On semi-convergence of a class of Uzawa methods for singular saddle-point problems

In this paper, a class of Uzawa methods are presented for singular saddle-point problems. These methods contain the recently proposed Uzawa-AOR and Uzawa-SAOR methods as special cases. The (1, 1)-block of the corresponding Uzawa preconditioner is positive definite. Both nonsingular and singular preconditioning matrices are considered. The semi-convergence of these methods is analyzed by using the techniques of singular value decomposition and Moore-Penrose inverse. Numerical results show that they need less workload per iteration step comparing with the GSOR and PIU methods in some situations, so they are feasible and effective for singular saddle-point problems in some situations.