On GPIU Method And Its Preconditioning Property For Nonsymmetric Saddle Point Problem

Saddle point problems arises in many scientific computations and engineering ap-plications, such as mixed finite element methods for solving elliptic partial differential equations and Stokes problems, constraint optimization, least-squrares problems, fluid dynamics, elasticity and so on. Because the saddle point problems have such a wide application source, it is of great interest to develop fast and efficient methods. The pa-rameterized inexact Uzawa method used to solve the saddle point problems is recently studied by many people.In paper [29], a parameterized inexact Uzawa algorithm for solving symmetric saddle point problems with the (1,2)-block being the transpose of the (2,1)-block and the (2,2)-block being zero was considered. In this paper, we extend it to nonsymmetric generalized saddle point case by allowing the (1,2)-block to be not equal to the transpose of the (2,1)-block and the (2,2)-block may not be zero. We proved that the iteration method is convergent under certain conditions, and conclude that the spectral radius of the preconditioned matrix is in the interval (0,2). With different of the parameter matrices in the matrix splitting, we get several algorithms for solving the nonsymmetric generalized saddle point problem. Numerical experiments are used to show that our method is feasible and effective.