Journal of Applied Mathematics

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Preconditioning Techniques for Sparse Linear Systems

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Volume 2012 |Article ID 402490 | https://doi.org/10.1155/2012/402490

Ning-Bo Tan, Ting-Zhu Huang, Ze-Jun Hu, "A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations", Journal of Applied Mathematics, vol. 2012, Article ID 402490, 12 pages, 2012. https://doi.org/10.1155/2012/402490

A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations

Academic Editor: Massimiliano Ferronato
Received08 Dec 2011
Revised02 Apr 2012
Accepted19 Apr 2012
Published28 Jun 2012

Abstract

A relaxed splitting preconditioner based on matrix splitting is introduced in this paper for linear systems of saddle point problem arising from numerical solution of the incompressible Navier-Stokes equations. Spectral analysis of the preconditioned matrix is presented, and numerical experiments are carried out to illustrate the convergence behavior of the preconditioner for solving both steady and unsteady incompressible flow problems.

1. Introduction

We consider systems of linear equations arising from the finite-element discretization of the incompressible Navier-Stokes equations governing the flow of viscous Newtonian fluids. The primitive variables formulation of the Navier-Stokes equations is 𝜕𝐮],[],[],𝜕𝑡−𝜐Δ𝐮+(𝐮⋅∇)𝐮+∇𝑝=𝐟onΩ×(0,𝑇(1.1)div𝐮=0onΩ×0,𝑇(1.2)𝐮=𝐠on𝜕Ω×0,𝑇(1.3)𝐮(𝐱,0)=𝐮0(𝐱)onΩ,(1.4) where Ω⊂ℝ2 is an open bounded domain with sufficiently smooth boundary 𝜕Ω, [0,𝑇] is an time interval of interest, 𝐮(𝐱,𝑡) and 𝑝(𝐱,𝑡) are unknown velocity and pressure fields, 𝜐 is the kinematic viscosity, Δ is the vector Laplacian, ∇ is the gradient, div is the divergence, and 𝐟, 𝐠, and 𝐮0 are given functions. The Stokes problem is obtained by dropping the nonlinearity (𝐮⋅∇)𝐮 from the momentum equation (1.1). Refer to [1] for an introduction to the numerical solution of the Navier-Stokes equations. Implicit time discretization and linearization of the Navier-Stokes equations by Picard or Newton fixed iteration result in a sequence of (generalized) Oseen problems. The Oseen problems by spatial discretization with LBB-stable finite elements (see [1, 2]) are reduced to a series of large sparse systems of linear equations with a saddle point matrix structure as follows: 𝐀𝐱=𝐛,(1.5) with î‚âŽ¡âŽ¢âŽ¢âŽ£ğ€=ğ€ğğ‘‡âŽ¤âŽ¥âŽ¥âŽ¦âŽ›âŽœâŽœâŽğ®ğ‘âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽğŸâŽžâŽŸâŽŸâŽ âˆ’ğ0,𝐱=,𝐛=−𝑔,(1.6) where 𝐮 and 𝑝 represent the discrete velocity and pressure, respectively. In two-dimensional cases, 𝐀=diag(𝐴1,𝐴2) denotes the discretization of the reaction diffusion, and each diagonal submatrix 𝐴𝑖 is a scalar discrete convection-diffusion operator represented as 𝐴𝑖=ğœŽğ‘‰+𝜐𝐿+𝑁𝑖(𝑖=1,2),(1.7) where 𝑉 denotes the velocity mass matrix, 𝐿 the discrete (negative) Laplacian, and 𝑁𝑖 the convective terms. The matrix 𝐀 is positive definite in the sense that 𝐀𝑇+𝐀 is symmetric positive definite. Matrix 𝐁𝑇=(𝐵𝑇1,𝐵𝑇2) denotes the discrete gradient with 𝐵𝑇1, 𝐵𝑇2 being discretizations of the partial derivatives 𝜕/𝜕𝑥, 𝜕/𝜕𝑦, respectively. 𝐟=(𝑓1,𝑓2)𝑇 and 𝑔 contain the forcing and boundary terms.

In the past few years, a considerable amount of work has been spent in developing efficient solvers for systems of linear equations in the form of (1.5); see [3] for a comprehensive survey. Here we consider preconditioned Krylov subspace methods, in particular preconditioned GMRES [4] in this paper. The convergence performance of this method is mainly determined by the underlying preconditioner employed. An important class of preconditioners is based on the block LU factorization of the coefficient matrix, including a variety of block diagonal and triangular preconditioners. A crucial ingredient in all these preconditioners is an approximation to the Schur complement 𝐒=𝐁𝐀−1𝐁𝑇. This class of preconditioners includes the pressure convection diffusion (PCD) preconditioner, the least-squares commutator (LSC) preconditioner, and their variants [5–7]. Somewhat related to this class of preconditioners are those based on the augmented Lagrangian (AL) reformulation of the saddle point problem; see [8–11]. Other types of preconditioners for the saddle point problems include those based on the Hermitian and skew-Hermitian splitting (HSS) [12–15] and the dimensional splitting (DS) [16] of the coefficient matrix 𝐀. In [17], a relaxed dimensional factorization preconditioner is introduced.

The remainder of the paper is organized as follows. In Section 2, we present a relaxed splitting preconditioner based on matrix splitting and prove that the preconditioned matrix has eigenvalue 1 of algebraic multiplicity at least 𝑛 (recall that 𝑛 is the number of velocity degrees of freedom). In Section 3, we show the results of a series of numerical experiments indicating the convergence behavior of the relaxed splitting preconditioner. In the final section, we draw our conclusions.

2. A Relaxed Splitting Preconditioner

2.1. A Splitting of the Matrix

In this paper, we limit to 2D case. The system matrix 𝐀 admits the following splitting: î‚âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´ğ€=10𝐵𝑇10𝐴2𝐵𝑇2−𝐵1−𝐵20⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝐴100000−𝐵1⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣0000𝐵𝑇10𝐴2𝐵𝑇20−𝐵20⎤⎥⎥⎥⎥⎦=𝐻+𝑆,(2.1) where 𝐴1∈ℝ𝑛1×𝑛1, 𝐴2∈ℝ𝑛2×𝑛2, 𝐵1∈ℝ𝑚×𝑛1, and 𝐵2∈ℝ𝑚×𝑛2. Thus, 𝐀∈ℝ(𝑛+𝑚)×(𝑛+𝑚) is of dimension 𝑛=𝑛1+𝑛2. Let 𝛼>0 be a parameter and denote by 𝐼 the identity matrix of order 𝑛1+𝑛2+𝑚. Then, 𝐻+𝛼𝐼 and 𝑆+𝛼𝐼 are both nonsingular, nonsymmetric, and positive definite. Consider the two splittings of 𝐀: 𝐀=(𝐻+𝛼𝐼)−(𝛼𝐼−𝑆),𝐀=(𝑆+𝛼𝐼)−(𝛼𝐼−𝐻).(2.2) Associated to these splittings is the alternating iteration, 𝑘=0,1,…, (𝐻+𝛼𝐼)𝐱𝑘+1/2=(𝛼𝐼−𝑆)𝐱𝑘+𝐛,(𝑆+𝛼𝐼)𝐱𝑘+1=(𝛼𝐼−𝐻)𝐱𝑘+1/2+𝐛.(2.3) Eliminating 𝐱𝑘+1/2 from these, we can rewrite (2.3) as the stationary scheme: 𝐱𝑘+1=𝑇𝛼𝐱𝑘+𝐜,𝑘=0,1,…,(2.4) where 𝑇𝛼=(𝑆+𝛼𝐼)−1(𝛼𝐼−𝐻)(𝐻+𝛼𝐼)−1(𝛼𝐼−𝑆)(2.5) is the iteration matrix and 𝐜=2𝛼(𝑆+𝛼𝐼)−1(𝐻+𝛼𝐼)−1. The iteration matrix 𝑇𝛼 can be rewritten as follows: 𝑇𝛼=(𝑆+𝛼𝐼)−1(𝐻+𝛼𝐼)−1(𝛼𝐼−𝐻)(𝛼𝐼−𝑆)=(𝑆+𝛼𝐼)−1(𝐻+𝛼𝐼)−1𝐀1(𝛼𝐼+𝐻)(𝛼𝐼+𝑆)−2𝛼=𝐼−2𝛼(𝐻+𝛼𝐼)(𝑆+𝛼𝐼)−1𝐀=𝐼−𝑃𝛼−1𝐀,(2.6) where 𝑃𝛼=(1/2𝛼)(𝐻+𝛼𝐼)(𝑆+𝛼𝐼).

Obviously, 𝑃𝛼 is nonsingular and 𝐜=𝑃𝛼−1𝐛. As in [18], one can show there is a unique splitting 𝐀=𝑃𝛼−𝑄𝛼 such that the iteration 𝑇𝛼 is the matrix induced by that splitting, that is, 𝑇𝛼=𝑃𝛼−1𝑄𝛼=𝐼−𝑃𝛼−1𝐀. Matrix 𝑄𝛼 is given by 𝑄𝛼=(1/2𝛼)(𝛼𝐼−𝐻)(𝛼𝐼−𝑆).

2.2. A Relaxed Splitting Preconditioner

The relaxed splitting preconditioner is defined as follows: âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´ğŒ=101𝛼𝐴1𝐵𝑇10𝐴2𝐵𝑇2−𝐵1−𝐵21𝛼𝐼−𝛼𝐵1𝐵𝑇1⎤⎥⎥⎥⎥⎥⎦.(2.7) It is important to note that the preconditioner 𝐌 can be written in a factorized form as 1𝐌=ğ›¼âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´1000𝛼𝐼0−𝐵1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣0𝛼𝐼𝛼𝐼0𝐵𝑇10𝐴2𝐵𝑇20−𝐵2⎤⎥⎥⎥⎥⎦=âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´ğ›¼ğ¼1000𝐼0−𝐵1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣10𝐼𝐼0𝛼𝐵𝑇10𝐴2𝐵𝑇20−𝐵2⎤⎥⎥⎥⎥⎦=âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´ğ›¼ğ¼1000𝐼0−𝐵1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣10𝐼𝐼0𝛼2𝐵𝑇110𝐼𝛼𝐵𝑇2⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼100𝐼𝛼2𝐵𝑇1𝐵200𝐴20⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣100𝛼𝐼𝐼000𝐼00−𝛼𝐵2𝐼⎤⎥⎥⎥⎥⎦,(2.8) where 𝐴2=𝐴2+(1/𝛼)𝐵𝑇2𝐵2. Note that both factors on the right-hand side are invertible provided that 𝐴1 have 𝐴2 have positive definite symmetric parts. Hence, the new preconditioner is nonsingular. This condition is satisfied for both Stokes and Oseen problems. We can see from (2.1) and (2.7) that the difference between 𝐌 and 𝐀 is given by ⎡⎢⎢⎢⎢⎢⎣1𝐑=𝐌−𝐀=00𝛼𝐴1𝐵𝑇1−𝐵𝑇1100000𝛼𝐼−𝛼𝐵1𝐵𝑇1⎤⎥⎥⎥⎥⎥⎦.(2.9) This observation suggests that 𝐌 could be a good preconditioner, since the appropriate values for the parameters involved in the new preconditioners are estimated. Furthermore, the structure of (2.9) somewhat facilitates the analysis of the eigenvalue distribution of the preconditioned matrix. In the following, we analyze the spectral properties of the preconditioned matrix 𝐓=𝐀𝐌−1.

Theorem 2.1. The preconditioned matrix 𝐓=𝐀𝐌−1 has an eigenvalue 1 with multiplicity at least 𝑛, and the remaining eigenvalues are 𝜆𝑖, where 𝜆𝑖 are the eigenvalues of an 𝑚×𝑚 matrix 𝑍𝛼∶=(1/𝛼)(𝑆1+𝑆2)−(1/𝛼2)𝑆2𝑆1 with 𝑆1=𝐵1𝐴1−1𝐵𝑇1 and 𝑆2=𝐵2𝐴2−1𝐵𝑇2.

Proof. First of all, from 𝐓∶=𝐌−1(𝐀𝐌−1)𝐌=𝐌−1𝐀 we see that the right-preconditioned matrix 𝐓 is similar to the left-preconditioned one 𝐓, then 𝐓 and 𝐓 have the same eigenvalues. Furthermore, we have 𝐓=ğˆâˆ’ğŒâˆ’ğŸğ‘âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£01=𝐈−𝐼000𝐼0𝛼𝐵2𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣1𝐼−𝛼2𝐵𝑇1𝐵2𝐴2−100𝐴2−10100ğ›¼ğ¼âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ£1𝐼0−𝛼2𝐵𝑇110𝐼−𝛼𝐵𝑇2âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦Ã—âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´00𝐼1−1𝐵000𝐼01𝐴1−1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣10𝐼00𝛼𝐴1𝐵𝑇1−𝐵𝑇1100000𝛼𝐼−𝛼𝐵1𝐵𝑇1⎤⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣1𝐼0∗0𝐼∗00𝛼𝑆1+𝑆2−1𝛼2𝑆2𝑆1⎤⎥⎥⎥⎥⎦.(2.10) Therefore, from (2.10) we can see that the eigenvalues of 𝐓 are given by 1 (with multiplicity at least 𝑛=𝑛1+𝑛2) and by the 𝜆𝑖’s.

Lemma 2.2. Let 𝐴𝛼=𝐴1−(1/𝛼)𝐵𝑇1𝐵1−(1/𝛼)𝐵𝑇1𝐵20𝐴2∈ℝ𝑛×𝑛, 𝛼>ğœŽmax(𝐵𝑇1𝐵1)/ğœŽmin(𝐴1), and 𝐴1, and 𝐴2 be positive definite. Then 𝐴𝛼 is positive definite.

Lemma 2.3. Let 𝐴𝛼∈ℝ𝑛×𝑛 and 𝐵∈ℝ𝑚×𝑛 (𝑚≤𝑛). Let 𝛼∈ℝ, and assume that matrices 𝐴𝛼, 𝐴𝛼+(1/𝛼)𝐵𝑇𝐵, 𝐵𝐴𝛼−1𝐵𝑇and 𝐵(𝐴𝛼+(1/𝛼)𝐵𝑇𝐵)−1𝐵𝑇 are all invertible. Then 𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵−1𝐵𝑇−1=𝐵𝐴𝛼−1𝐵𝑇−1+1𝛼𝐼.(2.11)

Theorem 2.4. Let 𝛼>ğœŽmax(𝐵𝑇1𝐵1)/ğœŽmin(𝐴1). The remaining eigenvalues 𝜆𝑖 of 𝑍𝛼 are of the form:𝜆𝑖=𝜇𝑖𝛼+𝜇𝑖,(2.12) where the 𝜇𝑖’s satisfy the eigenvalue problem: 𝐵𝐴𝛼−1𝐵𝑇𝜙𝑖=𝜇𝑖𝜙𝑖.

Proof. We note 𝑍𝛼=1𝛼𝑆1+𝑆2−1𝛼2𝑆2𝑆1=1𝛼𝐵1𝐵2î€¸âŽ›âŽœâŽœâŽğ´1−10−1𝛼𝐴2−1𝐵𝑇2𝐵1𝐴1−1𝐴2−1âŽžâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽğµğ‘‡1𝐵𝑇2⎞⎟⎟⎠=1𝛼𝐵1𝐵2î€¸âŽ›âŽœâŽœâŽğ´101𝛼𝐵𝑇2𝐵1𝐴2⎞⎟⎟⎠−1âŽ›âŽœâŽœâŽğµğ‘‡1𝐵𝑇2⎞⎟⎟⎠=1𝛼𝐵1𝐵2î€¸âŽ›âŽœâŽœâŽâŽ›âŽœâŽœâŽğ´1−1𝛼𝐵𝑇1𝐵1−1𝛼𝐵𝑇1𝐵20𝐴2⎞⎟⎟⎠+⎛⎜⎜⎝1𝛼𝐵𝑇1𝐵11𝛼𝐵𝑇1𝐵21𝛼𝐵𝑇2𝐵11𝛼𝐵𝑇2𝐵2⎞⎟⎟⎠⎞⎟⎟⎠−1âŽ›âŽœâŽœâŽğµğ‘‡1𝐵𝑇2⎞⎟⎟⎠=1𝛼𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵−1𝐵𝑇.(2.13) Thus, the remaining eigenvalues are the solutions of the eigenproblem: 1𝛼𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵−1𝐵𝑇𝜙𝑖=𝜆𝑖𝜙𝑖.(2.14) By Lemma 2.3, we obtain 1𝛼𝜙𝑖=𝜆𝑖𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵−1𝐵𝑇−1𝜙𝑖=𝜆𝑖𝐵𝐴𝛼−1𝐵𝑇−1𝜙𝑖+𝜆𝑖𝛼𝜙𝑖.(2.15) Hence, 𝜆𝑖=𝜇𝑖/(𝛼+𝜇𝑖), where 𝜇𝑖’s satisfy the eigenvalue problem 𝐵𝐴𝛼−1𝐵𝑇𝜙𝑖=𝜇𝑖𝜙𝑖.

In addition, we obtain easily that the remaining eigenvalues 𝜆𝑖→0 as ğ›¼â†’âˆž. Figures 1 and 2 show this behavior, that is, the nonunity eigenvalues of the preconditioned matrix are increasingly clustered at the origin as the parameters become larger.

2.3. Practical Implementation of the Relaxed Splitting Preconditioner

In this subsection, we outline the practical implementation of the relaxed splitting preconditioner in a subspace iterative method. The main step is applying the preconditioner, that is, solving linear systems with the coefficient matrix 𝐌. From (2.8), we can see that the relaxed splitting preconditioner can be factorized as follows: âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ´ğŒ=1000𝐼0−𝐵1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣10𝐼𝐼0𝛼2𝐵𝑇110𝐼𝛼𝐵𝑇2⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼100𝐼𝛼2𝐵𝑇1𝐵200𝐴20⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣100𝛼𝐼𝐼000𝐼00−𝛼𝐵2𝐼⎤⎥⎥⎥⎥⎦,(2.16) showing that the preconditioner requires solving two linear systems at each step, with coefficient matrices 𝐴1 and 𝐴2=𝐴2+(1/𝛼)𝐵𝑇2𝐵2. Several different approaches are available for solving linear systems involving 𝐴1 and 𝐴2. We defer the discussion of these to Section 3.

We conclude this section with a discussion of diagonal scaling. We found that scaling can be beneficial for the relaxed splitting preconditioner. Unless otherwise specified, we perform a preliminary symmetric scaling of the linear systems 𝐀𝐱=𝐛 in the form 𝐃−1/2𝐀𝐃−1/2𝐲=𝐃−1/2𝐛 with 𝐲=𝐃1/2𝐱, and 𝐃=diag(𝐷1,𝐷2,𝐼), where diag(𝐷1,𝐷2) is the main diagonal of the velocity submatrix 𝐀. Incidentally, it is noted that diagonal scaling is very beneficial for the HSS preconditioner (see [13]) and the DS preconditioner (see [16, 17]).

3. Numerical Experiments

In this section, numerical experiments are carried out for solving the linear system coming from the finite-element discretization of the two-dimensional linearized Stokes and Oseen models of incompressible flow in order to verify the performance of our preconditioner. The test problem is the leaky lid-driven cavity problem generated by the IFISS software package [19]. We used a zero initial guess and stopped the iteration when ||𝐫𝑘||2/||𝐛||2≤10−6, where 𝐫𝑘 is the residual vector. The relaxed splitting preconditioner is combined with restarted GMRES(m). We set 𝑚=30.

We consider the 2D leaky lid-driven cavity problem discretized by the finite-element method on uniform grids [1]. The subproblems arising from the application of the relaxed splitting preconditioner are solved by direct methods. We use AMD reordering technique [20, 21] for the degrees of freedom that makes the application of the Cholesky (for Stokes) or LU (for Oseen) factorization of 𝐴1 and 𝐴2 relatively fast. For simplicity, we use 𝛼=100 for all numerical experiments.

In Table 1, we show iteration counts (referred to as “its”) for the relaxed splitting preconditioned GMRES(30) when solving the steady Stokes problem on a sequence of uniform grids. We see that the iteration count is independent of mesh size involved in the Q2-Q1 and the Q2-P1 finite-element scheme. The Q2-P1 finite-element scheme has much better profile than the Q2-Q1 finite-element scheme.


GridQ2-Q1Q2-P1

16 × 162515
32 × 322612
64 × 642310
128 × 1281911
256 × 2561610

In Tables 2 and 3, we show iteration counts for the steady Oseen problem on a sequence of uniform grids and for different values of 𝜐, using Picard and Newton linearization of generalized Oseen problems, respectively. We found that the relaxed splitting preconditioner has difficulties dealing with low-viscosity, that is, the number of iterations increases with the decrease in the kinematic viscosity. In this case, it appears that the Q2-P1 finite-element scheme gives faster convergence results than the Q2-Q1 finite-element scheme.


Grid 𝜐 = 1 𝜐 = 0 . 1 𝜐 = 0 . 0 2
Q2-Q1Q2-P1Q2-Q1Q2-P1Q2-Q1Q2-P1

16 × 16271429164827
32 × 32271229145426
64 × 64231026124924
128 × 128191121103720
256 × 25616101081714


Grid 𝜐 = 1 𝜐 = 0 . 1 𝜐 = 0 . 0 2
Q2-Q1Q2-P1Q2-Q1Q2-P1Q2-Q1Q2-P1

16 × 16271529164626
32 × 32271229145324
64 × 64231026124623
128 × 128191121103418
256 × 25616101081513

Next, we report on analogous experiments involving the generalized Stokes problem and the generalized Oseen problem. As we can see from Table 4, for the generalized Stokes problem, the results are virtually the same as those obtained in the steady case. Indeed, we can see from the results in Table 1 that the rate of convergence for the relaxed splitting preconditioned GMRES (30) is essentially independent of mesh size involved in the Q2-Q1 and the Q2-P1 finite-element schemes.


GridQ2-Q1Q2-P1

16 × 162513
32 × 322612
64 × 642310
128 × 1281911
256 × 2561610

In Tables 5 and 6, for generalized Oseen problems, we compare our preconditioner with the RDF preconditioner in [17]. The RDF preconditioner can be factorized as follows: âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğµğ‘ƒ=𝐼0𝑇1ğ›¼âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ£îğ´0𝐼000𝐼1000𝐼0−𝐵1⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣0𝐴0𝐼𝐼002𝐵𝑇2⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣000𝛼𝐼𝐼000𝐼0−𝐵2ğ›¼ğ¼âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦,(3.1) where 𝐴1=𝐴1+(1/𝛼)𝐵𝑇1𝐵1 and 𝐴2=𝐴2+(1/𝛼)𝐵𝑇2𝐵2. It shows that RDF preconditioner requires solving two linear systems at each step. The new preconditioner requires solving linear systems with 𝐴1 and 𝐴2 at each step. We can see that the linear system with 𝐴1 is easier to solve than that with 𝐴1. From Tables 5 and 6, we can see for 128 × 128 grid with different viscosities that the RDF preconditioner leads to slightly less iteration counts than the new preconditioner, but the new preconditioner is slightly faster in terms of elapsed CPU time.


ViscosityRDFNew preconditioner
itsCPUitsCPU

0.11222.753741315.90265
0.01720.93871914.30008
0.001520.31654613.36891


ViscosityRDFNew preconditioner
itsCPUitsCPU

0.11222.866541316.12583
0.01721.02928814.34972
0.0011222.816741617.49554

From Figures 3 and 4, we found that for the relaxed splitting preconditioner the intervals containing values of parameter 𝛼 are very wide. Those imply that the relaxed splitting preconditioner is not sensitive to the value of parameter. Noting that the optimal parameters of the relaxed splitting preconditioner are always larger than 50, we can always take 𝛼=100 to obtain essentially optimal results.

4. Conclusions

In this paper, we have described a relaxed splitting preconditioner for the linear systems arising from discretizations of the Navier-Stokes equations and analyzed the spectral properties of the preconditioned matrix. The numerical experiments show good performance on a wide range of cases. We use direct methods for the solution of inner linear systems, but it is not a good idea to solve larger 2D or 3D problems at the constraint of memory and time requirement. In this case, exact solve can be replaced with inexact solve, which requires further research in the future.

Acknowledgments

This research is supported by NSFC (60973015 and 61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), and Sichuan Province Sci. & Tech. Research Project (12ZC1802).

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Copyright © 2012 Ning-Bo Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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