proves the equivalence between Krylov convergence and Schur-complement convergence under certain conditions, and analyzes the efficiency of various block preconditioners for nonsymmetric problems.

Findings

Effective Schur-complement preconditioners are necessary and sufficient for rapid GMRES convergence.

Approximate block-LDU preconditioners offer minimal iteration reduction compared to simpler block-triangular preconditioners.

Numerical experiments confirm the theory extends to inexact block inverses in practical PDE discretizations.

Abstract

The solution of matrices with $2\times 2$ 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.