On the generalized shift-splitting preconditioner for saddle point problems

TL;DR

This paper introduces a generalized shift-splitting preconditioner for saddle point problems, demonstrating its effectiveness through theoretical analysis and numerical experiments on finite element discretizations of the Stokes problem.

Contribution

It proposes a new preconditioner derived from a stationary iterative method, including a relaxed version, and analyzes its eigenvalue distribution for saddle point problems.

Findings

Preconditioner improves convergence for saddle point systems.

Eigenvalue distribution analysis supports effectiveness.

Numerical tests confirm practical efficiency.

Abstract

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.