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Journal of Applied Mathematics
Volume 2012, Article ID 402490, 12 pages
http://dx.doi.org/10.1155/2012/402490
Research Article

A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations

School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, Chengdu 611731, China

Received 8 December 2011; Revised 2 April 2012; Accepted 19 April 2012

Academic Editor: Massimiliano Ferronato

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.

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],[],[],𝜕𝑡𝜐Δ𝐮+(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 Ω2 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 𝐀=𝐀𝐁𝑇𝐮𝑝𝐟𝐁0,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=𝜎𝑉+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=𝐁𝐀1𝐁𝑇. This class of preconditioners includes the pressure convection diffusion (PCD) preconditioner, the least-squares commutator (LSC) preconditioner, and their variants [57]. Somewhat related to this class of preconditioners are those based on the augmented Lagrangian (AL) reformulation of the saddle point problem; see [811]. Other types of preconditioners for the saddle point problems include those based on the Hermitian and skew-Hermitian splitting (HSS) [1215] 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: 𝐴𝐀=10𝐵𝑇10𝐴2𝐵𝑇2𝐵1𝐵20=𝐴100000𝐵1+0000𝐵𝑇10𝐴2𝐵𝑇20𝐵20=𝐻+𝑆,(2.1) where 𝐴1𝑛1×𝑛1, 𝐴2𝑛2×𝑛2, 𝐵1𝑚×𝑛1, and 𝐵2𝑚×𝑛2. Thus, 𝐀(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)(𝛼𝐼𝑆),𝐀=(𝑆+𝛼𝐼)(𝛼𝐼𝐻).(2.2) Associated to these splittings is the alternating iteration, 𝑘=0,1,, (𝐻+𝛼𝐼)𝐱𝑘+1/2=(𝛼𝐼𝑆)𝐱𝑘+𝐛,(𝑆+𝛼𝐼)𝐱𝑘+1=(𝛼𝐼𝐻)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(𝛼𝐼𝐻)(𝐻+𝛼𝐼)1(𝛼𝐼𝑆)(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(𝛼𝐼𝐻)(𝛼𝐼𝑆)=(𝑆+𝛼𝐼)1(𝐻+𝛼𝐼)1𝐀1(𝛼𝐼+𝐻)(𝛼𝐼+𝑆)2𝛼=𝐼2𝛼(𝐻+𝛼𝐼)(𝑆+𝛼𝐼)1𝐀=𝐼𝑃𝛼1𝐀,(2.6) where 𝑃𝛼=(1/2𝛼)(𝐻+𝛼𝐼)(𝑆+𝛼𝐼).

Obviously, 𝑃𝛼 is nonsingular and 𝐜=𝑃𝛼1𝐛. As in [18], one can show there is a unique splitting 𝐀=𝑃𝛼𝑄𝛼 such that the iteration 𝑇𝛼 is the matrix induced by that splitting, that is, 𝑇𝛼=𝑃𝛼1𝑄𝛼=𝐼𝑃𝛼1𝐀. Matrix 𝑄𝛼 is given by 𝑄𝛼=(1/2𝛼)(𝛼𝐼𝐻)(𝛼𝐼𝑆).

2.2. A Relaxed Splitting Preconditioner

The relaxed splitting preconditioner is defined as follows: 𝐴𝐌=101𝛼𝐴1𝐵𝑇10𝐴2𝐵𝑇2𝐵1𝐵21𝛼𝐼𝛼𝐵1𝐵𝑇1.(2.7) It is important to note that the preconditioner 𝐌 can be written in a factorized form as 1𝐌=𝛼𝐴1000𝛼𝐼0𝐵10𝛼𝐼𝛼𝐼0𝐵𝑇10𝐴2𝐵𝑇20𝐵2=𝐴𝛼𝐼1000𝐼0𝐵110𝐼𝐼0𝛼𝐵𝑇10𝐴2𝐵𝑇20𝐵2=𝐴𝛼𝐼1000𝐼0𝐵110𝐼𝐼0𝛼2𝐵𝑇110𝐼𝛼𝐵𝑇2𝐼100𝐼𝛼2𝐵𝑇1𝐵200𝐴20100𝛼𝐼𝐼000𝐼00𝛼𝐵2𝐼,(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=𝐌𝐀=00𝛼𝐴1𝐵𝑇1𝐵𝑇1100000𝛼𝐼𝛼𝐵1𝐵𝑇1.(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=𝐀𝐌1.

Theorem 2.1. The preconditioned matrix 𝐓=𝐀𝐌1 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=𝐵1𝐴11𝐵𝑇1 and 𝑆2=𝐵2𝐴21𝐵𝑇2.

Proof. First of all, from 𝐓=𝐌1(𝐀𝐌1)𝐌=𝐌1𝐀 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=𝐈𝐌𝟏𝐑01=𝐈𝐼000𝐼0𝛼𝐵2𝐼1𝐼𝛼2𝐵𝑇1𝐵2𝐴2100𝐴210100𝛼𝐼1𝐼0𝛼2𝐵𝑇110𝐼𝛼𝐵𝑇2×𝐴00𝐼11𝐵000𝐼01𝐴1110𝐼00𝛼𝐴1𝐵𝑇1𝐵𝑇1100000𝛼𝐼𝛼𝐵1𝐵𝑇1=1𝐼00𝐼00𝛼𝑆1+𝑆21𝛼2𝑆2𝑆1.(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)𝐵𝑇1𝐵20𝐴2𝑛×𝑛, 𝛼>𝜎max(𝐵𝑇1𝐵1)/𝜎min(𝐴1), and 𝐴1, and 𝐴2 be positive definite. Then 𝐴𝛼 is positive definite.

Lemma 2.3. Let 𝐴𝛼𝑛×𝑛 and 𝐵𝑚×𝑛 (𝑚𝑛). Let 𝛼, and assume that matrices 𝐴𝛼, 𝐴𝛼+(1/𝛼)𝐵𝑇𝐵, 𝐵𝐴𝛼1𝐵𝑇and 𝐵(𝐴𝛼+(1/𝛼)𝐵𝑇𝐵)1𝐵𝑇 are all invertible. Then 𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵1𝐵𝑇1=𝐵𝐴𝛼1𝐵𝑇1+1𝛼𝐼.(2.11)

Theorem 2.4. Let 𝛼>𝜎max(𝐵𝑇1𝐵1)/𝜎min(𝐴1). The remaining eigenvalues 𝜆𝑖 of 𝑍𝛼 are of the form:𝜆𝑖=𝜇𝑖𝛼+𝜇𝑖,(2.12) where the 𝜇𝑖’s satisfy the eigenvalue problem: 𝐵𝐴𝛼1𝐵𝑇𝜙𝑖=𝜇𝑖𝜙𝑖.

Proof. We note 𝑍𝛼=1𝛼𝑆1+𝑆21𝛼2𝑆2𝑆1=1𝛼𝐵1𝐵2𝐴1101𝛼𝐴21𝐵𝑇2𝐵1𝐴11𝐴21𝐵𝑇1𝐵𝑇2=1𝛼𝐵1𝐵2𝐴101𝛼𝐵𝑇2𝐵1𝐴21𝐵𝑇1𝐵𝑇2=1𝛼𝐵1𝐵2𝐴11𝛼𝐵𝑇1𝐵11𝛼𝐵𝑇1𝐵20𝐴2+1𝛼𝐵𝑇1𝐵11𝛼𝐵𝑇1𝐵21𝛼𝐵𝑇2𝐵11𝛼𝐵𝑇2𝐵21𝐵𝑇1𝐵𝑇2=1𝛼𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵1𝐵𝑇.(2.13) Thus, the remaining eigenvalues are the solutions of the eigenproblem: 1𝛼𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵1𝐵𝑇𝜙𝑖=𝜆𝑖𝜙𝑖.(2.14) By Lemma 2.3, we obtain 1𝛼𝜙𝑖=𝜆𝑖𝐵𝐴𝛼+1𝛼𝐵𝑇𝐵1𝐵𝑇1𝜙𝑖=𝜆𝑖𝐵𝐴𝛼1𝐵𝑇1𝜙𝑖+𝜆𝑖𝛼𝜙𝑖.(2.15) Hence, 𝜆𝑖=𝜇𝑖/(𝛼+𝜇𝑖), where 𝜇𝑖s satisfy the eigenvalue problem 𝐵𝐴𝛼1𝐵𝑇𝜙𝑖=𝜇𝑖𝜙𝑖.

In addition, we obtain easily that the remaining eigenvalues 𝜆𝑖0 as 𝛼. 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.

fig1
Figure 1: Spectrum of preconditioned steady Oseen matrix, 32 × 32 grid with 𝜐=0.1.
fig2
Figure 2: Spectrum of preconditioned generalized steady Oseen matrix, 32 × 32 grid with 𝜐=0.001.
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: 𝐴𝐌=1000𝐼0𝐵110𝐼𝐼0𝛼2𝐵𝑇110𝐼𝛼𝐵𝑇2𝐼100𝐼𝛼2𝐵𝑇1𝐵200𝐴20100𝛼𝐼𝐼000𝐼00𝛼𝐵2𝐼,(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 𝐃1/2𝐀𝐃1/2𝐲=𝐃1/2𝐛 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||2106, 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.

tab1
Table 1: Iterations of preconditioned GMRES(30) for steady Stokes problem.

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.

tab2
Table 2: Iterations of preconditioned GMRES(30) for steady Oseen problems (Picard).
tab3
Table 3: Iterations of preconditioned GMRES(30) for steady Oseen problems (Newton).

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.

tab4
Table 4: Iterations of preconditioned GMRES(30) for generalized Stokes problems.

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: 𝐵𝑃=𝐼0𝑇1𝛼𝐴0𝐼000𝐼1000𝐼0𝐵10𝐴0𝐼𝐼002𝐵𝑇2000𝛼𝐼𝐼000𝐼0𝐵2𝛼𝐼,(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.

tab5
Table 5: Iterations of preconditioned GMRES(30) for generalized Oseen problem (Picard, Q2-Q1, 128 × 128 uniform grids).
tab6
Table 6: Iterations of preconditioned GMRES(30) for generalized Oseen problem (Newton, Q2-Q1, 128 × 128 uniform grids).

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.

fig3
Figure 3: Iteration number versus parameter, steady Oseen problem, with 𝜐=0.1. (a) 16 × 16 grid, (b) 32 × 32 grid.
fig4
Figure 4: Iteration number versus parameter, generalized Oseen problem, with 𝜐=0.001. (a) 16 × 16 grid, (b) 32 × 32 grid.

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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