Block factorized preconditioners for high-order accurate in time approximation of the Navier-Stokes equations

Block Factorized Preconditioners for High-orderAccurate in Time Approximation of theNavier-Stokes EquationsAlessandro VenezianiMOX, Modeling and Scientific Computing, Dipartimento di Matematica “F. Brioschi”,Politecnico di Milano, Via Bonardi 9, I-20133 MILANO, Italy

Received 8 January 2002; accepted 27 October 2002

DOI 10.1002/num.10060

Computationally efficient solution methods for the unsteady Navier-Stokes incompressible equations aremandatory in real applications of fluid dynamics. A typical strategy to reduce the computational cost is tosplit the original problem into subproblems involving the separate computation of velocity and pressure.The splitting can be carried out either at a differential level, like in the Chorin-Temam scheme, or in analgebraic fashion, like in the algebraic reinterpretation of the Chorin-Temam method, or in the Yosidascheme (see [1] and [19]). These fractional step schemes indeed provide effective methods of solutionwhen dealing with first order accurate time discretizations. Their extension to high order time discretizationschemes is not trivial. To this end, in the present work we focus our attention on the adoption of inexactalgebraic factorizations as preconditioners of the original problem. We investigate their properties andshow that some particular choices of the approximate factorization lead to very effective schemes. Inparticular, we prove that performing a small number of preconditioned iterations is enough to obtain a timeaccurate solution, irrespective of the dimension of the system at hand. © 2003 Wiley Periodicals, Inc. NumerMethods Partial Differential Eq 19: 487–510, 2003

I. INTRODUCTION

The numerical solution of the Navier-Stokes equations for incompressible fluids involves linearsystems of large dimensions, and therefore the setting up of efficient methods of solution ismandatory. A typical strategy to reduce the computational effort is to carry out a suitablesplitting of the original problem into smaller problems facing the computation of velocity andpressure separately. The celebrated Chorin-Temam scheme can be regarded in this perspective,being actually based on a differential splitting of the continuous problem. Other schemes relyon an inexact block factorization of the matrix obtained after the full discretization of the

Correspondence to: Alessandro Veneziani, MOX Modeling and Scientific Computing, Dipartimento di Matematica,“F. Brioschi”, Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italy (e-mail: [email protected])Contract grant sponsor: COFIN MURST Research Contracts, and CNR Special Project: “Metodi Matematici inFluidodinamica e Dinamica Molecolare”.

© 2003 Wiley Periodicals, Inc.

problem (see e.g., [1]). Stemming from first-order time discretization schemes, these inexactblock factorizations provide very effective methods, reducing the computational effort withoutaffecting the overall accuracy (see [2]). An important improvement of these methods would begiven by their extension to higher order time discretization schemes. A first contribution of thepresent work is the analysis of some possible high-order block factorization schemes, inparticular with respect to the issue of numerical stability (see Section III).

Another approach is based on adopting inexact block factorizations as a way for devisingpreconditioners for the Navier-Stokes problem. The analysis of this approach is the main goalof the present work. The Stokes (and Navier-Stokes) equations are relevant examples ofsaddle-point problems and this inhibits the adoption of the conjugate gradient method in theirnumerical solution. It is therefore necessary to consider stationary Richardson methods orgeneralized conjugate gradient type schemes, suitably preconditioned (see e.g., [3–12]). In thisarticle we consider specifically time-dependent (not necessarily symmetric) problems. We showthat the inexact block factorizations provide effective preconditioners. In particular, we willfocus our attention on the preconditioners obtained by a first-order factorization. We will showthat performing a small number of iterations of a Richardson scheme, preconditioned with suchinexact factorization, ensures a high-order time accuracy. This result will be rigorously justifiedfor the Stokes problem, by establishing a relationship between the spectral radius of the iterationmatrix of the preconditioned scheme and the time step size �t. Numerical tests suggest a similarresult also for the nonsymmetric Navier-Stokes problem.

The article is organized as follows. After briefly summarizing the basic issues aboutincompressible unsteady Navier-Stokes equations, in Section II, we recall some possiblesplitting strategies for the fully discrete Navier-Stokes problem. In particular, a first originalcontribution refers to the analysis of the stability properties of block factorization based onhigh-order approximations. In Section III we introduce the basic features of the block inexactfactorizations adopted as preconditioners in the Richardson iterative framework. The mainresults of the present work are in Section IV, where we analyze the simplest example ofpreconditioner arising from block factorizations and prove, in the case of the Stokes problem,the relationships between the number of preconditioned iterations and the time accuracy of theassociated preconditioned Richardson iterative method. Numerical results are shown in SectionV, where we also discuss some implementation details, aiming to suggest effective strategies forthe solution of the preconditioners introduced in Section IV.

A. The Navier-Stokes Equations and Their Discretization

Let � � �d (d � 2, 3) be a bounded domain and x � �. We consider a homogeneous,incompressible Newtonian isothermal fluid featuring, for t � 0, the velocity field u(x, t) and thepressure P(x, t), the constant density � and the dynamic constant viscosity �. Setting p(x, t)� P(x, t)/�, from the momentum and mass conservation principles, we obtain the Navier-Stokes(NS) problem:

��u�t

� �u � ��u � ��u � �p � f

� � u � 0for x � �, t � �0, T�,

u�t�0 � u0, �u � g for x � D,

pn � ��u�n

� h for x � N.(1.1)

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In (1.1) f(x, t) is a possible external forcing term, u0 the given initial condition; the boundaryof �, denoted by , is supposed to be split into two parts, D (Dirichlet boundary) and N

(Neumann boundary), where Dirichlet and Neumann conditions are assigned respectively, g andh being given functions for all t � 0 and n being the outward unit normal vector on . Wesuppose that meas(D) is always positive, while we do not exclude that meas(N) � 0. In thelatter case, D � and the compatibility condition � g � nd� � 0 has to be satisfied due to thecontinuity Equation (1.1)2.

Let us consider the numerical approximation of this problem.As for the time discretization, we will refer to classic backward finite differences schemes

(backward difference formulas, BDF; see [13]), which can be summarized as follows. The timeinterval (0, T] is subdivided into N subintervals with a positive time step �t � T/N. Then, wecollocate equations (1.1) at the time level tk � k�t, for k � 1, 2, . . . , N. The time derivative isapproximated as

�u�t�

t�tn 1

�1

�t �i�0

k�

�iu�x, tn 1i�, (1.2)

where the coefficients �i can be selected in order to make the approximation of order �(�tk�). We

resort to a semi-implicit treatment of the nonlinear convective term, amounting to the followingapproximation:

�un 1 � ��un 1 � �i�1

k�

i�un 1i � ��un 1, (1.3)

where the coefficients i are suitably chosen in order to make the extrapolation un 1 � ¥i�1k�

iun 1i accurate up to the order �(�tk�) (see [13]). This linearization of the problem does not

affect the global accuracy of the time discretization even if it introduces an upper bound on theallowable time step size �t. However, in many applications this bound is not too restrictive.

Concerning the space discretization issue, in the present work we will basically refer to thefinite element method (FEM), even if the subsequent analysis can be applied to other kinds ofapproximations. For any detail concerning the FEM discretization of the Navier-Stokes prob-lem, we refer, e.g., to [14–16]. We limit ourselves to point out that we consider only functionalspaces for the approximate velocity and pressure fields that satisfy the inf-sup or LBB condition(see e.g., [14]).

Discretization (in time and space) and the semi-implicit linearization (1.3) of Equations (1.1)yield the following algebraic system, to be solved at time step n 1:

�yn 1 � b, n � 0, . . . , N � 1, (1.4)

where

� � �C DT

D 0 � , yn 1 � �Un 1

Pn 1�, C �

�0

�tM � A, A � K � �

i�1

k�

iB�Un 1i�.

BLOCK FACTORIZED PRECONDITIONERS 489

Here, M denotes the mass matrix, K the stiffness matrix, B is related to the semi-implicitdiscretization of the convective term, Un 1, and Pn 1 denote the vectors of the nodal values ofthe velocity and pressure solutions at t � tn 1, respectively. Nu and Np are the number ofvelocity and pressure nodes respectively, so that C is Nu � Nu and D is Np � Nu. Havingassumed that the inf-sup condition is satisfied implies that ker(DT) � 0. The right-hand side b�[b1 b2]T depends on the discretization of the forcing term f, the boundary data and the solutionat the previous time steps, according to the adopted time discretization scheme. In the case ofthe Stokes problem, A actually coincides with the stiffness matrix K, which is s.p.d., so that Cturns out to be s.p.d. as well. In the sequel, for notational convenience, we will drop the timestep specification of the unknowns U and P, which is understood to be n 1, if not differentlyindicated.

II. BLOCK FACTORIZED SOLVERS FOR THE NS EQUATIONS

System (1.4) typically features large dimensions and bad conditioning properties. It is thereforemandatory to find efficient algorithms for its numerical solution. Some popular strategies arebased on subdividing the computation of velocity and pressure into the successive solution ofsubproblems having smaller dimensions. In particular, the pressure matrix method (see e.g.[16]) amounts to solving a system for the pressure unknown and then a system for the velocityfield, by a block Gaussian elimination carried out on the fully discrete problem. It formallycomputes the exact solution of system (1.4) by the successive solution of smaller subproblems.However, the computational cost can be very high. Indeed, the pressure is obtained by solvinga system associated with the pressure matrix � � DC1DT. In unsteady problems, � is badlyconditioned, the condition number increasing when the time step �t decreases (see e.g., [17]).Moreover, � is not explicitly computable, due to the presence of the C1 factor as well as tofill-in associated with the matrix product. Therefore, the solution of the pressure systemgenerally requires two levels of nested iterations (for more details, see e.g., [16]) and theselection of a good preconditioner for � is mandatory.

A different splitting method is the Chorin-Temam scheme. By using the Helmholtz decom-position principle, this method splits at a differential level the original problem into subproblemsinvolving respectively an intermediate velocity field, the pressure and the “end-of-step” velocity(see e.g., [16, 18]).

Another strategy, that actually embodies an algebraic counterpart of the Chorin-Temamapproach stems from the following LU-block factorization of �:

� � �C 0D ��I C1DT

0 I �, (2.1)

which formally allows to solving system (1.4) by means of successive smaller subproblems. Infact, this splitting corresponds exactly to the pressure matrix method. However, starting from(2.1), we can devise different schemes, achieving a reduction in the computational cost bysuitably approximating C1 with a matrix F in the L-block and G in the U-block, according tothe following inexact block LU factorization:

� � �C 0D DFDT��I GDT

0 I � � �C CGDT

D D�G � F�DT�. (2.2)

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For a discussion about this general factorization and its relationships with differential splittings,the reader is referred to [1]. Here, we simply observe that different choices can be pursued forthe matrices F and G. In particular we could take

C1 ��t

�0M1 � H1. (2.3)

This approximation can be regarded as a first order truncation of the Neumann Expansion (NEin the sequel). By denoting by I the Nu � Nu identity matrix, we have

C1 ��t

�0 I �

�0

�tM1A1

M1 ��t

�0�k�1

� �t

�0M1Ak1

M1. (2.4)

In particular, starting from a first order scheme (�0 � 1), we could select F � G � �tM1,which yields a scheme known as algebraic counterpart of the Chorin-Temam method (see [19]),because of the strong formal analogy with the original differential-splitting method. Otherwise,we could select F � �tM1 and G � C, which yields the so-called Yosida method introducedin [1] and analyzed in [2]. By extension, if H denotes a generic approximation of C1, we willcall in the sequel the ACT approach (Algebraic Chorin Temam) the choice F � G � H, whereasthe Y approach (Yosida) will denote the choice F � H and G � C1. Further, we will add theindex j to the matrix H to denote an approximant matrix obtained by truncating the NE to thejth term, as already done in (2.3) (with j � 1).

The effectiveness of these schemes is based on the following considerations. On one hand weachieve a reduction in the computational effort, since we are solving a sequence of smallersubproblems instead of the unsplit one. On the other hand, if we start from a first order timediscretization scheme, approximation (2.2)–(2.3), corresponding to a first order truncation of theNE, does not reduce the accuracy ([2]). The crucial point is now how to extend this result tomore accurate schemes (in time). A first natural approach consists of accounting for more termsin the NE. For example, we could consider a BDF and a semi-implicit linearization accurate upto the order k� � 2, i.e.,

�u�t�

t�tn 1

�1

�t 3

2un 1 � 2un �

1

2un1, �u � ��u�t�tn 1 � ��2un � un1��un 1. (2.5)

In that case,

C1 � 3

2�tM � A1

�2

3�t �

i�0

� 2

3�ti

�M1A�iM1, (2.6)

and we could take the first two terms of the expansion

C1 � 23

�tM1 �49

�t2M1AM1 � H2.


Since the truncation error in this case is �(�t2), it is to be expected that the resulting scheme issecond-order accurate. Preliminary numerical results confirm this conjecture. In particular thisapproach has been adopted both for second- and third-order BDF schemes and the correspond-ing truncation of the NE and it has been experimentally observed in [20] that, quite surprisingly,the second-order scheme requires an upper bound on �t, which is not required by the third-orderone. These results (that at the best of our knowledge have never been analyzed so far) revealsome drawbacks in the choice of this approach for achieving high-order accuracy, due topossible stability bounds. In the case of the Stokes problem this fact is highlighted by thefollowing Proposition.

Proposition 1. Let us suppose that A (and consequently C) is symmetric positive definite(s.p.d.). The inexact block LU factorization (2.2) (built according to the Y or the ACT strategies)lead to unconditionally stable schemes (in time) if j is odd, i.e., if an odd number of terms in theNE is retained.

Proof. First of all, we prove the following identities for all j 1:

CHj � I � �t

�0j

�AM1� j, C1 � Hj � C1�t

�0j

�AM1� j. (2.7)

Indeed, let us exploit the identity

I ��t

�0M1A � · · · �

�t

�0M1Aj1

� I ��t

�0M1A1I �

�t

�0M1Aj.

Since C � (�0 /�t) M(I (�t/�0) M1A), we have

CH � M I � �t

�0j

�M1A� jM1 � I � �t

�0j

�AM1� j.

The first relation in (2.7) is thus proven. The second derives immediately from the former byobserving that C1 Hj � C1(I CHj).

Now, consider the block factorized matrix (2.2) built according to the Y approach (i.e. withF � Hj and G � C1). By the second relation in (2.7), we have

� � �C DT

D �t

�0j

DC1�AM1� jDT�. (2.8)

Then, if j is odd

� � �C DT

D �t

�0j

DC1�AM1� jDT�. (2.9)

Matrix P � (�t/�0) jDC1(AM1) jDT is definite negative, i.e., there exists a suitable scalarproduct denoted by ( � , � )** such that:

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�y, Py�** � 0 @y � 0.

This is actually a consequence of the fact that D is a full-rank matrix and C, A and M are s.p.d.The definiteness of P leads to the stability of the scheme. Indeed, suppose to consider the Stokesproblem associated with homogenous boundary conditions and null forcing terms. In (1.4) theright-hand side vector will be

b � � 1

�t �i�1

k�

�iMUn 1i

0�.

Let y � [U, P]T be the solution of the system �y � b. Then, by multiplying the system on theleft by [UT, PT], we obtain

UTCU � PT�P�P �1

�t �i�1

k�

�iUTMUn 1i, (2.10)

for all �t � 0. Then, using standard arguments, the definiteness of the matrices C and P yieldsthe desired stability result, providing a bound on the solution at time tn 1 in terms of the solutioncomputed at the previous steps.

In the case of the ACT approach (F � G � Hj), we deal with the matrix:

� � �C CHjDT

D 0 � � �C I � �t

�0j

�AM1� jDT

D 0�. (2.11)

For j odd we obtain

� � �C I � �t

�0 j

�AM1� jDT

D 0� . (2.12)

In this case, it is useful to introduce the matrix

P � I � �t

�0 j

�AM1�j ,

which is s.p.d. (with respect to a suitable scalar product). Using similar arguments as in the Yapproach, we are led to the following inequality,

UT�I � P�1CUT 1

�t�I � P�1 �

i�1

p

�iUTMUn 1i.


Again, the definiteness of C and P yields the desired stability result, and the proof is thuscompleted. y

From the arguments of the previous Proposition (and the numerical results presented in [20]),we may desume that truncations based on an even number of terms of NE may suffer aconditional stability. Indeed, in this case, the (2,2) block in (2.8) is positive definite. In thecounterpart of the left hand side of (2.10), we have the sum of a positive term (UTCU) and anegative one, which is however �((�t)j). Therefore, only for a sufficiently small value of �t thefirst term dominates the second one, thus no unconditional stability is attainable.

III. BLOCK FACTORIZED PRECONDITIONERS FOR THE NS EQUATIONS

In this section, we investigate alternative ways to achieve high order in time stable solutions, stillbased on block factorizations. More precisely, we investigate the use of block factorizations aspreconditioners. We will start considering the inexact factorizations of matrix � as precondi-tioners of system (1.4). Then, we will investigate the role of inexact factorizations as a way forbuilding pressure matrix preconditioners for the pressure matrix method. In doing this, we willdistinguish the case of the Stokes problem from the general Navier-Stokes one.

A. Basic Facts

Suppose to use the matrix � in (2.2) as a preconditioner for system (1.4). For the sake ofsimplicity, we start considering a preconditioned Richardson method, with acceleration param-eter equal to 1. Precisely, at t � tn 1 we compute the solution y of (1.4) as follows: given theinitial estimate y(0), solve for k � 0, 1, . . . :

��y�k 1� � y�k�� b � �y�k�. (3.1)

In particular, referring to the block LU-factorization � � LU, each iterative step (3.1) amountsto solving:

�L�y�k 1� � y�k�� b � �y�k�

Uy�k 1� � y�k 1� , (3.2)

where we have set:

y�k� � Uy�k� � �I GDT

0 I �y�k�.

With obvious notation, (3.2) corresponds to solving:

C�U�k 1� � U�k�� b1 � CU�k� � DTP�k�,

DHDT�P�k 1� � P�k�� DU�k� � D�U�k 1� � U�k�� b2,

P�k 1� � P�k 1�, U�k 1� � U�k 1� � GDTP�k 1�. (3.3)

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In the sequel, we will denote by S the matrix DHDT, which is an approximant to the pressurematrix �. In particular, when H � Hj is obtained by a truncation of the NE up to the jth term,we will add the index j, i.e., Sj � DHjD

T.The iteration matrix associated with (3.1) is

� � I � �1� � �0 �C1DT � GDTS1��0 I � S1� � (3.4)

and has therefore Nu null eigenvalues. The other Np eigenvalues are those of I S1�. Thespectral radius associated with the iteration matrix � is therefore independent of the approx-imation G. In other terms, the spectral properties of the LU block factorized preconditioner areindependent of the approximating matrix G chosen in the U-block of (2.2). In the previoussection we have considered two possibilities, G � H and G � C1. Since G has no influenceon the convergence rate of the preconditioned scheme, it is useful to introduce a third choice,corresponding to set G � 0, yielding

� � �C 0D S�� I 0

0 I� � �C 0D S� � L (3.5)

that we will call the L-preconditioner, since it is actually a low triangular preconditioner. As anexample, in Table I, we indicate the number of iterations required to converge in the Kim andMoin test case (see Section V.B) for the ACT, the Y and the L preconditioners, for differentvalues of h and �t. The number is—at each step—exactly the same in the three cases.

Remark 1. The independence of the U-block with respect to the spectral properties of theblock factorization of � has been pointed out in the context of domain decompositiontechniques, where each block of the factorization corresponds to a subdomain and the interfaceequations are related to the Schur complement of the global problem (see e.g., [21], Chap. 2).

Remark 2. We have identified three convenient preconditioners arising from inexact blockfactorizations, namely the ACT, the Y and the L preconditioners, featuring the same conver-gence rate (in the framework of a stationary Richardson scheme). The L preconditioner seemstherefore preferable, having a lower computational cost. However, we point out that the threepreconditioners give rise to different dynamics for the residual. Actually, the ACT precondi-tioner reduces the residual of the mass conservation equation to the machine epsilon from thefirst iteration. Similarly, the Y preconditioner yields a null residual for the momentum equationstarting from the first iteration (see Fig. 1).

TABLE I. Number of iterations required by the ACT, Y and L preconditioners in a Kim and Moin testcase, for different values of the mesh size and of the time step.

h � 1/32 h � 1/64

�t � 1/100 29–31 11–12�t � 1/500 79–87 24–27

The number is exactly the same in the three cases independently of G.


B. Preconditioning the Pressure Matrix

Eliminating U(k 1) U(k), the second system in (3.3) can be rewritten as

S�P�k 1� � P�k�� P�k� � DC1b1 � b2, (3.6)

which is actually the stationary Richardson preconditioned iteration for the pressure matrixsystem obtained in the pressure matrix method. Eliminating U(k 1) in the last of (3.3) and takingthe converged pressure solution of (3.6), P(k 1) � P(k) � P, we finally recover the velocity field,by solving CU � b1 DTP. These considerations hold independently of the choice of G in theU-factor and suggest that the block factorizations can be reconsidered in the “two-step”perspective as a way of building pressure matrix preconditioners. In particular, Sj � DHjD

T isthe preconditioner obtained by truncating the NE.

The Stokes Problem As previously pointed out, in the case of the Stokes problem, thepressure matrix � is s.p.d., so the preconditioned Conjugate Gradient is applicable and theeffectiveness of Sj as preconditioner is subordinated to the fact that it is s.p.d. too. We investigatethis point in the following Proposition.

Proposition 2. If C is s.p.d., the matrices Sj are

● unconditionally s.p.d. for j odd, i.e. s.p.d. for all �t;● conditionally s.p.d. for j even, i.e. s.p.d. for �t sufficiently small.

Proof. From (2.4) we have for j 1:

Sj ��t

�0D �

k�1

j �t

�0M1Ak1M1DT.

FIG. 1. Residual history for the mass and momentum equations computed by the ACT and the Ypreconditioner. After the first iteration, the ACT preconditioner yields a residual for the mass equationbelow the machine epsilon. Similarly, the Y preconditioned solution after the first iteration yields a residualfor the momentum equation below the machine epsilon.

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The symmetry of this matrix is readily inferred from that of M and A.Now, we set Sj � (�t/�0)DM1/2M1/2WjM

1/2M1/2DT, where

Wj � �k�1

j �t

�0M1Ak1

.

If D is a full-rank matrix, the definiteness of Sj is subordinated to the definiteness of Wj. Moreprecisely, if the eigenvalues of Wj are positive, the (positive) definiteness of Sj follows. Denoteby {�i} the eigenvalues of M1A; they are by hypothesis positive, since M1A is s.p.d. withrespect to the scalar product ( � , � )M � (M � , � ). The eigenvalues �i of Wj will therefore havethe general form:

�i � �k�1

j �t

�0�ik1

Setting qi � (�t/�0)�i, which is positive for all i, we have therefore �i � [1 (qi)j]/(1 qi).

In the general case, we observe that, since for �t 3 0, qi 3 0 and �i 3 1, there exists aninterval (0, �] such that all the eigenvalues �i are positive for all �t belonging to such interval.In particular, for j odd, we have �i � (1 qi

j)/(1 qi), which is positive for any �t � 0 andthe thesis is then proven. y

We conclude that in the s.p.d. case, Sj is a good preconditioner for � for all values of �t ifj is odd. If j is even, the preconditioner Sj is s.p.d. only for a time step sufficiently small.

The General Case Let us consider the general, non symmetric, case. Even when reducing tosolve the pressure matrix system, the CG method is obviously not applicable. The choice ofsetting up a global preconditioner for � rather than of resorting to the preconditioned pressurematrix system will be actually driven by computational efficiency considerations.

Roughly speaking, let us denote by c1 the computational cost (in terms of flops) for thesolution of a system associated with the matrix C and, similarly, c2 will denote the cost of thesolution of a system associated with the pressure matrix preconditioner S. Let us assume that thenumber of iterations required to reach convergence when using L as a global preconditioner of� and when using S as preconditioner of � is the same and is denoted by N. This is reasonable,since in both cases the convergence rate is related to the eigenvalues of S1�, as we observedabove. If we adopt the L preconditioner, at each iteration we need to solve a system for C andone for S. The residual computation involves only matrix vector products, so we neglect its cost.The global cost at each time step is therefore N(c1 c2).

Consider now the computational cost of (3.6). In this case, each iteration for solving (3.6)2

requires the solution of a system for S and one for C. The latter is required in the residualcomputation for the system associated with �. Moreover, we have two further systems [(3.6)1

and (3.6)3], so the total cost amounts to (N 2)c1 Nc2, which is slightly greater than the oneassociated with the global preconditioner L.

However, consider a more efficient iterative method like the BiCGStab iterative framework(see e.g. [22]). The preconditioned BiCGStab iterative scheme (see e.g. [16], Chap. 2) for thegeneric system Ax � b, at each iteration requires to solve three times the system associated withthe preconditioner and to compute two times a matrix-vector product involving matrix A. With


considerations similar to those carried out for the stationary Richardson method, it is readilyverified that the computational cost of the L preconditioned scheme amounts to 3N(c1 c2),while the two-step strategy (3.6) requires a cost of 3Nc2 2(N 1)c1. Typically, it isreasonable to suppose N � 2 in real applications, which means that the preconditioned pressurematrix method turns out to be computationally convenient in the unsymmetric cases when theBiCGStab method is used. We conclude that approach (3.6) is preferable from the computationalviewpoint whenever a BiCGStab iterative scheme is adopted. Actually, in the numerical resultsof Section V we have pursued this strategy.

C. Other Preconditioners of the Pressure Matrix

The scheme (3.1) is very general and embodies different schemes for different choices of F andG, not necessarily derived from the truncation of the NE. For instance, we could select G � 0and

S � SUz � Mp, (3.7)

where Mp denotes the mass matrix for the pressure space �h, i.e. Mp,ij � �� i�jd�. In this case,the corresponding scheme reads

L�y�k 1� � y�k�� b � �y�k�f �CU�k 1� � b1 � DTP�k�

Mp�P�k 1� � P�k�� DU�k 1� � b2.

(3.8)

In (3.8) it is possible to recognize the well known Uzawa method for the solution of (1.4). Inthe perspective of the present work, the Uzawa method can therefore be regarded as a specificapplication of a sort of L preconditioner, where the (2,2) block is not obtained by a truncationof NE, but is given by the pressure mass matrix Mp.

Another possible choice for S considered in literature, which is a sort of combination of thefirst-order approximation of the pressure matrix S1 � (�t/�0)DM1DT, and (3.7), correspondsto setting:

SCC � ��M�p1 �

�0

�t�DM1DT�11

� ��M�p1 � S1

1�1. (3.9)

On the basis of spectral considerations, Cahouet and Chabard ([23]), justified this choice in aBiCGStab framework, providing an almost optimal preconditioner with respect to the spacediscretization (in the case of the generalized Stokes problem), in terms of number of iterationsrequired to reach convergence of the preconditioned scheme (see Fig. 2; see also [17]).

The preconditioners obtained by a simple NE truncation are actually suboptimal with respectto h (see Fig. 2), and however interesting features have to be pointed out with respect to the timeaccuracy of the solutions computed with such preconditioners (S1 in particular).

IV. ANALYSIS OF THE PRESSURE MATRIX PRECONDITIONER S1

Suppose to apply the preconditioner � built starting from a first-order time discrete scheme,according, e.g., to the Y strategy, i.e., with F � �tM1 and G � C1, and to perform just oneiteration. More precisely, we assume that

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Un 1�0� � Un 1

�0� � Un, Pn 1�0� � 0

Un 1 � Un 1�1� , Pn 1 � Pn 1

�1� . (4.1)

The obtained scheme reads:

�CUn 1 � b1

SPn 1 � DUn 1 � b2

CUn 1 � CUn 1 � DTPn 1 � b1 � DTPn 1.(4.2)

Note that this is exactly the Yosida scheme adopted as a solver of the Navier-Stokes problem.In [2] it has been proved, for the Stokes problem, that the error introduced by the inexactsplitting for this scheme is �(�t). In the perspective of the present work, we could regard thisresult in a slightly different way: just one iteration of the block factorized preconditioner yieldsa first-order time-accurate solution. Just one application of the preconditioner is thereforesufficient to guarantee the global accuracy of the scheme.

Now, a reasonable question is if a similar result holds in the general case whenever a timediscretisation of order k� is adopted, and the block factorized preconditioner is applied k� times.In the present section we provide a positive answer to this question for the Stokes problem byinvestigating the spectral properties of the matrix � in (3.4) when S � S1 � (�t/�0)DM1DT.The following proposition represents the main contribution of the present work.

Proposition 3. Let C be s.p.d. and denote by � the spectral radius of the iteration matrix �,obtained selecting S � S1 in (3.4). Then,

FIG. 2. Comparison on the Kim and Moin test case with a first order time discretization (�t � 1/100) ofthe max number of iterations required by the pressure matrix system (BiCGStab) with different precon-ditioners, the Cahouet Chabard preconditioner, the preconditioner based on S � �tDM1DT and the notpreconditioned case, for different values of the mesh size h.


a) � � 1-

b) there exist two constants �1 and �2 depending on the space discretization adopted, suchthat

� �1�t

1 � �2�t. (4.3)

Proof. We have already pointed out that the spectral radius of � coincides with that of the“reduced” matrix:

�r � INp� S1� � INp

��0

�t�DM1DT�1DC1DT. (4.4)

We are therefore interested in estimating the spectrum of �r. Let us denote by � the Nu � Np

matrix M1/2DT and consider its singular value decomposition, i.e., set

� :� M1/2DT � U�VT (4.5)

where U and V are orthogonal matrices, of sizes Nu � Nu and Np � Np, respectively, and � isthe singular values matrix:

� � � �0

0�NuNp��Np�,

where �0 � diag(�i) is the Np � Np diagonal matrix of the singular values �i (i � 1, 2, . . . ,Np). Since the LBB condition is satisfied, � is a full rank matrix, therefore �i � 0 for all i� 1, . . . , Np.

We have

S1 ��t

�0DM1/2M1/2DT �

�t

�0V�0

2VT.

Since C1 � (�0/�t)1 (INu (�t/�0) M1A)1M1, we obtain

�r � INp� �V�0

2VT�1DM1/2M1/2 INu ��t

�0M1A1

M1/2M1/2DT

� INp� �V�0

2VT�1V�TUTM1/2INu ��t

�0M1A1

M1/2U�VT. (4.6)

Let us set Y � (INu (�t/�0)M1A)1. Observe that (V�0

2VT)1 � V�02VT so that, denoting

�inv � � �01

0�NuNp��Np� � ��0

2,

we obtain

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�r � INp� V�inv

T UTM1/2YM1/2U�VT. (4.7)

�† � V �invT UT, is actually the Moore-Penrose pseudo-inverse of �, and �†� � INp

. Therefore,we get

�r � INp� �†M1/2YM1/2� � �†M1/2�INu � Y �M1/2�. (4.8)

Let us now compute the eigenvalues of �r, by computing the Rayleigh quotients:

�i �wi

T�rwi

wiTwi

i � 1, 2, . . . , Np, (4.9)

�i being the ith eigenvalue of �r and wi its corresponding eigenvector. Without loss ofgenerality, suppose that wi

Twi � 1. For the sake of simplicity, we will drop the specification ofthe index i, provided no ambiguity arises.

Set z1 � M1/2�†Tw, z2 � M1/2�w, B � (INu Y). With this notation, we obtain

� � z1TMBz2 � �z1, Bz2�M. (4.10)

Since M and A are symmetric positive definite matrices, and �0 � 0, then �01M1A is s.p.d.

with respect to ( � , � )M, therefore, it has real positive eigenvalues �i, and this is also the casefor Y and B. In particular, denote by �i the eigenvalues of B. The eigenvectors �i of B form anorthonormal basis with respect to ( � , � )M. Suppose to expand z1 and z2 with respect to this basis,i.e.,

zj � �i�1

Nu

�i,j�i j � 1, 2

Since w has unit norm, we have

�z1, z2�M � �i�1

Nu

�i,1�i,2 � wT�†M1/2M1/2�w � �w, w� � 1. (4.11)

Moreover, we have

�i,1�i,2 0, i � 1, . . . , Nu. (4.12)

Indeed, since {�i} are orthonormal with respect to ( � , � )M, we have:

�i,j � �zj, �i�M � j � 1, 2, i � 1, . . . , Nu�.

Therefore,

�i,1�i,2 � z1TM�i�i

TMz2 � wT�†M1/2�i�iTM1/2�w � wT�†�i�i

T�w,


having set �i � M1/2�i. Since � is a full rank matrix, by a singular value decomposition wehave that

�† � ��T��1�T.

Therefore,

�i,1�i,2 � wT��T��1�T�i�iT�w � wT�qiqi

Tw

where we have set � � (�T�)1, which is a s.p.d. matrix and qi � �T�i � �Np. Observe thatthe matrix �i � qiqi

T is symmetric and non negative so that

�i,1�i,2 � �w, �iw��,

and (4.12) follows from the non-negativity of �i (see the Appendix).Now, from (4.10) we obtain

� � �i�1

Nu

�i�i,1�i,2. (4.13)

Denote in particular by � the spectral radius of B. Then, from (4.13), (4.12), (4.11) we obtain

�� i�1

Nu

��i,2�i,1� � � �i�1

Nu

�i,2�i,1 � �. (4.14)

From the definition of B, denoting by �i the eigenvalues of �01M1A, we have

�i � 1 � �1 � �t�i�1 �

�t�i

1 � �t�i, (4.15)

which shows that for a fixed �t � 0 we have

�� 1. (4.16)

Since the spectral radii of � and �r actually coincide, this implies point (a) of the thesis.Now, in order to make the dependence of the upper bound of �� on �t explicit, let us denote

by �� and � the maximum and the minimum eigenvalues of �01M1A, respectively. From (4.16)

it follows that

� �t��

1 � �t�, (4.17)

which completes the proof. y

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Numerical assessments have been carried out that confirm Proposition 3, using matrices builtfrom the discretization of the unit square by unstructured meshes, with the mesh size of abouth � 1/8 and h � 1/16. In Fig. 3 we illustrate the eigenvalues computed with Matlab (TheMathworks, Inc.). Actually, in this case M1A is block diagonal with 2 blocks, since we aredealing with a 2D problem. Each block has the same eigenvalues and we plot the eigenvaluesof just one block. We have also computed the spectral radii of the iteration matrix �r for the twodifferent mesh sizes adopted and for different time steps. The results obtained are illustrated inTable II. In particular, we compare the values of the iteration matrix spectral radius with theupper bound provided by the Proposition and computed on the basis of the numerical valuesillustrated in Fig. 3. The spectral radii are in the second and fifth column of the table for the caseh � 1/8 and h � 1/16, respectively. The corresponding upper bounds are in the third and sixthcolumn, respectively. By the comparison, it is evident that the bound is largely verified. Anothernumerical validation, based on a heuristic argument, which however provides significant results,

FIG. 3. Eigenvalues �i of M1A for an unstructured mesh (with a lumped mass matrix) on a unit squarewith h � 1/8 (left) and h � 1/16 (right). The eigenvalues have been computed with the built-in functioneig of Matlab.

TABLE II. Numerical evaluation of the spectral radius of the iteration matrix � for different time stepsand two mesh sizes (second and fifth column respectively).

�t �h�1/8

�t�� 1/8

1 � �t�1/8

�t�*1/8

1 � �t�*1/8 �h�1/16

�t�� 1/16

1 � �t�1/16

�t�*1/16

1 � �t�*1/16

2 0.9482 47.3987 0.9565 0.9877 197.9074 0.98871 0.9025 27.5823 0.9166 0.9759 115.2450 0.9778

1/2 0.8249 15.0217 0.8461 0.9531 62.7914 0.95651/4 0.7079 7.8616 0.7333 0.9114 32.8699 0.91661/8 0.5571 4.0248 0.5789 0.8399 16.8301 0.84611/16 0.3962 2.3067 0.4074 0.7388 8.5174 0.7333The theoretical bounds provided for the spectral radius by Proposition 3, computed on the basis of the eigenvaluesreported in Fig. 3, are illustrated in the third and sixth columns. The values of the heuristic indicator (4.18) are in thefourth and seventh column, respectively.


very close to the computed spectral radii, is the following. As a numerical indicator, weintroduce the ratio

�t�*

1 � �t�*(4.18)

where �* is a sort of “average value” for the eigenvalues �i. By observing Fig. 3, we set

�* � 11 for h �18

, �* � 44 for h �116

.

The numerical values corresponding to the indicator (4.18) are reported in the fourth and seventhcolumns of Table II. A comparison of these values with the spectral radii of �r for both the twomesh sizes considered, shows a good agreement. This heuristic argument confirms the assertionof Proposition 3.

The non-optimality of S as preconditioner of � is evident: � is actually bounded in terms ofthe eigenvalues �i of M1A, which depend on the mesh size h (see e.g., [16] and Fig. 3). Thisis in agreement with Fig. 2.

Despite the non-optimality of the preconditioner, Proposition 3 has the following interestingCorollary.

Corollary 1. Performing k� iterations of the preconditioned scheme (3.6) in the case of theStokes problem, with S � S1 yields an error reduction of �(�t k�). Therefore, if (1.4) is obtainedusing a time discretization scheme of order k�, then k� iterations of the block preconditionedscheme are enough to ensure the scheme to be accurate up to the order k�.

Indeed, denote by (u, p) the solution of the Navier-Stokes problem, (uh,�t, ph,�t) thenumerical solution associated with the solution of the unsplit system (1.4) and by (uh,�t

(k) , ph,�t(k) )

the solution given at the kth iteration of the preconditioned iterative scheme. Then, set

eh,�t�k� � u � uh,�t

�k� , �h,�t�k� � p � ph,�t

�k� .

Let us focus our attention on the velocity error eh,�t(k) . Similar considerations hold for the pressure

error. Suppose that the time discretization ensures an order k� to the unsplit solution, i.e. u uh,�t L�(L2(�)) � �(�t k�). Since uh,�t and uh,�t

(k) both belong to the finite element space Vh,

uh,�t�k� � uh,�t L��L2�� max

0�t�T

M 2 U�k� � Uunsplit 2,

U(k) and Uunsplit being the vectors of the nodal values of the solution at the kth iteration, and ofthe unsplit solution, respectively. Now, in the s.p.d. case we have that

U�k� � Uunsplit �k U�0� � Uunsplit 2,

where � is the spectral radius of the iteration matrix. Using Proposition 3, we have that U(k)

Uunsplit c�t k U(0) U . We therefore conclude that after k� iterations we have

504 VENEZIANI

eh,�t�k�� L��L2�� u � uh,�t L��L2�� uh,�t

�k�� uh,�t L��L2�� t k��,

yielding the desired k� order in time.

V. IMPLEMENTATION DETAILS AND NUMERICAL RESULTSA. Solving the Pressure Matrix Preconditioner

Inexact block factorizations treated both as solvers or preconditioners require the solution ofsystems involving the matrix S � DHDT. In this section, we want to investigate the nontrivialtask of selecting efficient strategies for the solution of these linear systems. In fact, the matrixmultiplications of the three factors of S cannot be carried out, in practice, because of fill in, sowe resort to preconditioned iterative methods. In this case, the crucial issue is the selection ofa good preconditioner P. A possible preconditioner for S is the following one. Suppose that DT

is a full-rank matrix, and let DT � QR be its QR factorization, Q being an orthogonal Nu � Nu

matrix and R an upper triangular Nu � Np matrix, respectively. On the basis of this decompo-sition, we have S � RTQTFQR. Now, consider the preconditioner for S given by P � RTFR,obtained by neglecting the orthogonal factors in the previous factorization. Linear systemsassociated with P are readily solvable thanks to the triangular pattern of R. A practical issue tobe addressed for this preconditioner is obviously the computation of the QR factorization.Different strategies can be adopted. In particular, in [24], P. Matstoms introduces a multifrontalmethod in the context of least squares problems. This method has been implemented in theQR27 Library and is able to take advantage from the (unstructured) sparsity of the matrix andto reduce fill-in in the factorization. The computational cost is very satisfactory. In our context,this makes the QR preconditioner really effective.

Another choice for the preconditioner, that can be justified for S1, refers to the Chorin-Temam differential method and consists of assuming

P ��t

�0�p,h,

�p,h being the discretization of the Laplace operator for the pressure. The computationalconvenience of this choice relies on the good properties of �p,h, which is s.p.d. and can beeffectively solved by Cholesky decomposition.

In Figs. 4 and 5 we compare the performances of �p,h and the QR preconditioner for the Kimand Moin problem (see Section V) on a unit time interval. We show the max number ofiterations required in the nonpreconditioned case, and using the Laplace preconditioner (o) andthe QR preconditioner (�), respectively. More precisely, in Fig. 4 we considered a structuredmesh, and in Fig. 5 an unstructured one. The conclusion that we draw from the figures is thaton structured meshes the QR preconditioner performs really better than the Laplace precondi-tioner, while for unstructured meshes the two strategies are comparable. Actually, in compu-tations on more complex geometries we verified that the Laplace preconditioner performs better.

In some special cases, an alternative to the solution of S with iterative methods can berepresented by the use of direct methods. In particular, suppose to deal again with S1

� (�t/�0)DM1DT, where we assume to have a diagonal (lumped) mass matrix. Now, in a waysimilar to the one adopted for the QR preconditioner, we compute

��t

�0M1/2DT � QR, (5.1)


where again Q is orthogonal and R is a Nu � Np upper triangular matrix. The latter QRdecomposition is possible, since M is a s.p.d. matrix and DT is a full-rank matrix by hypothesis.Then, the solution of S1p � b can be reconducted to that of successive triangular systems bytaking advantage of the orthogonality of Q. This strategy has the same cost as the solution ofthe Laplace operator after a Cholesky decomposition. Again, the computation of the decompo-sition (5.1) and in particular of the R factor (the only matrix actually needed) can be effectivelycarried out by means of the QR27 Library. Indeed, if M is diagonal, the matrix product�tM1/2DT is trivial. Moreover, the QR factorization can be carried out once for all at thebeginning of the computations, thus strongly reducing the computational cost. In Table III weillustrate, for several test cases, the ratio between the CPU times required by the QR direct solverand the Laplace preconditioner. We have indicated the final time of the computation. Whenever

FIG. 4. Comparison among the nonpreconditioned case (solid line), the Laplacian (o) and the QR(�)preconditioners for the Kim and Moin problem on a structured grid.

FIG. 5. Comparison among the non preconditioned case (solid line), the Laplacian (o) and the QR(�)preconditioners for the Kim and Moin problem on an unstructured grid.

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the computations reached a steady solution, we have indicated the instant in which the steadystate has been reached.

We conclude that the direct strategy based on the QR factorization, whenever applicable, isthe most effective approach. In the other cases, an iterative method with the Laplace precon-ditioner provide a good strategy for unstructured grids, whereas the QR preconditioner providesgood results for structured meshes in simple geometries.

B. Numerical Results

Let � be the domain (0.25, 1.25) � (0.5, 1.5). We consider the Navier-Stokes problem withDirichlet conditions associated with the exact Kim and Moin solution:

u1 � cos�2�x�sin�2�y�e8� 2� t, u2 � sin�2�x�cos�2�y�e8� 2� t

p � 0.25�cos�4�x� � cos�4�y��e16� 2� t. (5.2)

Here, u1 and u2 denote the velocity components along x and y, respectively, p is the pressure and� (which we set equal to 0.01) is the kinematic viscosity. Actually, in this case, the matrix C isnot symmetric definite positive, so that the analysis of the previous section is not applicable.However, we tested how many iterations are needed for the preconditioner to ensure the desiredtime accuracy in time. The code is based on �1iso�2 �1 elements (see e.g. [16]) with masslumping for the velocity mass matrices. Figure 6 illustrates the results for the Kim and Moin testcase, both with a second- and a third order time discretizations and a mesh size of h � 1/64.More precisely, for different time steps, we illustrate the errors for the velocity solution (in theL�(0, 10; L2(�)) norm, i.e., maxt�[0,10] e L2(�)(t)) and the pressure solution (in the L2(0, 10;L2(�)) norm). The results illustrate that 3 preconditioned iterations carried out in the case of aBDF2 time discretization are enough to ensure a second-order time accuracy. Similarly, in thecase of a BDF3 time discretization, 6 preconditioned iterations ensure a third-order timeaccuracy. The accuracy reduction for �t small is due to the space discretization error (which isconstant since the mesh is fixed). These results suggest that also for the Navier-Stokes problema small number of iterations number is needed, at least for a not too high Reynolds numbers.More precisely, the results suggest that the error reduction for each preconditioned iteration is�(�t1/2), which would imply that about 2k� iterations are required to maintain the time accuracyof a time discretization of order k�. This result is inferred by numerical evidence and is not yetsupported by theoretical results.

TABLE III. Ratio of CPU times required by the QR solver and the Laplace preconditioner for severaltest cases.

Test case

CPUQRSolver

CPU�h,p

Kim and Moin h � 1/64, �t � 1/32, Tfin � 1 0.90207Kim and Moin h � 1/64, �t � 1/64, Tfin � 1 0.92882Groovy channel flow h � 0.1, �t � 0.1, Tsteady � 343.5 0.56405Lid driven cavity Re � 100, Tsteady � 22.24 0.89114Lid driven cavity Re � 1000, Tsteady � 109.08 0.83532Lid driven cavity Re � 10000 Tfin � 500 0.79906


In Fig. 7 we illustrate the solution obtained on a lid driven cavity problem with Re � 100.We have considered the CC preconditioner applied until convergence has been reached (thetolerance was fixed to 1010) as well as the S1 preconditioner, that we have applied 5 times. Inthe figures we illustrate the velocity and the pressure at the steady state (after 504 time stepiterations for both the methods with a tolerance of 108 for testing the steady state condition).The results obtained with the two methods actually coincide. In fact, the maximum differencesfor the computed solutions are max�u1

CC u1Y� � 5.1 � 1013, max�u2

CC u2Y� � 8.0 � 1013

and max�pCC pY� � 2.7 � 1012. However, the ratio between the computation times is

CPUS1

CPUCC� 0.3369.

FIG. 6. Velocity (left) and pressure (right) errors for the solution of the Kim and Moin problem usingboth a second order time discretization (3 preconditioned iterations) and a third-order time discretization(6 preconditioned iterations).

FIG. 7. Velocity vectors (left) and pressure contours (right) for the steady solution in a lid driven cavity(Re � 100) computed by the S1 preconditioner applied 5 times. The CC preconditioner gives the sameresults.

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These results illustrate that the preconditioned pressure matrix method, applied without reachingthe convergence, still provides good results and however with a strongly reduced computationaleffort.

The numerical results of this article have been obtained on a Digital Alpha Workstations with400 MHz and 0.5G Ram, with a code implemented by the author.

APPENDIX

Let us consider the n � n matrix B symmetric and non negative (i.e., xTBx 0 for every x� �n). Moreover, let R be a n � n s. p. d. matrix. Let us prove that

s � �x, Bx�R � xTRBx 0

for every x. Matrix R can be factorized as R � VT�V, V being orthonormal and � being adiagonal matrix with positive entries. Therefore s � xTVT�VBx � xVT�VBVTVx, having set y� Vx. Matrix C � VBVT is symmetric non-negative and we can write

s � yT�Cy � wT�1/2C�1/2w,

w � �1/2y being a generic vector. Let us factorize C using the singular value decompositionC � TT�T, where � is a diagonal matrix with non-negative entries �i, sos � wT�1/2TT�T�1/2w.

Now, set U � T�1/2 and observe that the matrix A � U1�U is by construction similar to �by means of its eigenvector matrix U. On the other hand, U is such that UTU � �1, that means thatthe eigenvectors of A are an orthogonal basis (even if not orthonormal), � being diagonal. Now, letus expand the generic vector w with respect to the eigenvectors {ui}. We have

s � wTAw � �i, j�1

n

wiuiTAwjuj � �

i�1

n

�iwi2�1,

which is clearly non-negative, since �i 0 and �i � 0.

The author is greatly indebted with Prof. Alfio Quarteroni and Prof. Fausto Saleri for manyfruitful discussions and useful suggestions and with Prof. Luca Formaggia and Prof. RiccardoSacco for carefully reading the manuscript. The author also thanks Prof. Michel Fortin forsuggesting the links with the Cahouet-Chabard preconditioner and Dr. Pontus Matstoms forkindly providing the QR27 Library.

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Block factorized preconditioners for high-order accurate in time approximation of the Navier-Stokes equations