# On nonlinear inexact Uzawa algorithms for stabilized saddle point problems

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## Abstract

In this paper, the nonlinear inexact Uzawa (NIU) methods for saddle point problems are studied. A further improved convergence result of the NIU algorithm for stabilized saddle point problems is presented, which results in much smaller convergence factor. This result can also be viewed as a generalization of some previous work for classical saddle point problems. Based on this result, we generalize some adaptive versions of the NIU algorithm with variable iteration parameters for classical saddle point problems to solve stabilized saddle point problems. Convergence analyses of these algorithms are presented. It shows that they converge under similar conditions as those in classical cases, which are more practical and without estimates on the extreme eigenvalues of the involved preconditioned systems. Numerical experiments are given to demonstrate the effectiveness of these algorithms over some original NIU algorithms.

### Keywords

Saddle point problem Uzawa algorithm Convergence analysis### Mathematics Subject Classification

65F10 65F15 65F50## Notes

### Acknowledgements

We would like to express our sincere thanks to the two unknown reviewers for their careful reading of the manuscript. Their useful comments and valuable suggestions greatly improve the quality of the paper.

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