On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems

On generalized local Hermitian and skew-Hermitian splitting iterative method for block two-by-two linear systems

“…It is evident that the iterative method presented above is a more generalized case of the local Hermitian and skew-Hermitian splitting (LHSS) and the modified LHSS (MLHSS) iterative methods used in [23]. Thus, the LHSS and MLHSS methods are both special cases of the GLHSS method.…”

“…The most important example is perhaps that of the linearized Navier-Stokes equations, where linearization has been obtained by Picard iteration or by some variant of Newton's method [7][8][9]. During the last decade, a large number of methods for solving linear systems of the form (1.1) have been studied in many papers, including Uzawa-type methods [4,2,[10][11][12][13][14], preconditioned Krylov subspace iterative methods [8,[15][16][17][18], Hermitian and skew-Hermitian splitting (HSS) method as well as its accelerated variants [3,[19][20][21][22][23][24], and restrictively preconditioned conjugate gradient methods [15,[25][26][27]. For designing preconditioners and analyzing the preconditioned matrices for the block 2 Â 2 linear system (1.1), some comprehensive and systematic approaches are presented in [3,7,8,15,21,25,26,29].…”

Semi-convergence analysis of the GPIU method for singular nonsymmetric saddle-point problems

“…Since the coefficient matrix is often large and sparse, iterative methods become more attractive than direct methods for the saddle point problem (1). Many efficient methods have been proposed, such as the Uzawa-type schemes ( [10,15,16,19,22]), the Krylov subspace methods ( [25,33,34]), the SOR-like methods ( [9,20,26,30,35]), the Hermitian and skew-Hermitian splitting (HSS) iteration methods ( [2,[4][5][6][7][8]29]) and some other iterative methods [15,28,42], we mention just a few.…”

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

“…For example, the matrix splitting iterative methods [7,11,18,36,40,45], Uzawa-type methods [12,17,21,22,25], HSS method and its variants [2,[4][5][6]8,9,28,29,38], Krylov subspace methods [1,10,[33][34][35]42], null space methods [24] and so on. When the saddle-point problem (1) is singular, there are also many relaxation iteration methods which have been established, e.g., the Uzawa-type methods [30,46,48,49] and the HSS-like methods [3,20,39].…”