Convergence behavior of generalized parameterized Uzawa method for singular saddle-point problems

In this paper, we will seek the least squares solution for singular saddle-point problems. The parameterized Uzawa (PU) method is further studied and a generalized PU (GPU) proper splitting is proposed. The convergence behavior of the corresponding GPU iteration is studied. It is proved that the GPU iteration method can converge to the best least squares solutions of the singular saddle-point problems. In addition, we prove that the GPU preconditioned GMRES for singular saddle-point problems will also determine the least squares solution at breakdown. The eigenvalue distributions of the GPU preconditioned matrix are derived. Numerical experiments are presented, which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the GPU iteration has better convergence behavior than the PU iteration, both as a solver and a preconditioner of GMRES.