A generalized shift-splitting preconditioner for saddle point problems

Abstract

Abstract. In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.

A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the Navier-Stokes equation

“…From the numerical examples in [42], we find the eigenvalues distribution of the preconditioned matrix P −1 GSS A gather more closely than those in [41], also the convergence performance is better.…”

“…The experiments have been carried out by MAT-LAB R2011b (7.13), Intel(R) Core(TM) i7-2670QM, CPU 2.20GHZ, RAM 8.GB PC Environment. Especially, when C = 0, i.e., the 2-by-2 block of the coefficient of (1.1) is zero, we compare the MPSS preconditioner with the RPSS in [43], GSS in [42] and no preconditioning situation. [4], the coefficient matrix have the following form…”

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

“…It is easy to see that P SS is a special case of P GSS when α = β. Numerical results in [21,20] confirmed that the GSS preconditioner is superior to the SS preconditioner.…”

An inexact relaxed DPSS preconditioner for saddle point problem

“…And Krylov subspace methods are very slow or even fail to converge if not conveniently preconditioned. Therefore, many researchers devote themselves to the preconditioned iterative methods (1.1) (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). And kinds of preconditioners for saddle point matrix are studied, such as symmetric indefinite preconditioners [28,29], inexact constraint preconditioners [29][30][31][32][33][34] and primal-based penalty preconditioners [35].…”