A modification of the generalized shift-splitting method for singular saddle point problems

TL;DR

This paper introduces a modified generalized shift-splitting method with a positive definite block diagonal matrix for better solving singular saddle point problems, demonstrating improved effectiveness through numerical experiments.

Contribution

The paper proposes a novel modification to the GSS method using a symmetric positive definite block diagonal matrix, enhancing convergence and robustness.

Findings

The modified GSS method shows superior performance over the classical GSS.

Numerical experiments confirm the effectiveness and robustness of the new preconditioner.

Eigenvalue analysis supports improved convergence properties.

Abstract

A modification of the generalized shift-splitting (GSS) method is presented for solving singular saddle point problems. In this kind of modification, the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric positive definite. Semi-convergence of the proposed method is investigated. The induced preconditioner is applied to the saddle point problem and the preconditioned system is solved by the restarted generalized minimal residual method. Eigenvalue distribution of the preconditioned matrix is also discussed. Finally some numerical experiments are given to show the effectiveness and robustness of the new preconditioner. Numerical results show that the modified GSS method is superior to the classical GSS method.