# Preconditioning of elliptic saddle point systems by substructuring and a penalty approach.

## Abstract

The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system. The approach can be applied to a variety of different saddle point problems. Although the preconditioner itself is symmetric and indefinite, all the eigenvalues of the preconditioned system are real and positive if certain conditions hold. Stronger conditions also ensure that the eigenvalues are bounded independently of mesh parameters. An interesting feature of the approach is that conjugate gradients can be used as the iterative solution method rather than GMRES. The effectiveness of the overall strategy hinges on the preconditioner for the primal Schur complement. Interestingly, the primary condition ensuring real and positive eigenvaluesmore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Laboratories

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 947799

- Report Number(s):
- SAND2005-0068C

TRN: US200905%%167

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Conference

- Resource Relation:
- Conference: Proposed for presentation at the 16th International Domain Decomposition Conference held January 12-15, 2005 in New York, NY.

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ELLIPTICAL CONFIGURATION; SADDLE-POINT METHOD; EIGENVALUES

### Citation Formats

```
Dohrmann, Clark R.
```*Preconditioning of elliptic saddle point systems by substructuring and a penalty approach.*. United States: N. p., 2005.
Web.

```
Dohrmann, Clark R.
```*Preconditioning of elliptic saddle point systems by substructuring and a penalty approach.*. United States.

```
Dohrmann, Clark R. Sat .
"Preconditioning of elliptic saddle point systems by substructuring and a penalty approach.". United States.
```

```
@article{osti_947799,
```

title = {Preconditioning of elliptic saddle point systems by substructuring and a penalty approach.},

author = {Dohrmann, Clark R},

abstractNote = {The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system. The approach can be applied to a variety of different saddle point problems. Although the preconditioner itself is symmetric and indefinite, all the eigenvalues of the preconditioned system are real and positive if certain conditions hold. Stronger conditions also ensure that the eigenvalues are bounded independently of mesh parameters. An interesting feature of the approach is that conjugate gradients can be used as the iterative solution method rather than GMRES. The effectiveness of the overall strategy hinges on the preconditioner for the primal Schur complement. Interestingly, the primary condition ensuring real and positive eigenvalues is satisfied automatically in certain instances if a Balancing Domain Decomposition by Constraints (BDDC) preconditioner is used. Following an overview of BDDC, we show how its constraints can be chosen to ensure insensitivity to parameter choices in the (2,2) block for problems with a divergence constraint. Examples for different saddle point problems are presented and comparisons made with other approaches.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2005},

month = {1}

}