A new class of regularized preconditioners for double saddle-point problems
Achraf Badahmane
TL;DR
The paper addresses solving 3x3 double saddle-point systems that arise in time-dependent Maxwell equations and liquid crystal director models. It introduces a regularized three-block preconditioner $\mathcal{P}_{R}$ (and the variant $\mathcal{P}_{RD}$) and provides a rigorous spectral analysis, deriving bounds for the eigenvalues of $\mathcal{P}_{R}^{-1}\mathcal{A}$ in terms of $S_C = C S^{-1} C^{T}$ and the regularization parameter $\alpha$. It also presents an explicit algorithmic implementation with inexact inner solves and reports extensive numerical experiments showing that $\mathcal{P}_{R}$ (and $\mathcal{P}_{RD}$ for symmetric problems) outperform existing preconditioners (BD, SS, RSS) in convergence and CPU time, especially at low viscosities (high Reynolds number). The results attribute the performance gains to eigenvalue clustering of the preconditioned operator and highlight the impact of the approximate Schur complement $\hat S$ on robustness.
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.