scholarly journals A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Wei-Hua Luo ◽  
Ting-Zhu Huang
Keyword(s):  
Saddle Point ◽  
Unit Circle ◽  
Iterative Methods ◽  
Krylov Subspace ◽  
Numerical Results ◽  
Saddle Point Problems ◽  
Good Convergence ◽  
Stokes Problems ◽  
Preconditioned Matrix ◽  
Generalized Saddle Point Problems

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that whenαis big enough, it has an eigenvalue at 1 with multiplicity at leastn, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameterα→0. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner.

A Modified Relaxed Positive-Semidefinite and Skew-Hermitian Splitting Preconditioner for Generalized Saddle Point Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 192-210 ◽  
Author(s):  
Yang Cao ◽  
An Wang ◽  
Yu-Juan Chen
Keyword(s):  
Saddle Point ◽  
Minimal Polynomial ◽  
Mixed Finite Element ◽  
Positive Semidefinite ◽  
Saddle Point Problems ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Oseen Equation ◽  
Preconditioned Matrix ◽  
Generalized Saddle Point Problems

AbstractBased on the relaxed factorization techniques studied recently and the idea of the simple-like preconditioner, a modified relaxed positive-semidefinite and skew-Hermitian splitting (MRPSS) preconditioner is proposed for generalized saddle point problems. Some properties, including the eigenvalue distribution, the eigenvector distribution and the minimal polynomial of the preconditioned matrix are studied. Numerical examples arising from the mixed finite element discretization of the Oseen equation are illustrated to show the efficiency of the new preconditioner.

scholarly journals A Note on the Eigenvalue Analysis of the SIMPLE Preconditioning for Incompressible Flow

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Shi-Liang Wu ◽  
Feng Chen ◽  
Xiao-Qi Niu
Keyword(s):  
Saddle Point ◽  
Incompressible Flow ◽  
Recent Article ◽  
Eigenvalue Analysis ◽  
Saddle Point Problems ◽  
Matrix Eigenvalue ◽  
Preconditioned Matrix ◽  
The Relationship ◽  
Generalized Saddle Point Problems

We consider the SIMPLE preconditioning for block two-by-two generalized saddle point problems; this is the general nonsymmetric, nonsingular case where the (1,2) block needs not to equal the transposed (2,1) block, and the (2,2) block may not be zero. The eigenvalue analysis of the SIMPLE preconditioned matrix is presented. The relationship between the two different formulations spectrum of the SIMPLE preconditioned matrix is established by using the theory of matrix eigenvalue, and some corresponding results in recent article by Li and Vuik (2004) are extended.

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