A Class of Generalized Shift-Splitting Preconditioners for Double Saddle Point Problems

TL;DR

This paper introduces a generalized shift-splitting preconditioner for double saddle point problems, providing convergence analysis, spectral bounds, and demonstrating superior performance through numerical experiments.

Contribution

It proposes a new generalized shift-splitting preconditioner with relaxed variants, along with convergence analysis and spectral bounds, improving solution efficiency for DSPPs.

Findings

Effective in solving DSPPs from PDE-constrained optimization and cavity problems.

Converges under established conditions with sharp spectral bounds.

Outperforms existing preconditioners in numerical tests.

Abstract

In this paper, we propose a generalized shift-splitting (GSS) preconditioner, along with its two relaxed variants to solve the double saddle point problem (DSPP). The convergence of the associated GSS iterative method is analyzed, and sufficient conditions for its convergence are established. Spectral analyses are performed to derive sharp bounds for the eigenvalues of the preconditioned matrices. Numerical experiments based on examples arising from the PDE-constrained optimization problem and the leaky lid-driven cavity problem demonstrate the effectiveness and robustness of the proposed preconditioners compared with existing state-of-the-art preconditioners.