Approximation of Navier Stokes IncompressibleFlow Using a Spectral Element Method with aLocal Discretization in Spectral SpaceKelly BlackKingsbury HallUniversity of New HampshireDurham, New Hampshire 03824

Received March 6, 1997; revised June 20, 1997

A spectral element technique is examined, which builds upon a local discretization within the spectral space.To approximate a given system of equations the domain is subdivided into nonoverlapping quadrilateralelements, and within each element a discretization is found in the spectral space. The difference is that thetest functions are divided into the higher-order polynomials, which have zero boundaries and lower-orderpolynomials, which are nonzero on one boundary. The method is examined for Navier–Stokes incompress-ible flow for fluid flow within a driven cavity and for flow over a backstep. c© 1997 John Wiley & Sons, Inc.Numer Methods Partial Differential Eq 13: 587–599, 1997

Keywords: Domain decomposition; spectral element; incompressible Navier Stokes; legendre polynomials

I. INTRODUCTION

A spectral-element technique is examined in which the local approximation is found from a linearcombination of Legendre polynomials rather than the Lagrange interpolants of the abscissa of theLegendre–Gauss– Lobatto quadrature. The approach is similar to the standard spectral elementapproach of Patera [1, 2], but within each subdomain the approximation is found in the spectralspace. Local expansions have been examined by Oden [3] and Israeli et al. [5], but the methodpresented here has more in common with the method proposed by Shen [5].

A variational form is constructed by dividing the test functions into the higher-order poly-nomials, which are zero on the boundaries and linear polynomials, which are nonzero on oneboundary. The result is a hybrid method that divides the computational domain in a similar man-ner as a standard spectral element method [6], but an approximation is constructed that yields anapproximation through a local spectral expansion.

c© 1997 John Wiley & Sons, Inc. CCC 0749-159X/97/060587-13

588 BLACK

The choice of basis functions is motivated by Shen [5], and the resulting stiffness matrix issymmetric, sparse, and nearly diagonal. Unlike spectral element approaches that make use ofLagrange interpolants of the abscissa of the Gauss–Lobatto quadrature, the individual entrieswithin the stiffness and mass matrices do not depend on the order of the discretization. Theresulting method is an h − p finite element method that greatly simplifies the process of p-refinement, and the system is algebraically identical to the spectral element method [2]. Moreover,because the entries in the matrices do not depend on the polynomial order, the method has a naturalextension to a multigrid solver.

The method is first introduced for a simple 1-D model problem, a Helmholtz equation. Oncedone, the approximation to the Navier–Stokes incompressible flow equations are examined. Theapproximation to the flow within a driven cavity is examined, and an example of flow over abackstep is also examined. In both examples, the pressure is calculated following the methodproposed by Karniadakis et al. [7].

II. MULTIDOMAIN METHOD

The computational domain is subdivided into nonoverlapping subdomains in order take advantageof the high accuracy and relatively small number of required degrees of freedom offered byspectral methods. On each subdomain an approximation is found that is a linear combinationof the Legendre polynomials up to a given degree. First, a simple 1-D Helmholtz equation isexamined:

uxx + λu = f, x ∈ (0, 1),

u(0) = u(1) = 0. (2.1)

The approximation is constructed by integrating against a sequence of test functions andconstructing a system of algebraic equations. In this case, a Galerkin method is examined in whichthe test functions and the trial functions are found from a sequence of Legendre polynomials. Sincea multidomain approximation is sought, the test functions, ψj(x), are defined using a sequenceof Legendre polynomials [5]:

ψ0(x) =1 + x

2,

ψ1(x) =1− x

2.

ψj(x) = Lj(x)− Lj−2(x), 2 ≤ j ≤ N. (2.2)

The test functions are zero on the endpoints for j = 2 · · ·N . For j = 0 and j = 1 the testfunctions are linear polynomials, and the span of all the test functions is the space of polynomialsup to a given degree, PN .

The computational domain is divided into M nonoverlapping subdomains. For a given sub-domain, r, the approximation is written as urN (x) ∈ PN and the left and right endpoints are Lr

and Rr, respectively (see Fig. 1). On each subdomain, the domain is mapped to the unit square[−1, 1] using a simple linear transformation:

xr = 2x− LrRr − Lr − 1, x ∈ [Lr, Rr]. (2.3)

LOCAL SPECTRUM SPECTRAL ELEMENTS 589

FIG. 1. A multidomain example in 1-D for subdomains l and r.

The approximation on subdomain r is written as a linear combination of the Legendre polynomialsand has support only on subdomain r:

urN (x) = ∑N

i=0 αrj ψi(x

r), x ∈ [Lr, Rr]0 otherwise.

(2.4)

In this example, x is in the computational domain defined in Eq. (2.1) and xr is found from Eq.(2.3). For 0 ≤ j ≤ N , the test functions for subdomain r are found in the same manner:

ψrj (x) =ψj(xr) x ∈ [Lr, Rr]0 otherwise.

(2.5)

With these definitions, the variational approximation can be examined. Except for the linearfunctions, the test functions are zero on the subdomain boundaries, and each test function hassupport on only one subdomain. The only inner-product that must be found is that part withinthe individual subdomain. The linear test functions, however, are not zero on the boundaries ofeach subdomain. If the linear functions on adjacent subdomains are examined, the result is asimple hat function (see Fig. 2). These composite test functions are used to ensure that the fluxis balanced across the subdomain interfaces.

Once the approximation and the test functions are defined on the two subdomains, the varia-tional approximation of Eq. (2.1) is constructed. For j = 2, . . . , N the following equations areenforced on each subdomain:∫ Rr

Lr−(d

dxurN (x)

)(d

dxψrj (x)

)+ λurN (x)ψrj (x) dx =

∫ Rr

LrfrN (x)ψrj (x) dx. (2.6)

Assuming that subdomain l is the subdomain to the left and adjacent to subdomain r, the equationsfor the linear test functions are constructed by integrating against the hat function:∫ Rl

Ll−(d

dxulN (x)

)(d

dxψl0(x)

)+ λurN (x)ψl0(x) dx

+∫ Rr

Lr−(d

dxurN (x)

)(d

dxψr1(x)

)+ λurN (x)ψr1(x) dx

=∫ Rl

Llf lN (x)ψl0(x) dx+

∫ Rr

Lr

frN (x)ψr1(x) dx. (2.7)

FIG. 2. Trial functions ψl0 (x) and ψr1 (x) combine on adjacent subdomains to assemble a ‘‘hat’’ function.

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Assuming that subdomain 0 is the far left subdomain and M is the subdomain on the far right,the boundaries are directly enforced:

u0N (0) = α0

1 = 0,

uMN (1) = αM0 = 0. (2.8)

The subdomain interfaces are enforced by simply requiring C0 continuity:

ulN (Rl) = urN (Lr),

αl0 = αr1. (2.9)

The solution to the system of equations in Eqs. (2.6)–(2.9) is the approximation to Eq. (2.1).Like spectral elements making use of Lagrange interpolants, the resulting spatial discretization

is an h − p finite element method. The method proposed here allows for simpler p-refinement,though. The difference is that, within each subdomain, the approximation is composed of anexpansion in terms of the Legendre polynomials, rather than the Lagrange polynomials interpo-lating a set of grid points. As will be shown in Section II.A, an individual entry in the stiffnessand mass matrices does not depend on the order of the discretization, and the system is easilyrefined (p-refinement) through a natural projection operator. Moreover, when examining theentries corresponding to a given subdomain, the submatrix is quite sparse, which offers a moreefficient implementation with an iterative solver.

A. Stiffness and Mass Matrices

Equations (2.6)–(2.9) are used to construct an approximation to Eq. (2.1). By substituting urN (x)from Eq. (2.4), the stiffness and the mass matrices can be constructed. A derivation for the entriesin the stiffness matrix is given, and the entries in the mass matrix are given.

The stiffness matrix is found by examining the variational form for the second derivative. Aftera substitution to allow for integration across [−1, 1], the variational form of the second derivativein Eq. (2.6) is derived for j = 0 · · ·N − 2:∫ Rr

Lr−(d

dxurN (x)

)(d

dxψrj (x)

)dx

=−2

Rr − Lr∫ 1

−1

(d

dxrurN

(Rr − Lr

2(xr + 1)− Lr

))(d

dxrψj(xr)

)dxr. (2.10)

The sum of Legendre polynomials is substituted for the approximationulN (x). For j = 0 · · ·N−2the result is∫ Rr

Lr−(d

dxurN (x)

)(d

dxψrj (x)

)dx

=N∑i=0

αri

( −2Rr − Lr

)∫ 1

−1

d

dxrψi(xr)

d

dxrψj(xr) dxr. (2.11)

The system of equations can be constructed through the use of a stiffness matrix, SN , by setting

LOCAL SPECTRUM SPECTRAL ELEMENTS 591

FIG. 3. Structure of the stiffness matrices for spectral-element schemes making use of Lagrange interpolantsand the proposed scheme. Each method consists of three subdomains with polynomial degree ten withineach subdomain. The same polynomial degree is found within each subdomain, but the equations for theboundaries are collapsed.

(SN )ji =−2

Rr − Lr∫ 1

−1

d

dxψi(x)

d

dxψj(x) dx,

=2

Rr − Lr∫ 1

−1(2i− 1)Li−1(x)(2j − 1)Lj−1(x) dx,

=2

Rr − Lr (2i− 1)δji, (2.12)

for 2 ≤ j ≤ N , and 0 ≤ i ≤ N .The equations for the interface are found by integrating against the hat function over adjacent

subdomains [just as in Eq. (2.7)]:∫ Rl

Ll−(d

dxulN (x)

)(d

dxψl0(x)

)dx+

∫ Rr

Lr−(d

dxurN (x)

)(d

dxψr1(x)

)dx

=N∑i=0

αli

(− 2Rl − Ll

)∫ 1

−1L′i(x

l)d

dxl

(1 + xl

2

)dxl+ (2.13)

N∑i=0

αri

(− 2Rr − Lr

)∫ 1

−1L′i(x

r)d

dxr

(1− xr

2

)dxr

= − 2Rl − Ll

(12αl0 −

12αl1

)+

2Rr − Lr

(−1

2αr0 +

12αr1

). (2.14)

The resulting stiffness matrix is a diagonal matrix for j = 2 · · ·N . As an example, the stiffnessmatrix for a spectral element using Lagrange interpolants is compared with the stiffness matrixof the proposed method in Fig. 3.

The mass matrix,MN , is found using the same approach, for 2 ≤ j ≤ N and 2 ≤ i ≤ N :

(MN )ji =Rr − Lr

2

∫ 1

−1ψi(xr)ψj(xr) dxr. (2.15)

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FIG. 4. Curvilinear mapping from a computational subdomain to a square, (x, y) ∈ [−1, 1] × [−1, 1].

The entries in the mass matrix can be found by taking advantage of the orthogonality of theLegendre polynomials, which results in a symmetric, tridiagonal matrix. The entries for themass matrix for the linear functions are found by integrating against the hat function over twosubdomains. The orthogonality of the Legendre polynomials allows for a closed form solutionto the integrals. The system of equations is constructed to reflect these constraints as well as theboundary conditions given in Eqs. (2.8) and (2.9).

B. Spatial Discretization for 2-D

For the 2-D equations, both the approximating and trial functions are found from a tensor productof the basis functions employed in the 1-D case. Within each subdomain an approximation isconstructed that is a linear combination of the basis functions in subdomain r:

urN (x, y) =N∑j=0

N∑i=0

αrijψi(xr)ψj(yr). (2.16)

The test functions are also found as a simple tensor product, ψrj (xr)ψri (y

r).Continuity across the subdomain interfaces are enforced by minimizing the difference between

the approximations on adjacent subdomains. For example, if subdomain r is to the right of sub-domain l, then on subdomain r the boundary xr = −1 is adjacent to the boundary on subdomainl when xl = 1. The continuity across this interface is enforced by requiring that the differencebetween the two approximations be orthogonal to the space of polynomials of degree N :∫ 1

−1(ulN (1, y)− urN (−1, y))Lj(y) dy = 0, (2.17)

which reduces to

αl0j = αr1j , j = 0 · · ·N,

for the special case of a conforming method.The approximation space within a given subdomain is the space of polynomials up to a given

degree. However, the basis functions defined in Eq. (2.15) are not adequate for anything otherthan the square (x, y) ∈ [−1, 1]× [−1, 1]. In general, each subdomain is mapped to this squaredomain, and the approximation is found through the mapping (see Fig. 4). A numerical mappingcan found through a Gordon–Hall transformation.

LOCAL SPECTRUM SPECTRAL ELEMENTS 593

The variational form of the resulting equations is expressed using the transformation Jacobian:

JN =

∣∣∣∣∣∂x∂η

∂x∂ξ

∂y∂η

∂y∂ξ

∣∣∣∣∣ , (2.18)

where it is assumed that the mapping is a polynomial of the same degree as the approximation,and the Jacobian evaluated at a specific point, (ηi, ξj), is denoted (JN )ij . The discrete divergenceoperator, ∇N , must also be defined with respect to the coordinate transformation, where

∇N (ψm(η)ψn(ξ)) =1JN

[∂y∂ξ −∂y∂η−∂x∂ξ ∂x

∂η

][ψ′m(η)ψn(ξ)

ψm(η)ψ′n(ξ)

]. (2.19)

On a given subdomain, Ωk, the discretized Laplacian can be expressed in terms of the mapping,∫ ∫Ωk∇N (ψm(x)ψn(y)) · ∇N (ψi(x)ψj(y)) dx dy

=∫ 1

ξ=−1

∫ 1

η=−1∇N (ψm(η)ψn(ξ)) · ∇N (ψi(η)ψj(ξ))JN dη dξ, (2.20)

which can be approximated through the use of the Gauss quadrature (just as is done with spectralelement methods, which make use of Lagrange interpolants [8]).

III. NAVIER--STOKES INCOMPRESSIBLE FLOW

The incompressible Navier–Stokes flow equations,

ut + (u · ∇)u +∇p1ρ

=1

Re∇2u,

subject to ∇ · u = 0, (3.1)

with no slip boundaries are examined. The geometries examined are for flow within a drivencavity as well as flow over a backstep. The spatial discretization employed is the same as examinedin the previous section. The temporal discretization is based on the splitting method [9] and themethods proposed by Karniadakis et al. [7]. The splitting method is a convenient scheme toseparate the actions of the two spatial operators acting on the velocity,

N (u) = 12 ((u · ∇)u +∇ · (uu)),

L(u) =1

Re∇2u. (3.2)

(The implementation employs the skew-symmetric form of the nonlinear operator.)In the following the method proposed by Karniadakis et al., a time averaged pressure is ap-

proximated [7, p. 416]:

1ρ

∫ tn+1

tn

∇p dt = ∇pn+1, (3.3)

594 BLACK

FIG. 5. Domain and boundaries for the flow within a driven cavity.

which acts as a Lagrange multiplier to maintain the constraint of a divergence-free vector field.The three relevant spatial operators can be isolated in three separate steps:

uN − unN = −∫ tn+1

tn

N (uN ) dt,

−uN + uN = −∇p, subject to ∇ · uN = 0,

un+1N − uN =

∫ tn+1

tn

L(uN ) dt. (3.4)

In the first step, the nonlinear term is integrated through the use of an explicit method suchas those from the Adams–Bashforth family of schemes, while the third step employs an implicitstep such as those found in the Adams–Moulton family of schemes. Because an explicit step istaken there is a restriction on the time step. However, the more restrictive time-step limitation ison the elliptic operator, and it can be mitigated through the use of an implicit scheme in the finalequation.

The second step requires an approximation of a Poisson equation, and the third step requiresthe approximation of a Helmholtz equation. Because of the boundary conditions, they are notnecessarily symmetric operators. The boundary conditions in the second step are found followingthe method proposed by Karniadakis [7] and the system is not symmetric. For a system that

FIG. 6. The domain is subdivided into quadrilateral elements.

LOCAL SPECTRUM SPECTRAL ELEMENTS 595

FIG. 7. Level curves of the pressure for Reynolds number 1000.

includes an inlet or outlet, the third equation may not be symmetric. A GMRES [10, 11, 14]iterative scheme is used to invert these linear systems.

A. Pressure Solves

The approximation of the pressure equation in Eq. (3.4) is carried out through the use of NeumannBoundary conditions [12] as proposed by Karniadakis et al. [7]. The boundary conditions yieldan unsymmentric, singular system. A GMRES method [10, 11, 14] is used to invert the systemto find the approximation of the pressure, pN . The system

LN pN = f (3.5)

is inverted, where LN is the discrete Laplace operator and f is the resulting right-hand sideafter the boundary conditions are applied. To accelerate convergence of the iterative scheme, thesystem is inverted through the use of a preconditioner:

L−1N LN pN = L−1

N f . (3.6)

Here LN is the decoupled discrete Laplacian applied to each subdomain with no intra-subdomaininformation.

FIG. 8. Level curves of the stream function for the Reynolds number Re = 1000.

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FIG. 9. Level curves of the stream function for the Reynolds number Re = 2000.

Using the GMRES method, an approximation is constructed from the span of a sequence ofvectors (the Krylov subspace). The space must be orthogonal to the kernel of the operator of thediscrete operator (which is a constant). On a given subdomain, r, the following test functions addto unity:

ψr1(x)ψr1(y) + ψr1(x)ψr0(y) + ψr0(x)ψr1(y) + ψr0(x)ψr0(y) = 1, (x, y) ∈ Ωr. (3.7)

Thus, over the entire domain,

M∑r=0

(∫Ωr−∇N pN · ∇N (ψr1(x)ψr1(y)) dω +

∫Ωr−∇N pN · ∇N (ψr1(x)ψr0(y)) dω

+∫

Ωr−∇N pN · ∇N (ψr0(x)ψr1(y)) dω +

∫Ωr−∇N pN · ∇N (ψr0(x)ψr0(y)) dω

)=

M∑r=0

∫Ωr−∇N pN · ∇N (ψr1(x)ψr1(y) + ψr1(x)ψr0(y) + ψr0(x)ψr1(y) + ψr0(x)ψr0(y)) dω

= 0, (3.8)

TABLE I. Comparison of vortex positions where ‘‘present’’ refers to the local spectral basis.

Re Reference Primary Secondary right Secondary left Secondary top left

100 Present 0.610, 0.750 0.950, 0.049 0.034, 0.034 —100 Pinelli 0.599, 0.750 0.959, 0.045 0.038, 0.038 —100 Phillips 0.598, 0.757 0.952, 0.048 0.038, 0.038 —100 Shen 0.609, 0.750 0.953, 0.047 0.031, 0.031 —

1000 Present 0.550, 0.580 0.870, 0.111 0.071, 0.067 —1000 Pinelli 0.540, 0.574 0.880, 0.114 0.067, 0.067 —1000 Phillips 0.549, 0.573 0.870, 0.113 0.071, 0.071 —1000 Shen 0.547, 0.578 0.922, 0.094 0.078, 0.063 —2000 Present 0.520, 0.548 0.854, 0.100 0.084, 0.095 0.032, 0.8902000 Pinelli 0.531, 0.548 0.859, 0.107 0.080, 0.097 0.045, 0.8982000 Phillips 0.525, 0.549 0.854, 0.113 0.084, 0.098 0.038, 0.8872000 Shen 0.531, 0.547 0.922, 0.094 0.078, 0.094 0.031, 0.902

LOCAL SPECTRUM SPECTRAL ELEMENTS 597

FIG. 10. Graph of the geometry for the backstep. The backstep has height 12 and the maximum velocity

at the inlet is 1. The domain is divided into 30 subdomains and a spectral approximation is constructedwithin each subdomain. Within each subdomain a Legendre polynomial approximation is employed withthe degree of the polynomial 6 in the x-direction and 6 in the y-direction.

and each member of the Krylov subspace must be orthogonal to the vector corresponding toEq. (3.8).

B. Driven Cavity

The first numerical test case is for the flow within a driven cavity. The flow within a simple square[0, 1]× [0, 1] is examined (see Fig. 5). The velocity satisfies zero boundaries at each wall exceptfor the horizontal component at the top boundary, which is given by (4x(1− x))2. The cavity isdivided into 16 subdomains with polynomials of degree 8 in both the x and the y-directions (seeFig. 6).

In the trials, Reynolds numbers of 100, 1000, and 2000 were examined. The level curves fora pressure calculation are shown in Fig. 7. The level curves of the stream functions are shownin Figs. 8 and 9. A comparison with the results of Pinelli and Vacca [6], Phillips [13], and Shen[15] are given in Table I.

C. Flow over a Backstep

The second numerical test is for the flow over a backstep. In the trial the backstep has height12 , the inlet has height 1

2 , and the outlet has height 1. The maximum velocity at the inlet is 1.In the trial the domain is divided into 30 subdomains (see Fig. 10), and within each subdomain

FIG. 11. Vector plot for the inlets for two simulations. The top simulation is from the test case Re = 200and the bottom simulation is from the test case Re = 400.

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FIG. 12. Vector plot focusing on the region behind the backstep for two simulations. The top simulation isfrom the test case Re = 200 and the bottom simulation is from the test case Re = 400.

the approximation utilizes a polynomial of degree 6 in both the x and the y directions. It is apressure-driven flow with no slip boundaries along the walls.

Two different trials were run with Reynolds numbers of 200 and 400, respectively. In the firsttrial, the initial condition was a zero velocity with a time step of ∆t = 10−3. The velocity fieldafter 6000 steps is shown in Figs. 11 and 12. The initial condition that was used for the secondtrial came from the data from the first trial. In the second trial, the results were found after 1500time steps.

IV. CONCLUSION

A novel spectral element technique is introduced and examples for Navier–Stokes incompressibleflow are given. The method departs from the standard spectral-element technique in that a localspectral basis is employed, but the resulting algebraic system is equivalent to the spectral elementmethod. However, the resulting stiffness and mass matrices demonstrate a sparsity that is similarto that found by Shen [5].

While the method is essentially a p-version finite element technique, the use of a local spectral-basis simplifies the way in which the entries in the mass and stiffness matrix are found and allowsfor a natural projection operator for use in a multigrid solver.

References

1. Anthony T. Patera, ‘‘A spectral element method for fluid dynamics: Laminar flow in a channel expan-sion,’’ J. Comput. Phys. 54, 468–488 (1984).

2. Y. Maday and A. Patera, ‘‘Spectral element methods for the incompressible Navier–Stokes equations,’’in State of the Art Surveys on Computational Mechanics, A. K. Noor and J. T. Oden, Eds., ASME, 1989,pp. 71–143.

3. J. T. Oden, ‘‘Theory and implementation of high-order adaptive h − p methods for the analysis ofincompressible viscous flows,’’ in Computational Nonlinear Mechanics in Aerospace Engineering,S. N. Atluri, Ed., AIAA Progress in Aerospace and Astronautics' Series, 146, 1992, pp. 321–363.

LOCAL SPECTRUM SPECTRAL ELEMENTS 599

4. M. Israeli, L. Vozovoi, and A. Averbach, ‘‘Spectral multidomain technique with local Fourier basis,’’J. Sci. Comput. 8, 135–149 (1993).

5. J. Shen, ‘‘Efficient spectral–Galerkin method I. Direct solvers of second and fourth-order equationsusing Legendre polynomials,’’ SIAM J. Sci. Comput. 15, 1489–1505 (1994).

6. A. Pinelli and A. Vacca, ‘‘Chebychev collocation method and multidomain decomposition for theincompressible Navier–Stokes equations,’’ Internat. J. Numer. Methods in Fluids 18, 781–799 (1994).

7. G. Karniadakis, M. Israeli, and S. Orszag, ‘‘High-order splitting methods for the incompressible Navier–Stokes equations,’’ J. Comput. Phys. 97, 414–443 (1991).

8. K. Z. Korczak and A. Patera, ‘‘An isoparametric spectral element method for solution of the Navier–Stokes equations in complex geometry,’’ J. Comput. Phys. 62, 361–382 (1986).

9. Y. Maday, A. Patera, and E. Ronquist, ‘‘An operator-integration-factor splitting method for time-dependent problems: Application to incompressible flow,’’ J. Sci. Comput. 5, 263–292 (1990).

10. Homer F. Walker, ‘‘Implementation of the GMRES method using householder transformations,’’ SIAMJ. Sci. Stat. Comput. 9, 152–163 (1988).

11. Gene H. Golub and Charles F. Van Loan, Matrix Computations, Second Ed., John Hopkins UniversityPress, Baltimore, MD, 1989.

12. Philip M. Gresho and Robert L. Sani, ‘‘On pressure boundary conditions for the incompressible Navier–Stokes equations,’’ Internat. J. Numer. Methods in Fluids 7, 1111–1145 (1987).

13. T. Phillips and G. Roberts, ‘‘The treatment of spurious pressure modes in spectral incompressible flowcalculations,’’ J. Comput. Phys. 105, 150–164 (1993).

14. Youcef Saad and Martin Schultz, ‘‘GMRES: A generalized minimal residual algorithm for solvingnonsymmetric linear systems,’’ SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).

15. J. Shen, ‘‘The Hopf bifuration on the unsteady regularized driven cavity flow,’’ J. Comput. Phys. 95,228–245 (1991).