A class of inexact block factorization preconditioners for indefinite matrices with a three-by-three block structure

TL;DR

This paper introduces eight inexact block factorization preconditioners tailored for indefinite matrices with a three-by-three block structure, enhancing the efficiency of Krylov subspace methods.

Contribution

The paper develops a unified theoretical framework for analyzing inexact block preconditioners and demonstrates their effectiveness through spectral analysis and numerical comparisons.

Findings

Three preconditioners achieve high-speed convergence

Spectral bounds are established for preconditioned matrices

Proposed preconditioners outperform existing methods in tests

Abstract

We consider using the preconditioned-Krylov subspace method to solve the system of linear equations with a three-by-three block structure. By making use of the three-by-three block structure, eight inexact block factorization preconditioners, which can be put into a same theoretical analysis frame, are proposed based on a kind of inexact factorization. By generalizing Bendixson Theorem and developing a unified technique of spectral equivalence, the bounds of the real and imaginary parts of eigenvalues of the preconditioned matrices are obtained. The comparison to eleven existed exact and inexact preconditioners shows that three of the proposed preconditioners can lead to high-speed and effective preconditioned-GMRES in most tests.