A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 211-226
Author(s):  
Yan Dou ◽  
Ai-Li Yang ◽  
Yu-Jiang Wu
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Spectral Properties ◽  
Numerical Results ◽  
Saddle Point Problems ◽  
Convergence Properties ◽  
Saddle Point Matrix

AbstractBased on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.

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 Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 172-191 ◽  
Author(s):  
Quan-Yu Dou ◽  
Jun-Feng Yin ◽  
Ze-Yu Liao
Keyword(s):  
Saddle Point ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Saddle Point Problems ◽  
Splitting Technique ◽  
Splitting Iteration Method

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

A New GSOR Method for Generalised Saddle Point Problems

2016 ◽  
Vol 6 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Na Huang ◽  
Chang-Feng Ma
Keyword(s):  
Saddle Point ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Saddle Point Problems ◽  
Successive Overrelaxation ◽  
Iteration Parameters

AbstractA novel generalised successive overrelaxation (GSOR) method for solving generalised saddle point problems is proposed, based on splitting the coefficient matrix. The proposed method is shown to converge under suitable restrictions on the iteration parameters, and we present some illustrative numerical results.

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