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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT The Krylov accelerated SIMPLER method for incompressible flow C. Vik and A. Saghir ISSN Reports of the Department of Applied Mathematical Analysis Delft 2002

2 Copyright 2002 by Department of Applied Mathematical Analysis, Delft, The Netherlands. No part of the Jornal may be reprodced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, withot the prior written permission from Department of Applied Mathematical Analysis, Delft University of Technology, The Netherlands.

3 [ The Krylov accelerated SIMPLER method for incompressible flow ] [C. Vik and A. Saghir] September 10, 2002 Abstract Nmerical modeling of the melting and combstion process is an important tool in gaining nderstanding of the physical and chemical phenomena which occr in a gasor oil-fired glass melting frnace. The incompressible Navier-Stokes eqations are sed to model the gas flow in the frnace. The discrete Navier-Stokes eqations are solved by the SIMPLER method. In or applications many SIMPLER iterations are necessary to obtain an accrate soltion. In this paper Krylov accelerated versions are proposed: GCR-SIMPLER. The properties of these methods are investigated for a simple two-dimensional flow. Thereafter the efficiency of the methods is compared for three-dimensional flows in indstrial glass melting frnaces. Keywords: SIMPLER method, Krylov acceleration, efficiency, incompressible Navier- Stokes 1 Introdction The increasing demand in qality, prodction efficiency and environmental isses drive glass prodcers in optimizing their melting frnaces. The qality demand is so high and the melting behavior so complex that a complete nderstanding of all important physical and chemical phenomena dring the melting process is reqired to make advances. At the TNO Institte of Applied Physics a CFD simlation model for gas- and oil-fired glass melting frnaces has been developed. This is a complete model for glass melting frnaces, describing the combstion space and glass bath, and predicting the effects on melting performance and glass qality. The model is sccessflly sed by the glass indstry and frnace manfactrers for prodct qality improvement, optimization of frnace designs and troble-shooting. The simlation of a complete glass melting frnace often reslts in large comptation times. One of the reasons for this is that the model ses the so-called SIMPLER pressrecorrection method to solve the incompressible Navier-Stokes eqations. It is well known 1

4 that the SIMPLER method often reqires many iterations. To redce the large comptation times of the SIMPLER method, a Krylov sbspace acceleration of the SIMPLER method has been developed. The GCR Generalized Conjgate Residals method has been sed for this prpose, since GCR can be applied to the non-symmetric matrices which reslt from discretization of the Navier-Stokes eqations. The new method presented in this paper is called GCR-SIMPLER. 2 Description of the mathematical model We compte nmerical soltions of the steady state Navier-Stokes eqations: ν + grad + gradp = f, div = 0, combined with appropriate bondary conditions. The vector field represents the velocity, p represents the pressre and ν is the viscosity. The Stokes eqations are the Navier-Stokes eqations withot the non-linear term. After discretization of the incompressible Stokes eqations one obtains the following linear system: Q G G T 0 p = b1 b 2, 1 where is the algebraic vector of velocity components and the algebraic vector p contains the pressre nknowns. In the remainder of this paper, this system is abbreviated as Ax = b. Discretization of the continity eqation leads to a zero block on the main diagonal of A, so the reslting linear system is indefinite and symmetric for the Stokes eqations. This leads to serios problems when linear solvers are sed. Varios methods are known to overcome these difficlties: the pressre-matrix method [4], Uzawa method [17, 21, 6, 8], SIMPLE-type methods[14, 10], penalty method [2], pressre correction method [20], PISO method [1], etc. For an overview of these methods we refer to [16] Section 9.6. In CFD packages, a poplar method is the SIMPLE method proposed by Patankar and Spalding [15] or one of its variants SIMPLER [14], SIMPLEST [26], or SIMPLEC [19]. In many applications the SIMPLE method needs many iterations before an accrate soltion is reached. Varios athors consider a mltigrid acceleration of the SIMPLE method [24, 11, 3]. In this paper we consider a Krylov sbspace acceleration of the SIMPLER method [23]. The reason for this is that Krylov methods have only a small amont of overhead costs and are easy to implement in an existing CFD package. Althogh the discretized Stokes eqation leads to a symmetric coefficient matrix, we se a Krylov sbspace method sitable for non-symmetric matrices, becase we also apply the reslting method to the discrete Navier-Stokes eqations, where a non-symmetric coefficient matrix occrs. Recently, saddle point preconditioners for Krylov sbspace methods have been proposed to solve the discretized Navier-Stokes eqations. For a srvey we refer to [9]. 2

5 3 SIMPLE type methods as distribtive iterative methods In order to combine the SIMPLE-type methods with a Krylov solver we first write these methods as a distribtive iterative method [26, 24]. 3.1 SIMPLE method The diagonal of the matrix Q is denoted by D and R G T D 1 G. The SIMPLE method as proposed by Patankar [14] is given by the following algorithm: SIMPLE algorithm 1. Choose an initial estimate p. 2. Solve Q = b 1 Gp. 3. Solve Rδp = b 2 G T. 4. Compte = D 1 Gδp and p := p + δp. 5. If not converged take p = p and go to 2. The soltions of the systems given in 2 and 3 are obtained by a small nmber of iterations with a Block Gass-Seidel method TDMA solver [14]. The SIMPLE method can also be seen as a distribtive iterative method. Instead of solving the system Ax = b the system AB R y = b, x = B R y will be solved. Choosing B R and M R as: I D B R = 1 G Q 0, M 0 I R = G T, 2 R and sing the splitting AB R = M R N R the following iteration is obtained SIMPLE method [24]: x k+1 = x k + B R M R 1 b Ax k, k = 1, 2,..., niter. In the SIMPLE method the matrix A is mltiplied from the right with B R and the reslting matrix is split. We can also se a mltiplication from the left with B L and se the splitting M L where the matrices are given by B L = I 0 G T D 1 I, M L = Q G 0 R Using the splitting B L A = M L N L we can se the iteration x k+1 = x k + M L 1 B L b Ax k, k = 1, 2,..., niter. This is a new SIMPLE type method, which we denote by SIMPLEL. 3. 3

6 3.2 SIMPLER method First the SIMPLER method is explained. Thereafter the SIMPLER method is written as a distribtive iterative solver. Sppose the velocity vector is known. Then an easy calclation shows that p is a soltion of the system: Rp = b 2 G T D 1 D Q + b 1. This idea is sed in the SIMPLER method. When k is known, p k+1 and k+1 are calclated as follows: SIMPLER algorithm 1. Solve Rp = b 2 G T D 1 D Q k + b Solve Q = b 1 Gp. 3. Solve Rδp = b 2 G T. 4. Compte k+1 = D 1 Gδp and p k+1 = p + δp. One iteration of the SIMPLER algorithm is approximately 1.3 times as expensive as one SIMPLE iteration. Steps 2, 3, and 4 of both methods are comparable. Using 3 it is easy to check that N L is given by N L = 0 0 G T D 1 D Q 0. 4 Now the first and second step of the SIMPLER algorithm can be written as: M L p = N L k p k + B L b1 b 2. 5 This is eqivalent to M L p = M L k p k + B L b1 b 2 M L N L k p k. 6 Mltiplying this eqation with M L 1 leads to p = k p k + M L 1 B L b1 b 2 A k p k. 7 After this step a SIMPLE iteration follows. So the new soltion is given by k+1 p k+1 = p 1 b1 + B R M R A b 2 p. 8 4

7 Combining both steps shows that one SIMPLER iteration can be denoted by where P S is given by k+1 p k+1 = k p k This expression can also be written as + P S b1 b 2 A k p k P S = B R M R 1 B R M R 1 AM L 1 B L + M L 1 B L. P S = B R M R 1 B L 1 Q 0 0 2R B R 1 M L 1 B L., 9 Smmarizing we note that the SIMPLER method can be written as a distribtive iterative method: x k+1 = x k + B R M 1 R B 1 L TB 1 R M 1 L B Lb Ax k, where T is the block diagonal part of the matrix M L + M R A. Note that the SIMPLER method is closely related to the Symmetric Block Gass-Seidel method. Frthermore, when Q is symmetric it follows from the definition that B L = B T R and M L = M T R, which implies that P S is symmetric. So for the incompressible Stokes eqation the MINRES method combined with the SIMPLER method as preconditioner can be sed. 4 A saddle point preconditioner In this section we describe one of the saddle point preconditioners proposed by Elman [7]. Thereafter we write the iteration matrices of this preconditioner and the SIMPLE preconditioner in the same way. This enables s to give a theoretical comparison of both preconditioners. In Section 6.2 these preconditioners are compared from nmerical point of view. In [7] saddle point preconditioners of the following form are considered: P E = Q G 0 X It is easy to show that AP E = I 0 G T Q 1 G T Q 1 GX 1, so that the eigenvales are: {1} σg T Q 1 GX 1. 5

8 In [7] varios choices for the matrix X are considered. It appears that a balance shold be fond between fast convergence, σg T Q 1 GX 1 {1}, and the amont of work to apply the preconditioner to a vector. For the SIMPLE preconditioner we can compte M R 1 : M R 1 = Q 1 0 R 1 G T Q 1 R 1 So the iteration matrix AB R M R 1 can be written as: I I QD 1 GR 1 G T Q 1 I QD 1 GR 1 0 I which implies that the eigenvales of AB R M R 1 are: {1} σi I QD 1 GR 1 G T Q 1. We note that the Elman preconditioner converges in one iteration when X = G T Q 1 G, whereas the SIMPLE preconditioner converges in one iteration when D = Q. In general X is an approximation of G T Q 1 G and D is an approximation of Q. Comparing both preconditioners, it appears that for the SIMPLE preconditioner the iteration matrix converges to the identity matrix for D Q, whereas the iteration matrix for the Elman preconditioner converges to a block lower trianglar matrix so the iteration matrix is non-normal for X G T Q 1 G.., 5 The GCR acceleration method In this section a Krylov acceleration of the preconditioners given in Section 3 and 4 is given. Many Krylov sbspace methods are known to solve non-symmetric linear systems. We choose the GCR method [5] becase the method is robst, minimizes the residal and allows a variable preconditioner [18, 22]. This final property is very important, since in practice the inverse of the matrices occrring in the preconditioners are only compted approximately. So, the preconditioner P k is a different operator in every iteration. GCR algorithm r 0 = b Ax 0 for k = 0, 1,...,ngcr s k+1 = P k r k v k+1 = As k+1 for i = 0, 1,...,k v k+1 = v k+1 v k+1, v i v i 6

9 s k+1 = s k+1 v k+1, v i s i end for v k+1 = v k+1 / v k+1 2 s k+1 = s k+1 / v k+1 2 x k+1 = x k + r k, v k+1 s k+1 r k+1 = r k r k, v k+1 v k+1 end for For P k one can choose B R M R 1 GCR-SIMPLE, P S GCR-SIMPLER, or P E GCR- Elman. De to the modified Gram-Schmidt orthogonalization, the amont of work and memory increases when the nmber of iterations grows. To bond these qantities the method is trncated or restarted after a small nmber of iterations. Comparing the amont of work with that of the SIMPLE method, we note that if the GCR-SIMPLE method is restarted after ngcr iterations, then ngcr 2 vector-pdates and ngcr 2 /2 inner-prodcts extra are reqired. Frthermore, an additional 2ngcr vectors shold be stored in memory. When ngcr is small these costs are negligible. For the SIMPLER preconditioners it appears that a diagonal scaling is beneficial for the convergence of the GCR-SIMPLER method. To implement this the following adaptations are made: compte D AB = diagab R and se r 0 = D 1 ABb Ax 0, s k+1 = BM 1 k D ABr k, and v k+1 = D 1 AB Ask+1. The discretization of the Navier-Stokes eqations gives a non-linear system de to the convection terms. The discretization eqations for the velocities can be written as follows: Q + Gp = b Varios methods can be chosen to linearize Q, like the Newton-Raphson method or the Picard iteration method. We have sed the Picard iteration method where Q k+1 is approximated by Q k. A non-symmetric linear system is obtained with the same strctre as the discretized Stokes eqations. Now, the GCR-SIMPLER algorithm for the Navier-Stokes eqations can be smmarized as follows: x 0 gessed vale for k = 0, 1, 2,..., niter solve Ax k x k+1 = b with GCR-SIMPLER end for The other preconditioners can be sed in the same way. Dring each iteration we do not need to solve this eqation ntil convergence becase the matrix A is defined sing an approximation of x k. This has the advantage that a small vale of ngcr can be chosen which leads to low memory reqirements. The optimal vale of ngcr can be different for each problem. 7

10 6 Nmerical experiments In this section attention will be given to the application of the varios methods to solve the incompressible Navier-Stokes eqations. We will first investigate the properties of SIMPLER and GCR-SIMPLER for a 2D Navier-Stokes flow between two flat plates. Thereafter we compare the GCR method sing the preconditioners proposed in Section 3 and 4 for a Poiseille flow and a backward facing step problem. Finally, in order to compare the efficiency for more realistic test problems, the Ford Nashville float glass frnace is sed. In the measrements the following qantities are sed: CP Utime: exection time of a sed method measred in seconds on an HP-735 in Section 6.1 and on an HP-J210 in Section 6.3, resid: absolte sm of residals for a given variable, niter: nmber of iterations. 6.1 Nmerical properties of the SIMPLE type methods In this section we present some reslts obtained when applying the SIMPLER and the GCR-SIMPLER method to the flow between two flat plates with distance D = 10 cm and length L = 500 cm. For this test problem an eqidistant grid will be sed. To apply the SIMPLER and the GCR-SIMPLER methods we first define some defalt vales of parameters sed in these methods. For both methods the termination criterion is: stop when the sm of the absolte residals of each variable is less than or eqal to The relaxation factor for the pressre is always 1. The SIMPLER method will be sed with relaxation factors eqal to 0.8 for the velocities 1 and 2. For the GCR-SIMPLER method ngcr is taken eqal to 3 and the relaxation factors for the velocities 1 and 2 are eqal to 1. The defalt TDMA solver is PLANE TDMA. For a motivation of these vales we refer to [23]. First we investigate the dependence of GCR-SIMPLER on the vale of ngcr. The reslts are given in Table 1. When ngcr increases the nmber of GCR-SIMPLER iterations decreases, bt every iteration becomes more expensive. On the grid we see that the CPU time is more or less the same for all vales of ngcr. For the grid there are larger differences. The choice ngcr = 14 leads to a minimal amont of CPU time; however, many vectors shold be stored in memory. Therefore the vale ngcr = 3 is a good compromise. When convergence problems occr for the GCR-SIMPLER method it helps to increase the vale of ngcr. In Table 2 the reslts are given for varios grid sizes. Both methods need more iterations when the grid size increases. For a small grid size the CPU times are comparable, whereas for a large grid size GCR-SIMPLER needs less CPU time than the SIMPLER method. For the grid the aspect ratio of the finite volmes is eqal to 50. 8

11 ngcr Grid Grid niter CPU time niter CPU time Table 1: Reslts of the GCR-SIMPLER method for varios vales of ngcr Grid size SIMPLER GCR-SIMPLER niter CPU time niter CPU time Table 2: Reslts for varios grid sizes 6.2 Comparison of the varios preconditioners In this section we present convergence reslts for the preconditioned GCR method, withot restarting or trncation, applied to some test problems. These experiments are done in Matlab. For the discretization we se the Matlab codes 1 given in [25]. In or SIMPLER preconditioners we always take D = diagq. From [7] it appears that X = hd I, for a ν niform grid of width h in d dimensions, is a good preconditioner as long as ν 1. Since this preconditioner is easy to se, we take X = γi, where γ is chosen sch that the convergence of GCR-Elman is optimal. We cannot se X = hd I becase the step-size in x and ν y-direction is different. We only report nmber of iterations to observe the convergence behavior. All inverse matrices in the preconditioners are compted exactly. In real applications approximate inverses shold be sed in order to save flops and memory. Channel flow Or first test problem in this section is a flow in a channel with width 2 and the length is varied. At the inflow bondary we se a parabolic profile, whereas a no-slip bondary condition is sed at the side walls. As otflow bondary conditions we se homogeneos Nemann conditions for both velocity components. For the discretization we se a staggered grid arrangement [25, 12]. The Dirichlet bondary conditions are inclded as extra eqations in the linear system. We start the GCR method with x 0 = 0 and stop if rk ɛ. b As a defalt we take ɛ = ta.twi.tdelft.nl/sers/wesselin/cfdbook.html 9

12 In many channel flows the length of the channel is mch larger than the width of the channel. From or experiments we know that a nmber of preconditioners break down or have a bad convergence behavior when stretched cells occrs. Therefore we vary the length of the channel two orders of magnitde and report the reslts for the proposed preconditioners in Table 3. These reslts are obtained for the Stokes eqations on a grid. Note that the nmber of iterations increases for the ILU and Elman preconditioners for increasing length, whereas the nmber of iterations is constant or decreasing for the SIMPLER preconditioners. Length ILU SIMPLE SIMPLER Elman Table 3: Nmber of iterations of the preconditioned GCR method for the Stokes eqations We have also sed the Oseen eqations: ν + w grad + gradp = f, div = 0, where w is a given velocity field. These eqations are sed dring a Picard iteration to solve the non-linear Navier-Stokes eqations. We take w eqal to the velocity of the Poiseille flow. From Table 4 and 5 it appears that the nmber of iterations increases for decreasing ν. The behavior for increasing length is comparable for the Stokes and Oseen eqations. Length ILU SIMPLE SIMPLER Elman Table 4: Nmber of iterations of the preconditioned GCR method for the Oseen eqations for ν = 1 Finally, solving the Navier-Stokes eqations with a Picard iteration the intermediate Stokes and Oseen eqations are solved with a relative large ɛ. Therefore we also give the reslts for ɛ = 10 2 in Table 6. It appears that for a low accracy there is a large growth in the nmber of iterations for the ILU and Elman preconditioners for increasing length of the channel. 10

13 Length ILU SIMPLE SIMPLER Elman Table 5: Nmber of iterations of the preconditioned GCR method for the Oseen eqations for ν = 0.1 Length ILU SIMPLE SIMPLER Elman Table 6: Nmber of iterations of the preconditioned GCR method for the Oseen eqations for ν = 0.1 and ɛ = 10 2 Backward facing step Or second test problem is a backward facing step problem. The length of the channel is 25 and the width is 2. At the inflow bondary we have a parabolic profile at the pper half part and a no-slip condition at the lower half part of the bondary. In the Oseen eqations we take w eqal to zero in the lower part of the channel and eqal to the Poiseille flow velocities in the pper part of the channel. The horizontal velocity contors of w and are given in Figre 1 and 2. The iteration conts are given in Table 7. Note that the SIMPLER preconditioner is insensitive to the choice of ν. For the other preconditioners the nmber of iterations increases for decreasing ν. This behavior is also reported in [7] for the Elman preconditioner. Stokes Oseen Preconditioner ν = 0.25 ν = ILU SIMPLE SIMPLER Elman Table 7: Nmber of iterations of the preconditioned GCR method for the backward facing step problem 11

14 Figre 1: Contor plot of horizontal component of w sed in the backward facing step problem Figre 2: Contor plot of the horizontal component of the soltion of the backward facing step problem 6.3 The Ford Nashville frnace In this section the SIMPLER and the GCR-SIMPLER method are sed to simlate the combstion chamber of the Ford frnace [13]. In the model for this frnace, the combstion of natral gas is described by the conserved scalar approach to high temperatre, nonpremixed combstion and the chemistry is described with a one-step global reaction. The geometry of the Ford frnace is sketched in Figre 3. The internal length, width and maximm height of the combstion chamber are m. The same convergence Figre 3: Geometry of the Ford float glass frnace criterion is sed for each method. In this problem the iteration process is stopped when the absolte sm of the residals of each variable is less than or eqal to The finite volme grid consists of = points. The same relaxation factors are sed for both methods. In the SIMPLER and the GCR-SIMPLER method the same SPACE TDMA solver is sed. The first simlation has been done sing the GCR-SIMPLER method. The reslts are: niter = 3390, CP U time 3.3 days. Using the SIMPLER method the simlation has been stopped after 7.5 days, becase the maximm nmber of iterations has been reached. We see again a large decrease in CPU time when the Krylov 12

15 acceleration is sed. The temperatre contors in a plane jst above the glass srface are shown in Figre 4. A comparison of the compted and measred qantities for this frnace is given in [13]. For more reslts concerning the convergence behavior of the varios methods applied to the IFRF frnace we refer to [23]. Flid Temperatre [ o C]: Figre 4: The temperatre contors of the Ford float frnace sing the GCR-SIMPLER method 6.4 Memory Using the GCR-SIMPLER method instead of the SIMPLER method leads to more memory. In Table 8 the memory reqirements are given for varios problems. For a threedimensional problem the increase is approximately 50 %. When ngcr is increased the CPU time may decrease bt the memory reqirements increase. problem SIMPLER GCR-SIMPLER ngcr = 3 Flat plates IFRF frnace Ford frnace Table 8: Memory reqirements for varios problems measred in Megabytes 7 Conclsions An efficient method to simlate glass-melting frnaces is considered. In this method the incompressible Navier-Stokes eqations are sed. SIMPLE-type methods are very poplar 13

16 to solve the discretized incompressible Navier-Stokes eqations. In this paper SIMPLE and SIMPLER are rewritten as classical distribtive iteration methods for linear systems. Two other preconditioners are considered ILU and Elman [7]. As Krylov acceleration the GCR method is sed. Firstly the dependence of the GCR-SIMPLER methods on grid-size and ngcr is investigated by nmerical experiments. These insights are sed to propose a nmber of defalt parameters ngcr = 3, TDMA solver, otlet bondary condition, etc. for these methods. Secondly the convergence behavior of the ILU, SIMPLER, and Elman preconditioner are compared for two test problems: a channel flow and a backward facing step flow. It appears that GCR-SIMPLER is only weakly dependent on the stretching of the grid and the viscosity. Points of crrent research are: inclsion of more advanced preconditioners proposed by [7] and comparison of the memory reqirements and CPU time of the proposed preconditioners for practical problems. Finally, the efficiency of SIMPLER and GCR-SIMPLER is compared sing a simlation of an indstrial frnace. For these simlations, where the grid has high aspect ratios, the GCR-SIMPLER method appears to be three times as fast as the SIMPLER method. Frthermore, larger relaxation factors can be sed for GCR-SIMPLER, which leads to a still higher efficiency. After convergence the qality of the compted reslts velocity, pressre, trblence qantities, etc is comparable. GCR-SIMPLER reqires more memory than the SIMPLER method. References [1] I.E. Barten. Comparison of SIMPLE- and PISO-type algorithms for transient flows. Int J. Nm. Meth. in Flids, 26: , [2] M. Bercovier. Pertrbation of a mixed variational problem, applications to mixed finite element methods. R.A.I.R.O. Anal. Nmér., 12: , [3] D. Braess and D. Sarazin. An efficient smoother for the Stokes problem. App. Nm. Math., 23:3 19, [4] J. Cahoet and J.P. Chabard. Some fast 3-D finite element solvers for the generalized Stokes problem. Int. J. Nm. Methods Flids, 8: , [5] S.C. Eisenstat, H.C. Elman, and M.H. Schltz. Variational iterative methods for nonsymmetric systems of linear eqations. SIAM J. Nm. Anal., 20: , [6] H.C. Elman. Mltigrid and Krylov sbspace methods for the discrete Navier-Stokes eqations. Int. J. Nm. Meth. Flids, 22: ,

17 [7] H.C. Elman. Preconditioning for the steady-state Navier-Stokes eqations with low viscosity. SIAM J. Sci. Compt., 20: , [8] H.C. Elman and G.H. Golb. Inexact and preconditioned Uzawa algorithms for saddlepoint problems. SIAM J. Nm. Anal., 31: , [9] H.C. Elman, D.J. Silvester, and A.J. Wathen. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes eqations. Nmer. Math., 90: , [10] J.H. Ferziger and M. Peric. Comptational methods for flid dynamics second edition. Springer, Berlin, [11] T. Gjesdal and M.E.H. Lossis. Comparison of pressre correction smoothers for mltigrid soltion of incompressible flow. Int. J. Nm. Meth. Flids, 25: , [12] F.H. Harlow and J.E. Welch. Nmerical calclation of time-dependent viscos incompressible flow of flid with a free srface. The Physics of Flids, 8: , [13] A.M. Lankhorst, G.P. Boerstoel, H.P.H. Mysenberg, and K.K. Koram. Complete simlation of the glass tank and combstion chamber of the former Ford Nashville float frnace inclding melter, refiner and working end. In Forth Int. Seminar on Mathematical Simlation in the Glass Melting, [14] S.V. Patankar. Nmerical heat transfer and flid flow. McGraw-Hill, New York, [15] S.V. Patankar and D.B. Spalding. A calclation procedre for heat and mass transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer, 15: , [16] A. Qarteroni and A. Valli. Nmerical approximation of partial differential eqations. Springer series in Comptational Mathematics 23. Springer Verlag, Berlin, [17] R. Temam. Navier-Stokes eqations. Theory and Nmerical Analysis, 3rd edition. North-Holland, Amsterdam, [18] H.A. van der Vorst and C. Vik. GMRESR: a family of nested GMRES methods. Nm. Lin. Alg. Appl., 1: , [19] J.P. Van Doormaal and G.D. Raithby. Enhancements of the SIMPLE method for predicting incompressible flid flows. Nm. Heat Transfer, 7: , [20] J. van Kan. A second-order accrate pressre-correction scheme for viscos incompressible flow. SIAM J. Sci. Stat. Compt., 7: ,

18 [21] C. Vincent and R. Boyer. A preconditioned conjgate gradient Uzawa-type method for the soltion of the Stokes problem by mixed Q1-P0 stabilized finite elements. Int. J. Nm. Meth. Flids, 14: , [22] C. Vik. New insights in GMRES-like methods with variable preconditioners. J. Comp. Appl. Math., 61: , [23] C. Vik, A. Saghir, and G.P. Boerstoel. The Krylov accelerated SIMPLER method for flow problems in indstrial frnaces. Int. J. for Nm. Meth. Flids, 33: , [24] P. Wesseling. An introdction to mltigrid methods. Wiley, Chichester, available at [25] P. Wesseling. Principles of Comptational Flid Dynamics. Springer Series in Comptational Mathematics, Vol.29. Springer, Heidelberg, [26] G. Wittm. Mlti-grid methods for Stokes and Navier-Stokes eqations with transforming smoothers: Algorithms and nmerical reslts. Nmer. Math., 54: ,