An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems

Abstract

In this paper, we propose an inexact Uzawa method with variable relaxation parameters for iteratively solving linear saddle-point problems. The method involves two variable relaxation parameters, which can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. This new algorithm has the advantage over most existing Uzawa-type algorithms: it is always convergent without any apriori estimates on the spectrum of the preconditioned Schur complement matrix, which may not be easy to achieve in applications. The rate of the convergence of the inexact Uzawa method is analysed. Numerical results of the algorithm applied for the Stokes problem and a purely linear system of algebraic equations are presented.

Some Observations on Generalized Saddle-Point Problems

“…The system (1.1)-(1.2) seems to be one of the most generalized saddle-point systems investigated in the literature. The case of bilinear forms c = 0 and b 1 = b 2 has been extensively studied [1,3,4,7,5,9,10]. Also, considerable research has been done on the system with b 1 = b 2 and c = 0 [4,11,14], while the well-posedness for the system with c = 0 but b 1 = b 2 was established in [13] and [2].…”

Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions

“…In recent years, there has been increasing interest in solving saddle-point problems like (3.1) by iterative methods; see, for example, [5], [6], [17], and [24]. But the most existing methods require the stiffness matrix corresponding to the primal variable u h above to be nonsingular, so they cannot be applied to solve the saddlepoint system (3.1) with γ 0 = 0, as the operatorĀ is singular in the space V h (Ω).…”

A variant of the AOR method for augmented systems

“…Let us consider the linear system of equations (1.1), given in [6], where the matrices A and B are defined as follows:…”

“…Many applications such as fluid dynamics, optimization and constrained or generalized least squares problems, image processing, linear elasticity and mixed Finite Element Method for elliptic equations [1]- [3], [6] and [15] lead us to a linear system of equations of the form:…”