Uzawa Method Research Articles

A Modified Nonlinear Inexact Uzawa Algorithm with a Variable Relaxation Parameter for the Stabilized Saddle Point Problem

This paper proposes a modified nonlinear inexact Uzawa (MNIU) algorithm for solving the stabilized saddle point problem, by introducing a variable overrelaxation parameter to speed up convergence. MNIU is an inner-outer iteration method with variable inner accuracy. We give a detailed error analysis for the convergence of MNIU, based upon a newly defined error norm which helps to handle variable inner accuracy for the Uzawa method. We also simply formulate the optimal overrelaxation parameter. Sufficient conditions are given for the convergence of MNIU. We show that MNIU converges in a relatively large range for the variable inner accuracy setting. For constant inner accuracy $\delta$, MNIU is convergent when $\delta<1$, too. Compared with the original nonlinear inexact Uzawa algorithm (NIU) that converges only for constant accuracy and $\delta<1/3$, this is a significant improvement. We also show a practical approach for estimating the optimal overrelaxation parameter for numerical computation. Numerical experiments of MNIU are given and compared with other methods. These results confirm the significant improvements of MNIU.

Block diagonally preconditioned PIU methods of saddle point problem

For large sparse saddle point problems, we firstly introduce the block diagonally preconditioned Gauss–Seidl method (PBGS) which reduces to the GSOR method [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38] and PIU method [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932] when the preconditioners equal to different matrices, respectively. Then we generalize the PBGS method to the PPIU method and discuss the sufficient conditions such that the spectral radius of the PPIU method is much less than one. Furthermore, some rules are considered for choices of the preconditioners including the splitting method of the (1, 1) block matrix in the PIU method and numerical examples are given to show the superiority of the new method to the PIU method.

A generalization of parameterized inexact Uzawa method for generalized saddle point problems

Recently, a class of parameterized inexact Uzawa methods has been proposed for generalized saddle point problems by Bai and Wang [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932], and a generalization of the inexact parameterized Uzawa method has been studied for augmented linear systems by Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. (2008)]. This paper is concerned about a generalization of the parameterized inexact Uzawa method for solving the generalized saddle point problems with nonzero (2, 2) blocks. Some new iterative methods are presented and their convergence are studied in depth. By choosing different parameter matrices, we derive a series of existing and new iterative methods, including the preconditioned Uzawa method, the inexact Uzawa method, the SOR-like method, the GSOR method, the GIAOR method, the PIU method, the APIU method and so on. Numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact Uzawa methods.

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

In this paper, we first present a local Hermitian and skew-Hermitian splitting (LHSS) iteration method for solving a class of generalized saddle point problems. The new method converges to the solution under suitable restrictions on the preconditioning matrix. Then we give a modified LHSS (MLHSS) iteration method, and further extend it to the generalized saddle point problems, obtaining the so-called generalized MLHSS (GMLHSS) iteration method. Numerical experiments for a model Navier–Stokes problem are given, and the results show that the new methods outperform the classical Uzawa method and the inexact parameterized Uzawa method.

Solving the quasi-variational Signorini inequality by the method of successive approximations

The method of successive approximations is examined as a tool for solving the semicoercive quasi-variational Signorini inequality. The auxiliary problems with given friction arising at each step of this method are solved using the Uzawa method with an iterative proximal regularization of the modified Lagrangian functional.

On semi-convergence of parameterized Uzawa methods for singular saddle point problems

Bai et al. recently proposed an efficient parameterized Uzawa method for solving the nonsingular saddle point problems [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters. The optimal iteration parameters and the corresponding optimal semi-convergence factor for the parameterized Uzawa method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the parameterized Uzawa method for solving singular saddle point problems.

Uzawa method on semi-staggered grids for unsteady Bingham media flows

A finite difference scheme on semi-staggered grids is used for numerical simulation of unsteady flows of an incompressible viscoplastic Bingham medium. The Duvaut–Lions variational inequality is considered as a mathematical model. The convergence of the Uzawa method is proved. The efficiency of the method is demonstrated on a model problem with a known exact solution (plane Poiseuille flow) and on the unsteady driven cavity problem.

A generalization of the inexact parameterized Uzawa methods for saddle point problems

For large sparse saddle point problems, Bai and Wang recently studied a class of parameterized inexact Uzawa methods (see Z.-Z. Bai, Z.-Q. Wang, On paramaterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932). In this paper, we generalize these methods and propose a class of generalized inexact parameterized iterative schemes for solving the saddle point problems. We derive conditions for guaranteeing the convergence of these iterative methods. With different choices of the parameter matrices, the generalized iterative methods lead to a series of existing and new iterative methods including the classical Uzawa method, the inexact Uzawa method, the GSOR method and the GIAOR method.

Iterative proximal regularization of the modified Lagrangian functional for solving the quasi-variational Signorini inequality

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).

Méthode du second membre modifié pour la gestion de rapports de viscosité importants dans le problème de Stokes bifluide

In this Note, we present a method to solve numerically the Stokes equation for the incompressible flow between two immiscible fluids presenting very different viscosities. The resolution of the finite element systems of equations is performed using Uzawa's method. The stiffness matrix conditioning problems related to the very important viscosity ratios are circumvented using an new iterative scheme. A numerical example is proposed to show the efficiency of this approach. To cite this article: T.T.C. Bui et al., C. R. Mecanique 336 (2008).

On parameterized inexact Uzawa methods for generalized saddle point problems

For the large sparse saddle point problems, Bai et al. recently studied a class of parameterized inexact Uzawa (PIU) methods [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang. On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. For a special case that the (1,1)-block is solved exactly, they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method. In this paper, we develop these methods to the large sparse generalized saddle point problems. For the obtained parameterized inexact Uzawa method, we prove its convergence under suitable restrictions on the iteration parameters. In particular, we determine its quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor for the saddle point problems. Furthermore, This PIU method is generalized to obtain a framework of accelerated variants of the parameterized inexact Uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration.

Duality scheme for solving the semicoercive signorini problem with friction

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).

A comparative study of efficient iterative solvers for generalized Stokes equations

AbstractWe consider a generalized Stokes equation with problem parameters ξ⩾0 (size of the reaction term) and ν&gt;0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess–Sarazin and Vanka‐type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters ξ and ν. We give an overview of the main theoretical convergence results known for these methods. For a three‐dimensional problem, discretized by the Hood–Taylor 𝒫2–𝒫1 pair, we give results of numerical experiments. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Convergence conditions for splitting iteration methods for non-Hermitian linear systems

Necessary and sufficient convergence conditions are studied for splitting iteration methods for non-Hermitian system of linear equations when the coefficient matrix is nonsingular. When this theory is specialized to the generalized saddle-point problem, we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods, which include the Uzawa and the inexact Uzawa methods as special cases. Moreover, we apply this theory to the two-stage iteration methods for non-Hermitian positive definite linear systems, and obtain sufficient conditions for guaranteeing the convergence of these methods.

On the convergence of iterative methods for stabilized saddle point problems

The convergence of the inexact Uzawa method for stabilized saddle point problems was analysed in a recent paper by Cao, Evans and Qin. We show that this method converges under conditions weaker than those stated in their paper.

An adaptive method for the Stefan problem and its application to endoglacial conduits

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the convergence of the method. As an application we model an endoglacial conduit in which a phase change phenomenon takes place.

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.

A corrected nonlinear Uzawa method for solving stabilized saddle point problems

In this paper, a corrected nonlinear Uzawa algorithm for solving stabilized saddle point problems is proposed and the convergence rate is analyzed. The results of numerical experiments are presented when we apply the algorithm to Stokes equations discretized by mixed finite elements.

Corrected Uzawa methods for solving large nonsymmetric saddle point problems

In this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods, which are similar in some respects to the Uzawa-type methods and are called corrected Uzawa methods. Their convergence rates are analyzed. The results of numerical experiments are presented when we apply them to Navier–Stokes equations discretized by mixed finite elements.

Gauge–Uzawa methods for incompressible flows with variable density

Two new Gauge–Uzawa schemes are constructed for incompressible flows with variable density. One is in the conserved form while the other is in the convective form. It is shown that the first-order versions of both schemes, in their semi-discretized form, are unconditionally stable. Numerical experiments indicate that the first-order (resp. second-order) versions of the two schemes lead to first-order (resp. second-order) convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water.