A new relaxed HSS preconditioner for saddle point problems

TL;DR

This paper introduces a new relaxed HSS preconditioner for saddle point problems, demonstrating its effectiveness through theoretical properties and numerical experiments on finite element discretizations of the Stokes problem.

Contribution

A novel relaxed HSS preconditioner derived from a convergent stationary iterative method for saddle point problems.

Findings

Effective preconditioning demonstrated on Stokes problem discretizations.

Eigenvalue distribution analysis supports the preconditioner's efficiency.

Numerical results show improved convergence of Krylov methods.

Abstract

We present a preconditioner for saddle point problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under a mild condition. Some properties of the preconditioner as well as the eigenvalues distribution of the preconditioned matrix are presented. The preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioner.