A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations

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+di𝐮𝐮v(𝐮(=𝐮𝐱⋅,𝐠∇=0))0𝐮=on+𝐮o𝜕₀∇nΩ𝑝Ω×=×[𝐟0[,0𝑇,o𝑇]n,],Ω×(0,𝑇],(((111...123)))(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: 𝐀𝐱=𝐛,(1.5) with 𝐀=⎡⎢⎢⎣𝐀𝐁^𝑇−𝐁0⎤⎥⎥⎦,𝐱=⎛⎜⎜⎝𝐮𝑝⎞⎟⎟⎠,𝐛=⎛⎜⎜⎝−𝐟𝑔⎞⎟⎟⎠,(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 𝐴_𝑖 is a scalar discrete convection-diffusion operator represented as 𝐴_𝑖=𝜎𝑉+𝜐𝐿+𝑁_𝑖(𝑖=1,2),(1.7) where 𝑉 denotes the velocity mass matrix, 𝐿 the discrete (negative) Laplacian, and 𝑁_𝑖 the convective terms. The matrix 𝐀 is positive definite in the sense that 𝐀^𝑇+𝐀 is symmetric positive definite. Matrix 𝐁^𝑇=(𝐵^𝑇1,𝐵^𝑇2) denotes the discrete gradient with 𝐵^𝑇1, 𝐵^𝑇2 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⁻¹𝐁^𝑇. 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: 𝐀=⎡⎢⎢⎢⎢⎣𝐴¹0𝐵^𝑇10𝐴₂𝐵^𝑇2−𝐵₁−𝐵₂0⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝐴¹−0𝐵₁000000⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣00𝐵^𝑇10𝐴₂𝐵^𝑇20−𝐵₂0⎤⎥⎥⎥⎥⎦=𝐻+𝑆,(2.1) where 𝐴₁∈ℝ^𝑛1×𝑛1, 𝐴₂∈ℝ^𝑛2×𝑛2, 𝐵₁∈ℝ^𝑚×𝑛1, and 𝐵₂∈ℝ^𝑚×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,…, (𝑆(𝐻++𝛼𝐼𝛼)𝐼𝐱)𝐱^𝑘+1/2_𝑘+1=(𝛼𝐼−𝑆)𝐱^𝑘+𝐛,=(𝛼𝐼−𝐻)𝐱^𝑘+1/2+𝐛.(2.3) Eliminating 𝐱^𝑘+1/2 from these, we can rewrite (2.3) as the stationary scheme: 𝐱^𝑘+1=𝑇_𝛼𝐱^𝑘+𝐜,𝑘=0,1,…,(2.4) where 𝑇_𝛼=(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 𝑇_𝛼 can be rewritten as follows: 𝑇_𝛼==((𝑆𝑆++𝛼𝛼𝐼𝐼))⁻¹₋₁(𝐻+𝛼𝐼)⁻¹(𝛼𝐼−𝐻)(𝛼𝐼−𝑆)=𝐼−1(𝐻+𝛼𝐼)⁻¹(𝛼𝐼+𝐻)(𝛼𝐼+𝑆)−2𝛼𝐀2𝛼(𝐻+𝛼𝐼)(𝑆+𝛼𝐼)⁻¹=𝐼−𝑃_𝛼−1𝐀𝐀,(2.6) where 𝑃_𝛼=(1/2𝛼)(𝐻+𝛼𝐼)(𝑆+𝛼𝐼).

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

2.2. A Relaxed Splitting Preconditioner

The relaxed splitting preconditioner is defined as follows: 𝐌=⎡⎢⎢⎢⎢⎢⎣𝐴¹01𝛼𝐴¹𝐵^𝑇10𝐴₂𝐵^𝑇2−𝐵₁−𝐵₂𝛼𝐼−1𝛼𝐵¹𝐵^𝑇1⎤⎥⎥⎥⎥⎥⎦.(2.7) It is important to note that the preconditioner 𝐌 can be written in a factorized form as 𝐌=1𝛼⎡⎢⎢⎢⎢⎣𝐴¹−0𝐵₁𝛼0𝐼000𝛼𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝛼𝐼0𝐵^𝑇10𝐴₂𝐵^𝑇20−𝐵₂𝛼𝐼⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝐴¹−0𝐵₁𝐼0000𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼01𝛼𝐵^𝑇10𝐴₂𝐵^𝑇2=⎡⎢⎢⎢⎢⎣𝐴0−𝐵₂𝛼𝐼₁⎤⎥⎥⎥⎥⎦−0𝐵₁𝐼0000𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣𝐼01𝛼²𝐵^𝑇10𝐼1𝛼𝐵^𝑇200𝐼⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼1𝛼²0𝐵𝐴^𝑇1₂𝐵₂000𝛼0𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼00−1𝐼000𝛼𝐵²𝐼⎤⎥⎥⎥⎥⎦,(2.8) where 𝐴₂=𝐴₂+(1/𝛼)𝐵^𝑇2𝐵₂. Note that both factors on the right-hand side are invertible provided that 𝐴₁ have 𝐴₂ 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 𝐑=𝐌−𝐀=⎡⎢⎢⎢⎢⎢⎣001𝛼𝐴¹𝐵^𝑇1−𝐵^𝑇10000𝛼𝐼−10𝛼𝐵¹𝐵^𝑇1⎤⎥⎥⎥⎥⎥⎦.(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=𝐀𝐌⁻¹.

Theorem 2.1. The preconditioned matrix 𝐓=𝐀𝐌⁻¹ has an eigenvalue 1 with multiplicity at least 𝑛, and the remaining eigenvalues are 𝜆_𝑖, where 𝜆_𝑖 are the eigenvalues of an 𝑚×𝑚 matrix 𝑍_𝛼∶=(1/𝛼)(𝑆₁+𝑆₂)−(1/𝛼²)𝑆₂𝑆₁ with 𝑆₁=𝐵₁𝐴¹⁻¹𝐵^𝑇1 and 𝑆₂=𝐵₂𝐴²⁻¹𝐵^𝑇2.

Proof. First of all, from 𝐓∶=𝐌⁻¹(𝐀𝐌⁻¹)𝐌=𝐌⁻¹𝐀 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==𝐈𝐈−−𝐌⎡⎢⎢⎢⎢⎣𝐼00^−𝟏1𝐑𝐼000𝛼𝐵²𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣𝐼−1𝛼²0𝐵𝐴^𝑇1𝐵₂₂₋₁𝐴²⁻¹00010𝛼𝐼⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣𝐼0−1𝛼²𝐵^𝑇10𝐼−1𝛼𝐵^𝑇2×⎡⎢⎢⎢⎢⎣𝐴00𝐼₁₋₁⎤⎥⎥⎥⎥⎥⎦𝐵0𝐼000₁𝐴¹⁻¹0𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣001𝛼𝐴¹𝐵^𝑇1−𝐵^𝑇10000𝛼𝐼−10𝛼𝐵¹𝐵^𝑇1=⎡⎢⎢⎢⎢⎣𝐼00𝐼001∗∗⎤⎥⎥⎥⎥⎥⎦𝛼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 𝜆_𝑖’s.

Lemma 2.2. Let 𝐴_𝛼=^{𝐴1−(1/𝛼)𝐵𝑇1𝐵1−(1/𝛼)𝐵𝑇1𝐵2}_0𝐴2∈ℝ^𝑛×𝑛, 𝛼>𝜎_max(𝐵^𝑇1𝐵₁)/𝜎_min(𝐴₁), and 𝐴₁, and 𝐴₂ be positive definite. Then 𝐴_𝛼 is positive definite.

Lemma 2.3. Let 𝐴_𝛼∈ℝ^𝑛×𝑛 and 𝐵∈ℝ^𝑚×𝑛 (𝑚≤𝑛). Let 𝛼∈ℝ, and assume that matrices 𝐴_𝛼, 𝐴_𝛼+(1/𝛼)𝐵^𝑇𝐵, 𝐵𝐴^𝛼−1𝐵^𝑇and 𝐵(𝐴_𝛼+(1/𝛼)𝐵^𝑇𝐵)⁻¹𝐵^𝑇 are all invertible. Then 𝐵𝐴^𝛼+1𝛼𝐵^𝑇𝐵⁻¹𝐵^𝑇⁻¹=𝐵𝐴^𝛼−1𝐵^𝑇⁻¹+1𝛼𝐼.(2.11)

Theorem 2.4. Let 𝛼>𝜎_max(𝐵^𝑇1𝐵₁)/𝜎_min(𝐴₁). The remaining eigenvalues 𝜆_𝑖 of 𝑍_𝛼 are of the form:𝜆_𝑖=𝜇^𝑖𝛼+𝜇_𝑖,(2.12) where the 𝜇_𝑖’s satisfy the eigenvalue problem: 𝐵𝐴^𝛼−1𝐵^𝑇𝜙_𝑖=𝜇_𝑖𝜙_𝑖.

Proof. We note 𝑍_𝛼=1𝛼𝑆¹+𝑆₂−1𝛼²𝑆₂𝑆₁=1𝛼𝐵¹𝐵₂⎛⎜⎜⎝𝐴¹⁻¹−10𝛼𝐴²⁻¹𝐵^𝑇2𝐵₁𝐴¹⁻¹𝐴²⁻¹⎞⎟⎟⎠⎛⎜⎜⎝𝐵^𝑇1𝐵^𝑇2=1⎞⎟⎟⎠𝛼𝐵¹𝐵₂⎛⎜⎜⎝𝐴¹10𝛼𝐵^𝑇2𝐵₁𝐴₂⎞⎟⎟⎠⁻¹⎛⎜⎜⎝𝐵^𝑇1𝐵^𝑇2=1⎞⎟⎟⎠𝛼𝐵¹𝐵₂⎛⎜⎜⎝⎛⎜⎜⎝𝐴¹−1𝛼𝐵^𝑇1𝐵₁−1𝛼𝐵^𝑇1𝐵₂0𝐴₂⎞⎟⎟⎠+⎛⎜⎜⎝1𝛼𝐵^𝑇1𝐵₁1𝛼𝐵^𝑇1𝐵₂1𝛼𝐵^𝑇2𝐵₁1𝛼𝐵^𝑇2𝐵₂⎞⎟⎟⎠⎞⎟⎟⎠⁻¹⎛⎜⎜⎝𝐵^𝑇1𝐵^𝑇2=1⎞⎟⎟⎠𝛼𝐵𝐴^𝛼+1𝛼𝐵^𝑇𝐵⁻¹𝐵^𝑇.(2.13) Thus, the remaining eigenvalues are the solutions of the eigenproblem: 1𝛼𝐵𝐴^𝛼+1𝛼𝐵^𝑇𝐵⁻¹𝐵^𝑇𝜙_𝑖=𝜆_𝑖𝜙_𝑖.(2.14) By Lemma 2.3, we obtain 1𝛼𝜙^𝑖=𝜆_𝑖𝐵𝐴^𝛼+1𝛼𝐵^𝑇𝐵⁻¹𝐵^𝑇⁻¹𝜙_𝑖=𝜆_𝑖𝐵𝐴^𝛼−1𝐵^𝑇⁻¹𝜙_𝑖+𝜆^𝑖𝛼𝜙^𝑖.(2.15) Hence, 𝜆_𝑖=𝜇_𝑖/(𝛼+𝜇_𝑖), where 𝜇_𝑖’s satisfy the eigenvalue problem 𝐵𝐴^𝛼−1𝐵^𝑇𝜙_𝑖=𝜇_𝑖𝜙_𝑖.

In addition, we obtain easily that the remaining eigenvalues 𝜆_𝑖→0 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.

(a)

(b)

(c)

Figure 1 

Spectrum of preconditioned steady Oseen matrix, 32 × 32 grid with 𝜐=0.1.

(a)

(b)

(c)

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: 𝐌=⎡⎢⎢⎢⎢⎣𝐴¹−0𝐵₁𝐼0000𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣𝐼01𝛼²𝐵^𝑇10𝐼1𝛼𝐵^𝑇200𝐼⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼1𝛼²0𝐵𝐴^𝑇1₂𝐵₂000𝛼0𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼00−1𝐼000𝛼𝐵²𝐼⎤⎥⎥⎥⎥⎦,(2.16) showing that the preconditioner requires solving two linear systems at each step, with coefficient matrices 𝐴₁ and 𝐴₂=𝐴₂+(1/𝛼)𝐵^𝑇2𝐵₂. Several different approaches are available for solving linear systems involving 𝐴₁ and 𝐴₂. 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 𝐀𝐱=𝐛 in the form 𝐃^−1/2𝐀𝐃^−1/2𝐲=𝐃^−1/2𝐛 with 𝐲=𝐃^1/2𝐱, 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 ||𝐫_𝑘||₂/||𝐛||₂≤10⁻⁶, where 𝐫_𝑘 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 𝐴₂ 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.

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.

Table 2 

Iterations of preconditioned GMRES(30) for steady Oseen problems (Picard).

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.

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: 𝑃=⎡⎢⎢⎢⎢⎢⎣𝐼0𝐵^𝑇100𝐼0𝐼𝛼0⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐴¹−0𝐵₁𝐼0000𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼0𝐴00²𝐵^𝑇200𝛼𝐼⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝐼00−𝐼0𝐵²00𝛼𝐼⎤⎥⎥⎥⎥⎦,(3.1) where 𝐴₁=𝐴₁+(1/𝛼)𝐵^𝑇1𝐵₁ and 𝐴₂=𝐴₂+(1/𝛼)𝐵^𝑇2𝐵₂. It shows that RDF preconditioner requires solving two linear systems at each step. The new preconditioner requires solving linear systems with 𝐴₁ and 𝐴₂ at each step. We can see that the linear system with 𝐴₁ is easier to solve than that with 𝐴₁. 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.

Table 5 

Iterations of preconditioned GMRES(30) for generalized Oseen problem (Picard, Q2-Q1, 128 × 128 uniform grids).

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.

(a)

(b)

Figure 3 

Iteration number versus parameter, steady Oseen problem, with 𝜐=0.1. (a) 16 × 16 grid, (b) 32 × 32 grid.

(a)

(b)

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).