A Modified Augmented Lagrangian Preconditioner For Nonsymmetric Saddle Point Problem

In large-scale scientific computing and engineering applications, many problems canbe transformed into large sparse saddle point problems. Such as fluid dynamics, optimiza-tion, Elasticity, etc. Therefore, the study of the numerical method of saddle point problemshas a very important theoretical significance and application value. Preconditioned methodis a hot topic for iterative solution of large sparse linear equations. This is due to a goodpreconditioner not only speed up the iteration speed, but also reduce the workload of eachiteration. Currently, many efcient preconditioners have been proposed for saddle pointproblems form diferent practical problems, such as Constraint preconditioner, HSS precon-ditioner, augmented Lagrangian preconditioner, and so on. Of course, construction of thepreconditioner is closely related to the structure of the problem.In this paper, we consider a class of non-symmetric saddle point problems. The coef-cient matrix has a special3×3block structure. For this kind of problem, we proposed a classof modified augmented Lagrangian preconditioners, which are easier to be implemmentedthan the augmented lagrangian preconditioner. Spectral properties of the preconditioned ma-trix are analyzed. Finally, numerical examples are illustrated to show the efectiveness of theproposed preconditioner. The theoretical analysis and numerical results extend those of theaugmented Lagrangian preconditioning techniques studied by Benzi.