Block Triangular Preconditioners For Generalized Saddle Point Problems

Saddle point problems arise in a variety of scientific and engineering applications,including optimization problem,computational fluid dynamics,construct analysis,and so on. So it is important to find the numerical solution.Recently,many researchers have posed a great many of effective algorithms.Such as,in the simple iterative methods,Uzawatype methods,inexact Uzawa methods,on parameterized inexact Uzawa methods,nonlinear Uzawa methods,SOR-like methods,HSS iteration methods and so on;in the preconditioned methods,block diagonal preconditioning,block triangular preconditioning,constrained preconditioning, HSS preconditioning,the restrictively preconditioned conjugate gradient methods and so on.Golub and Yuan proposed the ST decomposition for the nonsymmetric matrix in[19].Then,in[35],Wu,etc.applied the ST decomposition to slove the saddle point problems.In this paper,we extend the ST decomposition to the generalized saddle point problem and present three block triangular preconditioners.Then we take two of them and apply them to the generalized saddle point problem.The two preconditioned systems are symmetric and positive definite.Then we deduce the general properties and the upper bound of the condition number of the two preconditioned systems one by one.Finally,we propose two numerical examples,one is a particular linear system,the other is a linear algebra system based on the Stokes equation.The numerical results show that the two preconditioners given in the paper are effective.