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

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.