Some remarks on the nonlinear inexact Uzawa algorithm for saddle point problem

Abstract

In this paper, we discuss and analyze a slight modification of the nonlinear inexact Uzawa algorithm proposed by Bramble et al.. [Bramble, J. H., Pasciak, E. and Vassilev, A. T. (1997). Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal., 34, 1072–1092.] for solving saddle point problems. It is shown that the modified algorithm converges under a condition weaker than that given by Bramble et al. [Bramble, J. H., Pasciak, E. and Vassilev, A. T. (1997). Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal., 34, 1072–1092.] and Cheng [Cheng, X. L. (2000). On the nonlinear inexact Uzawa algorithm for saddle-point problems. SIAM J. Numer. Anal., 37, 1930–1934.]. In addition, we also compare the scheme to the iterative method with variable relaxation parameters proposed by Hu and Zou [Hu, Q. Y. and Zou, J. (2001). An iterative method with variable relaxation parameters for saddle-point problems. SIAM J. Matrix Anal. Appl., 23, 317–338.]. We remove the assumptions presented by Hu and Zou [Hu, Q. Y. and Zou, J. (2001). An iterative method with variable relaxation parameters for saddle-point problems. SIAM J. Matrix Anal. Appl., 23, 317–338.] for the preconditioner Q A .