A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems

Generalized ASOR and Modified ASOR Methods for Saddle Point Problems

“…Thus, ̸ = 0, and therefore is an eigenvalue of −1 −1 . Furthermore, let = 1; from (15) and (16), we have − = ( + ) and = 0. Thus, = − −1 by ̸ = 0 and the definiteness of , which yields that −1 = 0.…”

“…The saddle point problem (1) is important and arises in a variety of scientific and engineering applications, such as mixed finite element approximation of elliptic partial differential equations, optimal control, computational fluid dynamics, weighted least-squares problems, electronic networks, and computer graphics; see [1][2][3][4] and references therein. When the linear system (1) is nonsingular, a number of iterative methods and their numerical properties have been discussed to approximate the unique solution of nonsingular saddle point problem (1) in the literatures, such as SORlike methods [5][6][7][8][9][10][11], Uzawa-type methods [5,[12][13][14][15][16][17][18], Hermitian and skew-Hermitian splitting (HSS) methods and their variants [2,[19][20][21][22][23][24], restrictively preconditioned conjugate gradient (RPCG) methods [25,26], and preconditioned Krylov subspace iteration methods [27][28][29].…”

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

“…Recently, a lot of efforts have been spent on iteration methods for the problem (1.1). The list of the methods studied includes classical Uzawa iteration method [2] and its generalisations [13,20,28,31], Hermitian and skew-Hermitian splitting (HSS) iteration methods [11] and its variants [9,10,26,33,45], shift-splitting iteration methods [1,14,[22][23][24]29,41,42], residual reduction algorithms [3] and Krylov subspace iteration methods [40]. If is a non-Hermitian and/or ill-conditioned matrix, the preconditioning is often used to accelerate the convergence.…”

Generalized Shift-Splitting Preconditioner for Saddle Point Problems with Block Three-by-Three Structure

“…For instance, the Uzawa method [2] is a classical approach which is known for its low computational overhead and easy implementation. To improve the numerical performance, some Uzawa-type variants have been proposed recently which include the inexact Uzawa [5] [6] [7], Uzawa-SOR [8] and Uzawa-PSS [9], etc. The other way for solving the saddle point problems is to exploit the matrix splitting.…”