Constraint preconditioners for solving singular saddle point problems

On PSS-based constraint preconditioners for singular nonsymmetric saddle point problems

“…Assume that the conditions of Lemma 3.1 are satisfied, then G is positive definite. From Lemma 2.4 and Theorem 2.9 in [32] we immediately have the following result. …”

“…For this case, there are also many efficient methods, for which we refer to, e.g., [27][28][29][30][31][32]. Recently, Yang et al in [33] proposed a singular constraint preconditioner P and the corresponding preconditioned iteration method, where (1.2) and the preconditioned iteration method is described as follows: (k) ), k = 1, 2, .…”

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

“…We call (1.1) the singular saddle-point problem. A number of effective methods have been proposed in the literature to solve the singular saddle-point problems, such as the Uzawa-type methods [8][9][10][11], Krylov subspace methods [12,13] and matrix splitting iteration methods [14][15][16][17][18] Based on the above splitting, Bai et al [19] proposed an HSS iteration method for solving non-Hermitian positive definite system of linear equations. The iteration scheme of the HSS method used for solving Au ¼ q can be written as ðaI þ HÞu ðkþ1=2Þ ¼ ðaI À SÞu ðkÞ þ q; ðaI þ SÞu ðkþ1Þ ¼ ðaI À HÞu ðkþ1=2Þ þ q; where a is a positive iteration parameter and TðaÞ ¼ ðaI þ SÞ À1 ðaI À HÞðaI þ HÞ À1 ðaI À SÞ ¼ ðaI þ SÞ À1 ðaI þ HÞ À1 ðaI À HÞðaI À SÞ;…”

On semi-convergence of a class of relaxation methods for singular saddle point problems

“…We call (1.1) the singular saddle point problem. A number of effective methods have been proposed in the literature to solve the singular saddle point problems, such as the Uzawa-type methods [22,32,35], Krylov subspace methods [26,33] and matrix splitting iteration methods [13,14,17].…”