A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

Abstract

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.

Keywords:

• saddle point matrices
• Krylov subspace methods
• generalized saddle point matrices
• minimal polynomial
• preconditioners

2010 AMS Subject Classifications:

• 65F10
• 65F15

Acknowledgements

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC (61203179, 61202098, 61170309), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2013GGJS-142), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Scientific and Technological Key Project of Education Department of Henan Province (12B110028, 13B430355), Basic and Advanced Technological Research Project of Henan Province (132300410373) and School Youth Fund (2012113004).