Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman … · 2014-05-12 · Advances in Fluid Mechanics III 183 whereas the discrete pressure belongs to the space Qh :— J space

On the performance of smoothers in

coupled multigrid methods for the

solution of incompressible Navier-Stokes

equations on parallel computers

V.JohnInstitut fur Analysis undNumerik,Otto-von-Guericke-Universitdt Magdeburg, Germany.

Abstract

We present a comparison of Vanka and Braess-Sarazin smoothers in coupled mul-tigrid methods for the solution of incompressible Navier-Stokes equations describ-ing two-dimensional flows past obstacles. The numerical tests are based on theCrank-Nicolson time discretization and the nonconforming Pi/Po-finite elementdiscretization in space on unstructured grids. All computations were performed on32 processors of a MIMD parallel computer. The Vanka smoother turned out to bein general more efficient in this framework.

1 Introduction

The efficient numerical solution of incompressible Navier-Stokes equations playsan important role in the numerical simulation of many complex problems in fluidmechanics. Benchmark computations (Schafer and Turek [1]) show that coupledmultigrid methods are one of the best class of solvers known so far for the regimeof laminar flows. The efficiency of multigrid methods is considerably determinedby the smoother. There are essentially two classes of smoothers considered inthe literature. The first class, called Vanka-type (Vanka [2]), local (Turek [3]),coupled or segregated (Paisley and Bhatti [4]), can be considered as block Gauss-Seidel method where one block consists of a small number of degrees of freedom.The characteristic feature of this type of smoother is that in each smoothing stepa large number of small linear systems of equations has to be solved. The secondclass, called Braess-Sarazin-type (Braess and Sarazin [5]), global, decoupled or

Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9

182 Advances in Fluid Mechanics III

segregated smoothers, solves in each smoothing step one large linear system ofequations. This system is solved with a pressure Schur complement approach.

Numerical studies of the behaviour of both types of smoothers in differentframeworks can be found in several papers. However, we could find direct com-parisons only in [4] and John and Tobiska [6]. Both studies came to somewhatdifferent results, for a detailed discussion of these results see Section 3, Remark 2.Thus, smoothers in coupled multigrid methods need still much more investigationsin order to achieve a general evidence about their behaviour and efficiency.

2 The discretization of the Navier-Stokes equations

We consider the incompressible Navier-Stokes equations

-^ -z/Au+ (u- V)u+ Vp = f in nx (0,71,

V-u = 0 inn x (0,T],

-z/Vu • nano + P̂ d̂ o = 0 on dtto x (0, T],u = UQ in n for t = 0,

where n denotes a bounded domain in IR^ with boundary dtl — 8&D U d£lo,dtin H 3n<3 = 0, u the velocity, p the pressure, z/ the kinematic viscosity of thefluid, g a Dirichlet boundary condition on the Dirichlet boundary <9n#, d^o theoutflow boundary with the outward directing normal n^no' ̂o an initial velocity,T the end of the time interval, and f represents exterior forces. We study also thesteady state Navier-Stokes equations where du/dt = 0 in eqn. (1).

First, eqn. (1) is discretized in time by the Crank-Nicolson scheme. Thearising nonlinear equation in each time step is linearized by a fixed point iteration.In each nonlinear iteration step, a so-called Oseen equation

in n,

in n, (2)

= 0 on^o,

with the given iterate (u", p") has to be solved. Here, A/ denotes the length of thecurrent time step and f is the sum off and terms arising in the temporal discretiza-tion of the equation. Problem (2) is discretized in space by the nonconformingjPi/jPo-finite element discretization from Crouzeix and Ravi art [7]. To apply thisdiscretization, Q is decomposed into triangles and, if necessary, the boundary <9Qis approximated by a polygon with the vertices of the polygon on <9Q. The discretevelocity is computed in the finite element space

_ f space of vector-valued piecewise linear functions which arer h I \

1 continuous at the midpoints of edges of the triangulation


Advances in Fluid Mechanics III 183

whereas the discrete pressure belongs to the space

Qh :— J space of piecewise constant functions >.

An additional stabilization becomes necessary for convection dominated flows.We use the Samarskij upwinding stabilization analyzed by Schieweck and Tobiska[8] for the nonconforming Pi/Po-ftnite element discretization of the steady stateNavier-Stokes equations.

Altogether, the linearization and discretization of the incompressible Navier-Stokes equations require the approximate solution of large and sparse saddle pointproblems of the form

u \ _ ( A B\( u \ _ f fP ) " \ B? 0

3 The coupled multigrid solver

Coupled multigrid methods compute the solution for both types of unknowns (ve-locity and pressure) of eqn. (3) simultaneously. The parallel efficiency of multigridmethods is not always convincing. Often, the multigrid V-cycle shows a good par-allel but a not satisfying numerical efficiency whereas the situation in the W-cycleis vice versa. Numerical studies, e.g. in [6], show that the F-cycle is a good com-promise on parallel computers. For this reason, we will present only results for theF-cycle.

3.1 The Vanka smoother

The (cell wise) Vanka smoother, originally proposed by Vanka [2] for finite dif-ference discretizations, can be considered as block Gauss-Seidel method where ablock corresponds to all degrees of freedom which are connected with one meshcell K. For the nonconforming P]/Po-finiteelement discretization, these are sixvelocity degrees of freedom (two at the midpoint of each edge) and one pressuredegree of freedom. Thus, a smoothing step with the Vanka smoother consists ina loop over all triangles where in each triangle a 7 x 7 linear system of equationshas to be solved. The degrees of freedom are updated in a Gauss-Seidel manner.Since each velocity degree of freedom in the interior of Q belongs to two triangles,it is updated twice in one smoothing step.

We denote by AK the block of the matrix A which is connected with thedegrees of freedom of the triangle K, i.e. the intersection of the rows and columnsof A which are associated with A'. We define

, _


1 84 Advances in Fluid Mechanics III

Similarly, we denote by (•)# the restriction of a vector on the rows correspondingto the degrees of freedom connected with K. The Vanka smoother updates thevelocity and pressure values in the triangle K by

P K V P K \\ 3 P K

In order to avoid communications within a smoothing iteration, we apply theVanka smoother in parallel on each processor. After each smoothing iteration,the values at the interfaces are averaged by one communication step. The num-bering of the triangles influences the behaviour of the Vanka smoother. In ourtests, the triangles are ordered lexicographically: a triangle with barycentre coor-dinates (%o, 2/0) is predecessor of a triangle with barycentre coordinates (#1,2/1)

iff (#o < %i) V ((#o = %i) A (2/0 < 2/i))-

3.2 The Braess-Sarazin smoother

This smoother was introduced by Braess and Sarazin in [5]. Let vf* be given. Then,the Braess-Sarazin smoother solves in one smoothing step the global linear system

aC B \ / %k+i \ ̂ / y _ (^ _ #C)̂

B? 0 /\ /+!/ \ 2

where (aC)~* is an approximation to A~*. The matrix C should be easier invert-ible than A. Solving the upper equation for !/+* and inserting the result into thelower equation, we have to solve the pressure Schur complement system

(BFc~*B)p*+* — -(B^C~^(Au* - f) + a(g - B̂ V')). (5)

The efficient solution of eqn. (5) is the core of an efficient solution of eqn. (4). Inthe application of the Braess-Sarazin smoother, system (5) needs to be solved onlyapproximately. After having computed p^+* from eqn. (5), i^+* can be obtainedfrom the upper equation of system (4).

The Braess-Sarazin smoother belongs to the class of pressure Schur comple-ment schemes whose most famous representative is SIMPLE by Patankar andSpalding [9]. SIMPLE is also often used as smoother in coupled multigrid meth-ods. However, both types of smoothers differ in important properties, e.g. theerror in the Braess-Sarazin smoother depends only on the old velocity u* but noton the old pressure p** like in SIMPLE. The convergence of the coupled multigridW-cycle with the Braess-Sarazin smoother can be proven for the Stokes equa-tions, C — I (identity), and inf-sup stable conforming and nonconforming finiteelement discretizations if a is chosen large enough, [5, 10]. It can be observed alsofor the Navier-Stokes equations that a has to be sufficiently large, see [6]. In [5]there is an example which shows that SIMPLE in general does not possess goodsmoothing properties.

An important point is the choice of the approximation (aC)~* of A~*. Inthe steady state Oseen equations, A is in general nonsymmetric. We used in the


Advances in Fluid Mechanics III 185

numerical tests presented below (aC)~* — [ILU(Q)(A)]~*, i.e. a block ILU-decomposition of A without additional fill-in, and a — 1. A block corresponds toall velocity degrees of freedom which are stored on one processor and within theblocks a lexicographical ordering is used. The arising Schur complement matrixB̂ [ILU(0}(A)}~̂ -B cannot be stored explicitely and system (5) is solved ap-proximately with 3 steps of GMRES. Since the system matrix in (5) is only knownimplicitly, we could not construct an appropriate preconditioner for GMRES. An-other drawback of this approach are additional memory requirements which areconnected with GMRES.

The situation for the time dependent Oseen equations is much different. There,the matrix A is, at least for small time steps, diagonally dominant. Thus, wecan use (aC~*) = diag(A)~*. In addition, the nonconforming Pi/Po-finite el-ement discretization offers an easy way to store the Schur complement matrixJ3̂ (diag(A))~ij0 explicitly. This matrix is symmetric and positive definite. Forthe approximate solution of eqn. (5), we used 3 steps of a preconditioned conjugategradient method with diagonal preconditioner.

The application of the Braess-Sarazin smoother requires the choice of muchmore parameters than the use of the Vanka smoother. A good choice of this set ofparameters might be difficult in practice and can be accomplished often only bynumerical tests.

The grid transfer operations are usual defect restriction and function prolonga-tion. The system on the coarsest grid is solved approximately with 10 iterations ofthe Vanka smoother. That means, we used the same coarse grid solver in all numer-ical tests. Since the unsatisfying parallel efficiency of multigrid methods originatesfrom the fact that the numerical work (flops) on all grids which are coarser thanthe finest one is in general small compared to the amount of communications, wefound it advantageous for enhancing the parallel efficiency to limit the iterationson the coarsest grid to a small number, see John [11, 12].

Remark 1 On the nonconforming Pi/Po-finite element discretization. Thenonconforming Pi/fo-Anite element discretization fulfills the inf-sup or Babus-ka-Brezzi stability condition uniformly with respect to the mesh size and the shaperegularity constant of the mesh. This guarantees the unique solvability of the aris-ing discrete systems. Besides this favourable analytical property, it has also ad-vantages from the point of view of the implementation on a parallel computer. Thedegrees of freedom of the velocity are associated with the midpoints of the edgesof the triangles. Thus, those degrees of freedom which belong to an interface be-tween two subdomains, each of which stored on a separate processor, have to bestored on two processors only. For this reason, only one communication is neededto interchange information between the same degree of freedom on different pro-cessors. This is not true for conforming finite element spaces where the degreesof freedom are connected with the vertices of mesh cells. In this case, the degreesof freedom at so-called crosspoints belong to and have to be stored on more thantwo processors. As a consequence, a complete interchange of information in caseof crosspoints requires more than one communication. This results in an increase



of the communication overhead.A drawback of this discretization is that it is only of first order accuracy. A

recent study on a benchmark problem for the steady state Navier-Stokes equations(John and Matthies [13]) showed that results of the same accuracy could be ob-tained with higher order finite element discretizations with much less degrees offreedom. But a difficulty of higher order discretizations might be the solution ofthe discrete systems. In [13] it was also found that a multigrid solver with thehigher order discretization on the top of the multigrid (finest level) and the remain-ing multigrid hierarchy built by a first order nonconforming upwind discretizationcould overcome this difficulty in many situations. One core of the success of sucha multigrid solver is a highly efficient multigrid method for first order noncon-forming upwind discretizations as it is studied here.

Remark 2 Comparisons of smoothers in the literature. Vanka and pressureSchur complement smoothers have been compared recently in at least two papers[4, 6]. The framework of these comparisons as well as the results of the numericaltests were different. In [6], the same framework as in the present paper was usedand the Vanka smoother behaved more efficient than the Braess-Sarazin smootherin all tests. In [4], the steady state Navier-Stokes equations were considered withan additional transport equation. The equations were discretized by a finite volumemethod on rectangular grids. The convective terms were dealt with a second orderflux-limited scheme. SIMPLE is used as a pressure Schur complement smootherand numerical results for the W(l,l)-cycle on a workstation were presented. Inboth numerical studies, the Vanka smoother, called SCGS in [4], behaved betterfor the lid driven cavity problem. In [4], SIMPLE was the best smoother in com-putations of flows past obstacles. For some reasons, SIMPLE was compared inthese computations to some Vanka-type smoothers but not to a mesh cell Vankasmoother. The reason for the worse efficiency of the Vanka-type smoothers arelong computing times for assembling the local equations, see [4].

We will present numerical studies of flows in a channel past obstacles wherethe cell wise Vanka smoother is compared to the Braess-Sarazin smoother.

4 Numerical studies

All computations were performed on 32 processors of a parallel computer ParsytecGCPowerPlus (80 MHz, 9.2 MFlops / processor (LINPACK), 35 MB/s communi-cation, 5/is message setup time, 60/̂ s minimal network latency). This computerpossesses a fast floating point operation speed compared to its communicationspeed. The implementation of the algorithms was done using modules of the pro-gram package ugpl.Q which is an early version of the now available program UG3from Bastian et al. [14]. In particular, data structures, load balancing routines andthe parallel environment of uypl.Q were used. However, the data structures had tobe extended in order to handle multidimensional data, nonconforming finite ele-ments, Navier-Stokes equations, and time dependent problems.

For the steady state equations, the initial guess for the fixed point iteration onthe finest level was the interpolation of the solution on the previous level. The


Advances in Fluid Mechanics 1 187

nonlinear iteration was stopped for an Euclidean norm of the residual less than10~*°. In the time dependent problems, the solution of the previous time step wasused as initial iterate for the nonlinear iteration and this iteration was stopped foran Euclidean norm of the residual less than 10~^. The linear saddle point problem(3) was solved in each step of the fixed point iteration up to a reduction of theEuclidean norm of the residual by the factor 10.

4.1 Channel flow past a vertical barrier

-20 0 60

Figure 1: Domain and initial grid for Example 4.1 (not to scale).

We consider the flow through a channel with a vertical barrier of height 1 as pre-sented in Figure 1. On the inlet, a parabolic inflow profile with maximal value 1and on the outlet outflow boundary conditions are prescribed. On all other bound-aries, no-slip conditions are applied. The geometry of this example is the sameas in [4], however, the boundary conditions are somewhat different. We have usedthe initial grid (level 0) shown in Figure 1 and present results on refinement level6, i.e. on the finest grid are 943 808 velocity degrees of freedom and 315 392pressure degrees of freedom (total % 1.26 million).

Computing times for the steady state Navier-Stokes equations and differentvalues of v are presented in Table 1. The Vanka smoothers was faster than theBraess-Sarazin smoother in all tests. The communication overhead was muchhigher using the Braess-Sarazin smoother. In addition, this smoother does notbecome more advantageous for smaller kinematic viscosities. We have done alot of other tests with different parameters in the Braess-Sarazin smoother (e.g.with different a, different numbers of GMRES iterations and C~^ as ILU(Q)(A)-iteration) which are not reported in this paper. We could reach the computingtimes of the Vanka smoother in none of these tests. But we want to emphasizethat the worse computing times of the Braess-Sarazin smoother are not causedby its smoothing property but they come from the high computational costs of itsapplication. To reduce these costs is the key of enhancing its efficiency.

The time dependent Navier-Stokes equations were solved for i/ = 10"^, T =10s, the constant length of the time step A/ = 0.01, and UQ was chosen as alreadydeveloped flow field. The computing times, Table 2, show that the problem couldbe solved faster again with the Vanka smoother. But its superiority is not as largeas in the steady state equations. The Braess-Sarazin smoother demonstrates here abetter smoothing property than the Vanka smoother since the F(2,2)-cycle workedwell whereas the same cycle failed to converge with the Vanka smoother. TheBraess-Sarazin smoother suffers again from higher computational costs.



Table 1: Channel flow past a vertical barrier, steady state equation.

smoothercyclez/-i = 100time (s)commun. (%)y-i =200time (s)commun. (%)(/-̂ =300time (s)commun. (%)

VankaF(2,2)

116512.5

275912.5

8313125

F(3,3)

110212.4

250312.4

7522123

F(4,4)

124312.4

3029124

8498126

Braess-SarazinF(2,2)

178423.2

487723.2

1441023.4

F(3,3)

250523.8

6836218

1964024.0

F(4,4)

195323.6

896724.4

24620243

Table 2: Time dependent channel flow past a vertical barrier.

smoothercycletime (s)commun. (%)

VankaF(3,3)35745

9.9

F(4,4)4179110.3

F(5,5)4794910.6

Braess-SarazinF(2,2)4635915.5

F(3,3)6030916.6

F(4,4)7434917.3

4.2 Channel flow past a bump

-20 0 60

Figure 2: Domain and initial grid for Example 4.2 (not to scale).

The domain of this example is presented in Figure 2 where the bump is given by0.5(1 -f cos(7r#/1.8)), —1.8 < x < 1.8. All other parameters in the definition ofthe Navier-Stokes equations and in the numerical tests are the same as in Example4.1. Again, this domain is taken from [4]. The initial grid is shown in Figure 2.We present results on refinement level 6, i.e. with 845 760 velocity and 282 624pressure degrees of freedom (total # 1.13 million).

The results for the steady state Navier-Stokes equations and different valuesof v are presented in Table 3. The behaviour of the smoothers is similar as inExample 4.1. Again, the Vanka smoother showed a higher efficiency in all tests.

In the time dependent equations, see Table 4, the F(2,2)-cycle with the Braess-Sarazin smoother was a bit faster than all computations with the Vanka smoother.


Advances in Fluid Mechanics III

Table 3: Channel flow past a bump, steady state equation.

189

smoothercycle

i/-i = 100time (s)commun. (%)z/-i = 200time (s)commun. (%)z/-i = 300time (s)commun. (%)

VankaF(2,2)

92712.5

152812.4

345412.4

F(3,3)

81112.2

155912.2

331812.2

F(4,4)

91212.1

185412.3

411812.3

Braess-SarazinF(2,2)

100923.3

305223.6

677023.6

F(3,3)

137323.8

297623.9

657323.9

F(4,4)

169924.2

383524.1

841424.2

But for all other cycles, the Vanka smoother turned out to be more efficient.

Table 4: Time dependent channel flow past a bump, T — 10s, At — 0.01.

smoothercycletime (s)commun. (%)

VankaF(2,2)45302

9.8

F(3,3)46162

9.7

F(4,4)4525810.0

Braess-SarazinF(2,2)4139015.8

F(3,3)5410216.9

F(4,4)6683617.7

Acknowledgement. The work of the author was supported by the DeutscheForschungsgemeinschaft (DFG) under the grant JO 329/2 - 1 .

References

[1] M. Schafer and S. Turek. The benchmark problem "Flow around a cylinder".In E.H. Hirschel, editor, FlowSimulationwithHigh-Performance Computers//, volume 52 of Notes on Numerical Fluid Mechanics, pages 547 - 566.Vieweg, 1996.

[2] S. Vanka. Block-implicit multigrid calculation of two-dimensional recircu-lating flows. Comp. A/Wz. App/. Mzc/z. &%g., 59(l):29-48, 1986.

[3] S. Turek. Efficient Solvers for Incompressible Flow Problems: An Algorith-mic and Computational Approach, volume 6 of Lecture Notes in Computa-tional Science and Engineering. Springer, 1999.

[4] M.F. Paisley and N.M. Bhatti. Comparison of multigrid methods for neutraland stably stratified flows over two-dimensional obstacles. J. Comput. Phys.,142:581-610, 1998.



[5] D. Braess and R. Sarazin. An efficient smoother for the Stokes problem.AppliedNumer. Math., 23(1):3- 19, 1997.

[6] V. John and L. Tobiska. Smoothers in coupled multigrid methods for theparallel solution of the incompressible Navier-Stokes equations. Int. J. Num.Meth. Fluids, 2000. to appear.

[7] M. Crouzeix and P.-A. Ravi art. Conforming and nonconforming finite ele-ment methods for solving the stationary Stokes equations I. R.A.LR.O. Anal-yse Numerique, 7:33-76, 1973.

[8] F. Schieweck and L. Tobiska. An optimal order error estimate for an upwinddiscretization of the Navier-Stokes equations. Num. Meth. Part. Diff. Equ.,12:407-421, 1996.

[9] S.V. Patankar and D.B. Spalding. A calculation procedure for heat and masstransfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer,15:1787-1806, 1972.

[10] V. John and L. Tobiska. A coupled multigrid method for nonconformingfinite element discretizations of the 2d-Stokes equation. Computing, 2000.to appear.

[11] V. John. On the parallel performance of coupled multigrid methods for thesolution of incompressible Navier-Stokes equations. In M. Griebel, O.P.Iliev, S.D. Margenov, and PS. Vassilevski, editors, Large-Scale ScientificComputations of Engineering and Environmental Problems, volume 62 ofNotes on Numerical Fluid Mechanics, pages 269 - 280. Vieweg, 1998.

[12] V. John. A comparison of parallel solvers for the incompressible Navier-Stokes equations. Comput. Visual. ScL, 1(4): 193 - 200, 1999.

[13] V. John and G. Matthies. Improved reference values for a benchmark prob-lem for the incompressible Navier-Stokes equations. Preprint, Fakultat furMathematik, Otto-von-Guericke-Universitat Magdeburg, 2000.

[14] P. Bastian, K. Birken, K. Johannsen, S. Lang, N. NeuB, H. Rentz-Reichert,and C. Wieners. UG - a flexible software toolbox for solving partial differ-ential equations. Comput. Visual. Sci., 1(1):27-40, 1997.


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Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman … · 2014-05-12 · Advances in Fluid Mechanics III 183 whereas the discrete pressure belongs to the space Qh :— J space