A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations

Applied Mathematics Letters 55 (2016) 18–26

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Applied Mathematics Letters www.elsevier.com/locate/aml

A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations✩ Hong-Tao Fan a,∗ , Xin-Yun Zhu b a b

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China Department of Mathematics, University of Texas of the Permian Basin, Odessa, TX 79762, USA

article

info

Article history: Received 27 October 2015 Received in revised form 22 November 2015 Accepted 22 November 2015 Available online 7 December 2015 Keywords: Generalized saddle point problems Modified relaxed splitting preconditioner Matrix splitting Eigenvalue distribution Navier–Stokes equations

abstract Based on a new matrix splitting of the original coefficient matrix, a modified relaxed splitting (MRS) preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations is considered. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied. The proposed preconditioner is closer to the original matrix than the generalized relaxed splitting (GRS) preconditioner in the sense of certain norm, which straightforwardly results in a MRS iteration method. Finally, numerical results are given to demonstrate the theoretical analysis. The results show that this novel preconditioner is competitive with and more effective than some of the best existing preconditioners. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we are interested in solving the linear systems arising from 2D linearized Navier–Stokes equations with the following structure [1–3]:      u1 A1 0 B1T f1      Au≡ 0 (1.1) A2 B2T  u2  =  f2  ≡ b, −B1 −B2 C p −g where A1 ∈ Rn1 ×n1 , A2 ∈ Rn2 ×n2 are nonsymmetric positive definite matrices, B1 ∈ Rm×n1 , B2 ∈ Rm×n2 have full row ranks, and C ∈ Rm×m is a symmetric positive semi-definite matrix. These assumptions guarantee the existence and uniqueness of the solution of the system (1.1). Here it is worth noting that the linear system (1.1) is actually a special case of generalized saddle point problems. ✩

This work is supported by the National Natural Science Foundation of China (Nos. 11171371, 11571004). ∗ Corresponding author. E-mail addresses: [email protected] (H.-T. Fan), zhu [email protected] (X.-Y. Zhu).

http://dx.doi.org/10.1016/j.aml.2015.11.011 0893-9659/© 2015 Elsevier Ltd. All rights reserved.

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Generalized saddle point problems are often generated in various scientific and engineering applications such as constrained optimization, mixed finite element analysis of elliptic PDEs, and constrained least squares problems; for detailed elaborations see [4–6]. In the past few decades, tremendous effort has been invested in the development of fast solution methods for generalized saddle point problems. Most of the work has been aimed at developing efficient preconditioners for Krylov subspace methods; see [4] for a comprehensive survey. In this paper, we focus on preconditioned Krylov subspace methods, especially preconditioned GMRES [7]. Some famous preconditioners have been proposed for generalized saddle point problems, such as HSS-based preconditioners [8–12], block-based preconditioners [13–17], constraint preconditioners [18], dimensional split preconditioners [1,2], augmented Lagrangian-based preconditioners [19], and so on. Though a variety of splitting-based preconditioners are available in the literature for solving such problems as the generalized Stokes problem and the rotation form of the Navier–Stokes equations, most of them are not well-suited for the standard (convection) form, see [1,20]. Recently, to solve the linear system (1.1) with C = 0, Benzi et al. [1] first split the coefficient matrix A of (1.1) into the following two parts:       A1 0 B1T A1 0 B1T 0 0 0       A = 0 (1.2) A2 B2T  =  0 0 0  + 0 A2 B2T  ≡ A¯1 + A¯2 . −B1 −B2 0 −B1 0 0 0 −B2 0 Based on the splitting (1.2), by utilizing the alternating direction implicit-like (ADI-like) stationary iteration method, a dimensional splitting (DS) preconditioner [1] and a relaxed dimensional factorization (RDF) preconditioner [3] can be obtained. Due to the high efficiency of the DS preconditioner, Cao et al. [2], by adding a nonzero block C to the (3, 3)-position of the matrices A and A¯2 , respectively, proposed a modified dimensional splitting (MDS) preconditioner to solve the linear system (1.1). More recently, to further improve the efficiency of both the DS preconditioner and the RDF preconditioner used for solving generalized saddle point problems with C = 0, Tan et al. [21] proposed another splitting of the coefficient matrix A :       A1 0 B1T A1 0 0 0 0 B1T       A = 0 (1.3) 0 0 + 0 A2 B2T  ≡ A1 + A2 . A2 B2T  =  0 −B1 −B2 0 −B1 0 0 0 −B2 0 Combining the splitting (1.3) with the classic alternating direction implicit iteration approach (ADI), the authors in [21] presented a relaxed splitting (RS) preconditioner for solving the problem (1.1) with C = 0. The greatest strength of the RS preconditioner is that it is easier to be implemented and more economical than the DS preconditioner and the RDF preconditioner in practical applications; see [21]. As a result of this, Cao et al. [22] extended this method to the general case with C ̸= 0 and presented a generalized relaxed splitting (GRS) preconditioner for solving generalized saddle point problems. In [22] the GRS preconditioner is defined by: 

PGRS =

A1 1  0 α −B1

0 αI 0

 0 αI  0  0 αI 0

0 αA2 −B2

B1T B2T αI + C





A1

0

  =  0

A2

−B1

−B2

 1 A1 B1T α  . B2T  1 αI + C − B1 B1T α

(1.4)

The GRS preconditioner PGRS has several potential advantages. One is that the structure of the GRS preconditioner can provide a convenient way to analyze the spectral distribution of the preconditioned matrix −1 PGRS A . Another is when using Krylov subspace methods to solve we only need to solve two  such problems,  subsystems of linear equations with coefficient matrices A1 and

A2 −B2

while we need to solve two subsystems with coefficient matrices



T

B2 αI + C

αI + A1 −B1

for the GRS preconditioner,    T αI + A2 B2 and for αI −B2 αI + C

T B1

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the MDS preconditioner. This implies that less computational cost is required compared to the MDS preconditioner. As we know, the initial basic theory behind generating an effective preconditioner is that it should be (ideally) as close as possible to the original coefficient matrix A . However, from (1.4) and (1.1), we can see that the difference between the preconditioner PGRS and the coefficient matrix A is given by:        1  1 T T T 0 0 A − I B A1 0 A1 B 1 1 1  0 B1  α α    A1   T T  = − RGRS =  0 A B 0 A B   . (1.5)   2 2 0 0 0 2 2     1 1 T −B1 −B2 C T −B1 −B2 αI + C − B1 B1 B B 0 0 αI − 1 1 α α We note that both the (1, 3)-block and the (3, 3)-block in RGRS tend to +∞ as α → 0+ . As the parameter α → +∞, the (1, 3)-block in RGRS tends to −B1T while the (3, 3)-block goes to +∞. Hence, the parameter α still needs to be found in order to judge and weigh the proportion of the two nonzero sub-blocks. Motivated by this, an additional new parameter shall be introduced to overcome the above difficulty and a modified relaxed splitting (MRS) preconditioner will be proposed in this paper. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are analyzed. Numerical results are provided to validate the theoretical analysis. The results prove that the new preconditioner is more competitive and effective compared to the GRS preconditioner, the MDS preconditioner, and the HSS preconditioner in light of the iteration numbers and elapsed CPU times. 2. The modified relaxed splitting preconditioner Let α, β > 0. On the basis of the generalized relaxed splitting method [22], a new modified relaxed splitting for the generalized saddle point matrix A will be established:       1 1 T 0 0 A1 − I B1T A1 0 A1 B 1  α α     T − A = PM RS − RM RS =  (2.1) 0 A B  . 2 0 2   0 0  1 1 T −B1 −B2 βI + C − B1 B1 0 0 βI − B1 B1T α α To facilitate discussion, we find that the matrix PM RS can be decomposed as follows:    1 T A1 0 0 I B 0 α 1     PM RS =  0 (2.2) I 0  B2T  .  0 A2 −B1 0 I 0 −B2 βI + C The splitting (2.1) naturally leads to the following modified relaxed splitting iteration method for the generalized saddle point problem (1.1). The modified relaxed splitting iteration method: Let α and β be two given positive constants, T T T and let (u01 , u02 , p0 )T be an initial guess vector. For k = 0, 1, 2, . . . , until a certain stopping criterion is satisfied, compute:       1  k   1 k+1  T 0 0 A1 − I B1T u1 u1 A1 0 A1 B 1 f 1   α α    k+1     k    0  u A2 B2T  u2  +  f2  . 0 0 0   2 =   1 1 −g1 T pk+1 pk −B1 −B2 βI + C − B1 B1T 0 0 βI − B B 1 1 α α The above iteration scheme can also be written in a fixed point form uk+1 = Γ uk + c,

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where 

Γ =

−1 PM RS RM RS

A1   = 0 −B1

0 A2 −B2

   −1  1 1 T 0 0 A1 − I B1T A1 B 1  α α      B2T   0 0 0    1 1 T T βI + C − B1 B1 0 0 βI − B B 1 1 α α

(2.3)

−1 is the iteration matrix and c = PM RS b. Hence, the generalized saddle point linear system A u = b is −1 equivalent to the linear system PM RS A u = c. If we use a Krylov subspace method such as the GMRES method or its restarted variant to approximate the solution of this linear system, then the matrix PM RS can serve as a preconditioner. We refer to PM RS as the modified relaxed splitting preconditioner (MRS) for generalized saddle point problems.

Remark 2.1. Unlike DS and MDS, the preconditioners, including RS, RDF, GRS, and MRS, no longer relate to the classic alternating direction iteration (or ADI-like) approach. This fact is of no consequence when PM RS is used as a preconditioner for a Krylov subspace method like GEMRES [7]. In the following part, we will elaborate on specific implementation issues. At each step of the MRS iteration or applying the MRS preconditioner PM RS with a Krylov subspace method, we need to consider the following two aspects. One is the choice of the parameters α and β. In actual applications, the selection of the involved parameters is an extremely difficult task and the optimal parameters will be most often chosen experimentally. The other is how to solve a linear system of the form PM RS z = r for a given r at each iteration. With the special decomposition of (2.2), we can derive the following algorithmic version of the modified relaxed splitting iteration method. T T T T T T ˆ Algorithm 2.1.   For a given vector r = (r1 , r2 ) , where r1 = u1 and r2 = (u2 , p ) , if we let A2 = T A2 B2 1 T T T T T T ˆ ˆ , B1 = (0 , −B1 ) and B2 = (0, α B1 ), then the vector z = (z1 , z2 ) can be computed by −B2 βI + C the following steps:

(1) solve A1 t1 = r1 ; ˆ1 ; (2) solve Aˆ2 z2 = r2 − B ˆ2 z2 . (3) z1 = t1 − B Here, it is worth mentioning that in matrix RM RS the (1, 3)-block approaches −B1T and the (3, 3)-block becomes zero, respectively, as α → +∞ and β → 0+ , while both the (1, 3)-block and the (3, 3)-block tend to +∞ as α → 0+ and β → 0+ . This observation suggests that PM RS may be a better preconditioner than PGRS , since it gives a better approximation of the coefficient matrix A in some degree. Analogous to (1.5), the structure of (2.1) somewhat facilitates the analysis of the eigenvalue distribution of the preconditioned matrix. Next, we will proceed to discuss the eigenvalue distribution and an upper bound of the degree of the −1 minimal polynomial of the preconditioned matrix PM RS A . Theorem 2.1. Let the MRS preconditioner PM RS be defined as in (2.1). Then the preconditioned matrix −1 PM RS A has an eigenvalue of 1 with multiplicity at least n1 + n2 , and the remaining eigenvalues λ satisfy the following generalized eigenvalue problem: −1 T T T −1 T (C + B1 A−1 1 B1 + B2 A2 B2 )z = λ(βI + C + B2 A2 B2 )z.

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Proof. From (2.1), we get 

A1

0

  PM RS =  0  −B1

A2 −B2

 1 A1 B1T α   B2T .  1 βI + C − B1 B1T α

(2.4)

−1 Let PM RS = (Cij ). Then using the formula to calculate the inverse of block matrix, we obtain that

C11 = A−1 1 −

1 T T −1 B (βI + C + B2 A−1 B1 A−1 2 B2 ) 1 , α 1

1 T −1 B2 A−1 C12 = − B1T (βI + C + B2 A−1 2 B2 ) 2 , α

1 −1 T −1 T −1 T , C21 = −A−1 B1 A−1 C13 = − B1T (βI + C + B2 A−1 2 B2 ) 2 B2 (βI + C + B2 A2 B2 ) 1 , α −1 T −1 T −1 −1 T −1 T C22 = A−1 B2 A−1 C23 = −A−1 , 2 − A2 B2 (βI + C + B2 A2 B2 ) 2 , 2 B2 (βI + C + B2 A2 B2 ) T −1 C31 = (βI + C + B2 A−1 B1 A−1 2 B2 ) 1 ,

T −1 C32 = (βI + C + B2 A−1 B2 A−1 2 B2 ) 2 ,

T −1 C33 = (βI + C + B2 A−1 . 2 B2 ) −1 Multiplying A by PM RS , it follows that

 I  −1 PM A = 0  RS 0

0 I 0

 Θ1  Θ2  , Θ3

(2.5)

where  Θ1 =

A−1 1 −

 1 1 T −1 T I B1T + B1T (βI + C + B2 A−1 (βI − B1 A−1 2 B2 ) 1 B1 ), α α

−1 T −1 T T Θ2 = A−1 (βI − B1 A−1 2 B2 (βI + C + B2 A2 B2 ) 1 B1 ), T −1 T Θ3 = I − (βI + C + B2 A−1 (βI − B1 A−1 2 B2 ) 1 B1 ). −1 From (2.5), we can easily see that the preconditioned matrix PM RS A has an eigenvalue of 1 with multiplicity at least n1 + n2 , and the remaining eigenvalues are the eigenvalues of the matrix Θ3 . Thus, the remaining −1 m eigenvalues of the preconditioned matrix PM RS A satisfy the following generalized eigenvalue problem −1 T T T −1 T (C + B1 A−1 1 B1 + B2 A2 B2 )z = λ(βI + C + B2 A2 B2 )z, −1 where λ is an eigenvalue of the preconditioned matrix PM RS A , and z is the corresponding eigenvector.



In a manner similar to Bai et al. [23] Proposition 2.2, using Theorem 2.1, we can conclude the following: Theorem 2.2. Let the MRS preconditioner be defined as in (2.1). Then the degree of the minimal polynomial −1 of the preconditioned matrix PM RS A is at most m + 1. Therefore, the dimension of the Krylov subspace −1 K(PM A , b) is at most m + 1. RS Remark 2.2. Perhaps the MRS iteration method, as seen from Theorem 2.2, is not convergent; however, it is of no great consequence to use PM RS as a preconditioner, since a cluster of the eigenvalues of the preconditioned matrix around a constant results in a favorable convergence rate of the preconditioned Krylov subspace method. 3. Numerical experiments In this section, we use a numerical example to examine the effectiveness of the proposed preconditioner. We consider the generalized saddle point problem from the discretization of the 2D linearized steady-state

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Table 1 GMRES(20) numerical results for the Oseen problem with ν = 0.1. Grids 16 × 16

32 × 32

64 × 64

(α, β) IT CPU (α, β) IT CPU (α, β) IT CPU

MRS

GRS

MDS

HSS

(10, 1) 13 0.1790 (1, 0.008) 11 0.4975 (1, 1×10−5 ) 12 2.5356

(1, –) 21 0.3110 (1.5, –) 24 1.0412 (0.5, –) 20 3.7707

(0.03, –) 23 0.3177 (0.02, –) 22 1.0626 (0.0004, –) 22 3.8048

(0.06, –) 48 0.8556 (0.033, –) 117 11.7048 (0.005, –) 262 137.2467

Table 2 GMRES(20) numerical results for the Oseen problem with ν = 0.05. Grids 16 × 16

32 × 32

64 × 64

(α, β) IT CPU (α, β) IT CPU (α, β) IT CPU

MRS

GRS

MDS

HSS

(8, 1) 13 0.1698 (1000, 10) 15 1.1165 (1000, 35) 13 3.7755

(10, –) 21 0.2627 (20, –) 21 1.4702 (4, –) 23 6.9657

(0.02, –) 23 0.2701 (0.002, –) 22 1.4930 (0.0005, –) 24 7.0663

(0.033, –) 47 0.8199 (0.002, –) 105 8.4210 (0.005, –) 260 138.1838

Navier–Stokes equation, i.e., the steady Oseen equation of the form  −ν∆u + ω · ∇u + ∇p = f, in Ω , ∇ · u = 0,

(3.1)

where Ω is a bounded domain, ν > 0 is the viscosity, and ω is the viscosity field. The vector field u stands for the velocity, and p represents the pressure. We use the IFISS software package developed by Elman et al. [24] to generate discretizations of the “regularized” two-dimensional lid-driven cavity problem for the Oseen equation (3.1). The mixed finite element used here is the bilinear pressure Q1 –P0 pair with local stabilization, where 0.25 is used as the stabilization parameter. We use GMRES(20) in conjunction with the preconditioners GRS [22] and MDS [2]. We also compare the results of the GRS and MDS preconditioners with those of the Hermitian and skew-Hermitian (HSS) preconditioner [9]. All runs are started from the initial zero vector and terminated if the current iterations satisfy ∥b − A uk ∥2 /∥b∥2 < 10−6 or if the prescribed iteration number κmax = 1000 is exceeded. In actual computations, the subsystems of linear equations arising in the applications of the preconditioners are solved by the LU factorization. The symbols IT and CPU stand for the iteration counts and total CPU time respectively. All experiments were run on a PC using MATLAB 2012a under the Windows 7 operating system. To implement all the preconditioners mentioned above efficiently, we need to choose the parameters α and β appropriately, since the analytic determination of the parameters which results in the fastest convergence of the preconditioned GMRES iteration appears to be quite a difficult problem, especially in the case of the Oseen problem. In our experiment, the parameters are the experimentally optimal ones such that the total iteration counts of the corresponding iteration processes are minimized. In Table 1, we report numerical results of exact solvers for the Oseen problem with ν = 0.1 on uniform grids of increasing size. From this table, we see that the iteration numbers and the elapsed CPU times of the MRS preconditioned GMRES method are much smaller than those of the GRS, MDS, and HSS preconditioned GMRES methods. The reasons may be that either the modified relaxed splitting iteration has considerably less computing workloads than the others or that the MRS preconditioner could bring a

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Fig. 1. The eigenvalue distribution of the original and four preconditioned matrices (32 × 32 uniform grids, ν = 0.1).

Fig. 2. The eigenvalue distribution of the original and four preconditioned matrices (32 × 32 uniform grids, ν = 0.05).

good approximation of the original matrix. To better show the great advantage of the MRS preconditioner, we also examined the case with ν = 0.05; a similar result can be seen in Table 2. It is worth noting that with α → ∞ and β → 0, like (α, β) = (1, 5 × 10−5 ) for 64 × 64 grids, the superiority of MRS preconditioner becomes more and more apparent, which in turn confirms the theoretical analysis presented in Section 2.

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In order to further illustrate its advantage over the other preconditioners (as previously described), the eigenvalue distributions of the original and four preconditioned matrices are drawn in Figs. 1 and 2. It can −1 be observed that eigenvalues of preconditioned matrix PM RS A are more tightly clustered around one than −1 −1 −1 those of PGRS A , PM DS A and PHSS A . Therefore, the modified relaxed splitting iteration method is very effective as a preconditioner for solving generalized saddle point problems. Acknowledgments The authors are very much indebted to the anonymous referees and editor Prof. Alan Tucker for their constructive suggestions and insightful comments, which greatly improved the original manuscript of this paper. The authors thank Prof. Bing Zheng for his constant support and encouragement. The authors are very grateful for the many efforts and contributions from Prof. Jun-Ying Wang. The authors are also grateful to Rachel Harris and William Vanderzyden for the careful proof reading of a preliminary version of the manuscript. References [1] M. Benzi, X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations, Appl. Numer. Math. 61 (2011) 66–76. [2] Y. Cao, L.-Q. Yao, M.-Q. Jiang, A modified dimensional split preconditioner for generalized saddle point problems, J. Comput. Appl. Math. 250 (2013) 70–82. [3] M. Benzi, M. Ng, Q. Niu, Z. Wang, A relaxed dimensional factorization preconditioner for the incompressible Navier–Stokes equations, J. Comput. Phys. 230 (2011) 6185–6202. [4] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005) 1–137. [5] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15, Springer-Verlag, New York, 1991. [6] H.C. Elman, D.J. Silvester, A.J. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, New York, 2006. [7] Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., SIAM, Philadelphia, PA, 2003. [8] M. Benzi, G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26 (2004) 20–41. [9] Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626. [10] Z.-Z. Bai, G.H. Golub, J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004) 1–32. [11] Z.-Z. Bai, G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007) 1–23. [12] L.A. Krukier, B.L. Krukier, Z.-R. Ren, Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems, Numer. Linear Algebra Appl. 21 (2014) 152–170. [13] Z.-Z. Bai, Structured preconditioners for nonsigular matrices of block two-by-two structures, Math. Comp. 75 (2006) 791–815. [14] X.-N. Wu, G.H. Golub, J.A. Cuminato, J.-Y. Yuan, Symmetric-triangular decomposition and its applications Part II: Preconditioners for indefinite systems, BIT 48 (2008) 139–162. [15] H.-T. Fan, B. Zheng, X.-Y. Zhu, A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for nonHermitian generalized saddle point problems, Numer. Algorithms (2016). http://dx.doi.org/10.1007/s11075-015-0068-5. [16] Y. Cao, M.-Q. Jiang, Y.-L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18 (2011) 875–895. [17] Y. Cao, J.-L. Dong, Y.-M. Wang, A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation, J. Comput. Appl. Math. 273 (2015) 41–60. [18] Z.-Z. Bai, M.K. Ng, Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31 (2009) 410–433. [19] M. Benzi, Z. Wang, Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations, SIAM J. Sci. Comput. 33 (2011) 2761–2784. [20] M. Benzi, J. Liu, An efficient solver for the incompressible Navier–Stokes equations in rotation form, SIAM J. Sci. Comput. 29 (2007) 1959–1981. [21] N.-B. Tan, T.-Z. Huang, Z.-J. Hu, A relaxed splitting preconditioner for the incompressible Navier–Stokes equations, J. Appl. Math. (2012). http://dx.doi.org/10.1155/2012/402490. [22] Y. Cao, S.-X. Miao, Y.-S. Cui, A relaxed splitting preconditioner for genenralized saddle point problems, Comput. Appl. Math. (2015). http://dx.doi.org/10.1007/s40314-014-0150-y.

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[23] Z.-Z. Bai, M.K. Ng, On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput. 26 (2005) 1710–1724. [24] H.C. Elman, A. Ramage, D.J. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Software 33 (2007) Article 14.