A relaxed HSS preconditioner for saddle point problems from meshfree discretization

Yang Cao*, Linquan Yao, Meiqun Jiang, Qiang Niu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

78 Citations (Scopus)


In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS preconditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS preconditioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.

Original languageEnglish
Pages (from-to)398-421
Number of pages24
JournalJournal of Computational Mathematics
Issue number4
Publication statusPublished - Jul 2013


  • Element-free Galerkin method
  • HSS preconditioner
  • Krylov subspace method
  • Meshfree method
  • Preconditioning
  • Saddle point problems

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