Semi-convergence analysis of Uzawa methods for singular saddle point problems

Recently, Zheng, Bai and Yang studied the parameterized Uzawa method for solving singular saddle point problems (B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009) 808-817). In this paper, we discuss the inexact Uzawa method, which covers the Uzawa method, the preconditioned Uzawa method, and the parameterized Uzawa method to solve the singular saddle point problems. We prove the semi-convergence result under restrictions by verifying two necessary and sufficient conditions, that is, all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear, and the pseudo-spectral radius of the iterative matrix is less than 1. Sufficient conditions for the semi-convergence of several Uzawa-type methods are also provided. In addition, numerical examples are given to demonstrate the semi-convergence of Uzawa-type methods.