A new class of regularized preconditioners for double saddle-point problems

TL;DR

This paper introduces a new class of block preconditioners for double saddle-point problems, demonstrating improved performance over existing methods especially at high Reynolds numbers, with thorough theoretical and numerical analysis.

Contribution

The paper proposes a novel three-by-three block preconditioner specifically designed for double saddle-point problems, addressing limitations of previous methods at high Reynolds numbers.

Findings

Preconditioner outperforms existing methods at high Reynolds numbers.

Theoretical analysis confirms favorable spectral properties.

Numerical experiments verify efficiency and bounds.

Abstract

The block structure of double saddle-point problems has prompted extensive research into efficient preconditioners. This paper introduces a novel class of three-by-three block preconditioners tailored for such systems from the time-dependent Maxwell equations or liquid crystal director modeling. The main motivation of this work is to highlight the limitations of recent preconditioners under high Reynolds numbers, as the original studies did not explore this scenario, and to demonstrate that our preconditioner outperforms the existing ones in such regimes. We thoroughly analyze the convergence and spectral properties of the proposed preocnditioner. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds.