# The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations

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## Abstract

In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally, a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity *n*, and the other *m* eigenvalues are of the form \(1-\lambda \) with \(|\lambda |<1\), independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved.

## Keywords

Nonsymmetric generalized saddle point problem Generalized double shift-splitting Krylov subspace method Convergence## Mathematics Subject Classification

65F10 65F50## Notes

### Acknowledgements

The authors are very much indebted to the anonymous referees for their constructive suggestions and insightful comments. The incorporation of these suggestions has greatly improved the original manuscript of this paper. This work is supported by the National Natural Science Foundation of China (nos. 11571004, 11701456).

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