A New Constraint Preconditioner For The Saddle Point Problem

We consider a class of saddle point problems which represent discrete Stokes equations. The(1,1)-block A of the system matrix has special structure which makes the linear system Az=r can be solved exactly.We present a kind of constraint preconditioner for the saddle point problems by making use of the special structure of A.The difference between our preconditioner and other preconditioners is that we use the exact (1,1)-block while other preconditioners usually use an approximate of A.We study the special structure of preconditioned system and design a direct and fast algorithm to solve the system Az=r.The main advantage of our new preconditioner is that we take advantage of the special structure of the(1,1)-block as possible as we can.In numerical experiments,we solve an Poiseuille flow problem with the preconditioned GMRES method.We compare our new preconditioner with some other preconditioners, and also compare the effects of different variants of our preconditioner. Numerical results show that our new preconditioner can greatly reduces the number of iterations and CPU time,which means that our preconditioner is very effective.Finally, we make further generalization of our study.