scholarly journals New Preconditioners for Nonsymmetric Saddle Point Systems with Singular (1,1) Block

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qingbing Liu
Keyword(s):  
Saddle Point ◽  
Krylov Subspace ◽  
Optimal Parameter ◽  
Spectral Characteristics ◽  
Ill Conditioning ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Saddle Point Matrix ◽  
Large Linear Systems ◽  
Point Type

We investigate the solution of large linear systems of saddle point type with singular (1,1) block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented.

scholarly journals New Block Triangular Preconditioners for Saddle Point Linear Systems with Highly Singular (1,1) Blocks

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
TingZhu Huang ◽  
GuangHui Cheng ◽  
Liang Li
Keyword(s):  
Saddle Point ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Optimal Parameter ◽  
Spectral Characteristics ◽  
Saddle Point Problems

We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

scholarly journals An Alternative HSS Preconditioner for the Unsteady Incompressible Navier-Stokes Equations in Rotation Form

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Jia Liu
Keyword(s):  
Iterative Method ◽  
Krylov Subspace ◽  
Stokes Equations ◽  
Mesh Size ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

We study the preconditioned iterative method for the unsteady Navier-Stokes equations. The rotation form of the Oseen system is considered. We apply an efficient preconditioner which is derived from the Hermitian/Skew-Hermitian preconditioner to the Krylov subspace-iterative method. Numerical experiments show the robustness of the preconditioned iterative methods with respect to the mesh size, Reynolds numbers, time step, and algorithm parameters. The preconditioner is efficient and easy to apply for the unsteady Oseen problems in rotation form.

scholarly journals The alternate direction iterative methods for generalized saddle point systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shuanghua Luo ◽  
Angang Cui ◽  
Cheng-yi Zhang
Keyword(s):  
Saddle Point ◽  
Iterative Methods ◽  
Iterative Schemes ◽  
Numerical Example ◽  
Convergence Results ◽  
Alternate Direction ◽  
Saddle Point Matrix

Abstract The paper studies two splitting forms of generalized saddle point matrix to derive two alternate direction iterative schemes for generalized saddle point systems. Some convergence results are established for these two alternate direction iterative methods. Meanwhile, a numerical example is given to show that the proposed alternate direction iterative methods are much more effective and efficient than the existing one.

scholarly journals A saddle point type solution for a system of operator equations

2021 ◽  
pp. 1-12
Author(s):  
Piotr Kowalski
Keyword(s):  
Saddle Point ◽  
Minimax Theorem ◽  
Dirichlet Problems ◽  
Type Solution ◽  
Operator Equations ◽  
Potential Operators ◽  
System Of Operator Equations ◽  
Ky Fan ◽  
Point Type

Let Ω⊂Rn n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where N:R×R→R is C1 and is partially convex-concave and A:W01,p(Ω)→(W01,p(Ω))* B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

Verification of Iterative Methods for the Linear Complementarity Problem

2016 ◽  
pp. 545-580 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Linear Complementarity Problem ◽  
Numerical Experiments ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Convergence Results ◽  
Fixed Point Principle ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.

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