Some preconditioning techniques for a class of double saddle point problems

TL;DR

This paper analyzes spectral properties of block preconditioners for double saddle point problems, introducing an inexact triangular preconditioner that accelerates FGMRES convergence, supported by theoretical and numerical evidence.

Contribution

It introduces and analyzes an inexact block triangular preconditioner for double saddle point problems, demonstrating its spectral properties and efficiency.

Findings

Eigenvalues lie in a circle of radius 1 centered at (1,0)

Inexact preconditioners lead to faster FGMRES convergence

Numerical results confirm theoretical spectral bounds

Abstract

In this paper, we describe and analyze the spectral properties of a number of exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the FGMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1,0) and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.