The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems

In this paper, based on the positive-definite and positive-semidefinite splitting (PPS) iteration scheme, we establish a class of Uzawa-PPS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite. Theoretical analyses show that the convergence and semi-convergence properties of the proposed methods can be guaranteed under suitable conditions. Furthermore, we consider acceleration of the Uzawa-PPS methods by Krylov subspace (like GMRES) methods and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners.