Augmentation-Based Preconditioners for Saddle-Point Systems with Singular Leading Blocks

TL;DR

This paper introduces a novel preconditioning technique for symmetric saddle-point matrices with singular leading blocks, improving iterative solution efficiency by achieving a limited eigenvalue spectrum.

Contribution

It proposes a new ideal positive definite block diagonal preconditioner tailored for singular leading blocks in saddle-point systems, with practical implementation strategies.

Findings

Preconditioned operator has four distinct eigenvalues.

Numerical experiments demonstrate improved convergence.

Preconditioner effectively handles singular leading blocks.

Abstract

We consider the iterative solution of symmetric saddle-point matrices with a singular leading block. We develop a new ideal positive definite block diagonal preconditioner that yields a preconditioned operator with four distinct eigenvalues. We offer a few techniques for making the preconditioner practical, and illustrate the effectiveness of our approach with numerical experiments.