A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM MESHFREE DISCRETIZATION

• Published in: Journal of computational mathematics 2013-07, Vol.31 (4), p.398-421
• Main Authors: Cao, Yang, Yao, Linquan, Jiang, Meiqun, Niu, Qiang
• Format: Article
• Language: English
• Subjects: Krylov子空间方法; 多项式预处理; 无网格; 松弛; 离散化方法; 鞍点问题; 预条件子; 高速钢
• ISSN: 0254-9409; 1991-7139
• DOI: 10.4208/jcm.1304-m4209

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 pre- conditioner to accelerate the convergence of some Krylov subspace methods （like GMRES） is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned 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.