On generalized parameterized inexact Uzawa method for a block two-by-two linear system

Recently, Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765-771] presented a parameterized inexact Uzawa (PIU) algorithm for solving symmetric saddle point problems, where the (1, 2)- and the (2, 1)-blocks are the transpose of each other. In this paper, we extend the PIU method to the block two-by-two linear system 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 prove that the iteration method is convergent under certain conditions. With different choices of the parameter matrices, we obtain several new algorithms for solving the block two-by-two linear system. Numerical experiments confirm our theoretical results and show that our method is feasible and effective.