# On nonlinear inexact Uzawa algorithms for stabilized saddle point problems

## 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.

- Authors:

- Lanzhou University, School of Mathematics and Statistics (China)

- Publication Date:

- OSTI Identifier:
- 22769320

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; CONVERGENCE; EIGENVALUES; NONLINEAR PROBLEMS

### Citation Formats

```
Liang, Zhao-Zheng, and Zhang, Guo-Feng.
```*On nonlinear inexact Uzawa algorithms for stabilized saddle point problems*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0440-2.

```
Liang, Zhao-Zheng, & Zhang, Guo-Feng.
```*On nonlinear inexact Uzawa algorithms for stabilized saddle point problems*. United States. doi:10.1007/S40314-017-0440-2.

```
Liang, Zhao-Zheng, and Zhang, Guo-Feng. Tue .
"On nonlinear inexact Uzawa algorithms for stabilized saddle point problems". United States. doi:10.1007/S40314-017-0440-2.
```

```
@article{osti_22769320,
```

title = {On nonlinear inexact Uzawa algorithms for stabilized saddle point problems},

author = {Liang, Zhao-Zheng and Zhang, Guo-Feng},

abstractNote = {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.},

doi = {10.1007/S40314-017-0440-2},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}