A Note on an Accelerated High-Accuracy MultigridSolution of the Convection-Diffusion Equationwith High Reynolds NumberJun ZhangDepartment of Computer ScienceUniversity of Kentucky773 Anderson HallLexington, Kentucky 40506-0046

Received August 18, 1997; accepted July 22, 1999

We present a new strategy to accelerate the convergence rate of a high-accuracy multigrid method for thenumerical solution of the convection-diffusion equation at the high Reynolds number limit. We proposea scaled residual injection operator with a scaling factor proportional to the magnitude of the convectioncoefficients, an alternating line Gauss–Seidel relaxation, and a minimal residual smoothing accelerationtechnique for the multigrid solution method. The new implementation strategy is tested to show an improvedconvergence rate with three problems, including one with a stagnation point in the computational domain.The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourth-order compact discretization are investigated numerically. c© 2000 John Wiley & Sons, Inc. Numer Methods PartialDifferential Eq 16: 1–10, 2000

Keywords: multigrid method; fourth-order compact discretization schemes; residual transfer operators;convection-diffusion equation

I. INTRODUCTION

In this article, we study the problem of developing an efficient multigrid method for the numericalsolution of the two-dimensional convection-diffusion equation

−(uxx + uyy + p(x, y)ux + q(x, y)uy) = f(x, y), (x, y) ∈ Ω. (1)

Here Ω is a continuous convex domain with appropriate boundary conditions. p(x, y), q(x, y), andf(x, y) are assumed to be sufficiently smooth, and u(x, y) is differentiable with respect to x andy in Ω. Convection-diffusion problems like Eq. (1) occur in many applications of computationalfluid dynamics, such as in the transport of air and groundwater pollutants, in the equations ofdrift-diffusion models for semiconductor modeling, and in Navier–Stokes equations.

Correspondence to: Jun Zhang; e-mail: jzhang@cs.uky.edu. URL: http://www.cs.uky.edu/ jzhang.c© 2000 John Wiley & Sons, Inc. CCC 0749-159X/00/010001-10

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To facilitate the discussion, we define the Reynolds number as

Re = sup(x,y)∈Ω

|p(x, y)|, |q(x, y)|,

which is used to measure the ratio of the convection to diffusion. For small Re, say Re < 103,Eq. (1) is said to be diffusion-dominated; otherwise, it is convection-dominated.

The very first problem encountered when one tries to seek the numerical solution of Eq. (1) ishow to discretize the continuous problem. In the context of finite difference discretizations, it iswell known that the central difference scheme, although very accurate for small Re, may produce anumerical solution with undesirable oscillations. Furthermore, the linear systems arising from thecentral difference scheme (with large Re) are very hard to solve with most iterative methods. Onthe other hand, the standard upwind difference scheme has the effect of suppressing oscillationsby adding large artificial viscosity, but yields numerical solution of lower accuracy. It seems thatthere is no generally accepted best discretization scheme. Recently, there is growing interest indeveloping higher-order compact discretization schemes for problems similar to Eq. (1) [1, 2].These schemes have the properties of providing high-accuracy numerical solutions for small tomoderate Re and are stable for large Re. For convection-dominated problems, the fourth-ordercompact schemes can be viewed as a compromise between the central difference scheme andthe upwind scheme. In fact, the level of difficulty of solving the linear systems arising from thefourth-order compact schemes by iterative methods is between that of the central difference andthe upwind schemes.

A numerical solution of the convection-diffusion equation also provides a simple but nontrivialchallenge to classical multigrid methods, represented by the introductory article of Brandt [3]. Ithas been known for years that the convergence rate of the standard multigrid methods deteriorates(or does not even exist) for solving Eq. (1) as Re increases. Over the past 20 years, thanks tothe continuous efforts of multigrid method researchers and practitioners, geometric and algebraicmultigrid methods have been developed to compute a numerical solution of Eq. (1) with reasonableconvergence rates, see [4–6] and the references therein. However, most of these approaches solvedlinear systems discretized by the upwind (or similar) scheme, which are relatively less challenging,as we shall see later. Studies on using the multigrid method to solve linear systems arising fromthe fourth-order compact scheme (for convection-dominated problems) are relatively new [7, 8].

In [7], we proposed an accelerated multigrid method for high-accuracy numerical computationof Eq. (1). The proposed method employs a fourth-order 9-point compact discretization scheme ofGupta et al. [1] and a standard multigrid algorithm accelerated by a minimal residual smoothing(MRS) technique [9]. It was shown that the method yields high-accuracy numerical solutions,and the acceleration technique is cost effective.

However, even with the MRS acceleration, the convergence rate of the multigrid solutionmethod was shown to deteriorate at the high Reynolds number limit (Re ≥ 106). More than 1000iterations were needed to reduce the residual norm by a factor of 1010. Furthermore, the standardmultigrid method does not converge for solving Eq. (1) at the high Reynolds number limit, ifthe computational domain contains some stagnation points. In this article, a stagnation point isdefined as a point in Ω, where both convection coefficients p(x, y) and q(x, y) vanish. By thisdefinition, a problem with stagnation points may or may not model a recirculating flow [4].

On the other hand, two results from our recent work [10] on analytic investigation of thefourth-order compact scheme may help improve the numerical solution method. For solving highReynolds number convection-diffusion equations by the classical multigrid method, it was shownthat alternating line Gauss–Seidel relaxation is a robust smoother, and the residual must be scaledby a scaling factor before it is projected to the coarse grid. These two results, together with the

ACCELERATED HIGH-ACCURACY MULTIGRID SOLUTION. . . 3

MRS acceleration scheme, are employed to design a new multigrid solution strategy for solvingEq. (1) at the high Reynolds number limit.

In this article, classical, standard, and geometric multigrid methods all mean a similar approach,in which the coarse grid operator is formed by applying the same discretization scheme on thecoarse level grids. Algebraic multigrid methods mean that the coarse grid operator is generatedby the Galerkin coarse-grid approximation technique.

II. NEW STRATEGIES

It was shown in [10] that both point and line Gauss–Seidel relaxations fail to damp error com-ponents along certain characteristics directions, and the alternating line Gauss–Seidel relaxationmust, therefore, be employed as a robust relaxation method. The alternating line Gauss–Seidelrelaxation in lexicographic order performs one sweep of line Gauss–Seidel relaxation along thex-coordinate direction, followed by another sweep of line Gauss–Seidel relaxation along they-coordinate direction.

However, it was shown in [10] that merely using the alternating line Gauss–Seidel relaxationin a standard multigrid method does not provide fast convergence for high Reynolds numberproblems. The coarse-grid solution, with a large amount of artificial viscosity, may not providea meaningful correction to the fine-grid approximate solution with a small amount of artificialviscosity. This problem can be fixed by properly scaling the residual that is projected to the coarsegrid [11].

Residual scaling techniques are studied extensively in [11, 12]. Similar techniques have alsobeen studied by Brandt and Yavneh [4], and Kouatchou [13]. The theoretical basis for suchtechniques is not clear. Heuristic argument can be based on the fact that both the fourth-ordercompact scheme and the upwind scheme add artificial viscosity to the difference equations [10].The amount of artificial viscosity is proportional to the discretization parameter h (the meshsize). Since different h’s are used at different levels, a geometric multigrid method that usesa similar discretization scheme at all levels solves different problems at different levels. Thesolutions obtained on different levels do not have the same scale. Hence, without specific treatmentof the scale of the solutions, the coarse-grid solution cannot be used to improve the fine-gridapproximation solution. One cure for this problem is to scale the residual explicitly before it isprojected to the coarse grid, or after the coarse-grid correction is transferred back to the fine grid.The so-called prescaling and postscaling techniques are mathematically equivalent, if they usethe same scaling factor [11].

The demand of residual scaling and the lack of comprehensive knowledge of the scalingfactors make some powerful relaxation schemes useless. For example, experience suggests thatan alternating line relaxation in zebra order usually has a better smoothing effect than an alternatingline relaxation in lexicographic order that we used. The fact that we have not been able to find asuitable method to scale the residual from the zebra order relaxation properly forced us to abandonthe potentially more efficient relaxation scheme.

It was also indicated in [12] that the residual injection operator instead of the full weightingoperator should be used in the standard multigrid method for solving convection-dominatedproblems with stagnation points in their computational domain. The reason for this treatmentis that the standard weighting operators cannot represent the complex behavior of the residualaround the stagnation points. The residual values at different grid points around a stagnationpoint may have different signs, the weighting schemes may remove this feature by computing anaverage of residual values of different signs (cancellation effect). For the alternating line Gauss–

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Seidel relaxation, our numerical experience suggested that there is no significant difference inconvergence by using a residual injection operator or the full weighting operator, if Ω does notcontain any stagnation point. At the high Reynolds number limit, both operators need to be scaledby the same scaling factor as remarked above.

A. Minimal Residual Smoothing Acceleration

Since the convection coefficients may vary substantially in the computational domain, any pre-determined scaling parameters may need a tolerance of small variation. The MRS accelerationtechnique, which is introduced in [9] as a multigrid acceleration method, may be used for thispurpose. Suppose that uk is a sequence of approximate solutions and rk is the correspondingresidual sequence. The minimal residual smoothing technique generates a new sequence ukwith a smoothed residual sequence r from the original sequences in the form

uk = uk−1 + βk(uk − uk−1),rk = rk−1 + βk(rk − rk−1),

with u0 = u0, r0 = r0. The MRS parameter βk at each step is chosen to minimize the newresidual norm ‖rk‖ in some sense; for example,

βk = − rTk−1(rk − rk−1)‖rk − rk−1‖22

, (1)

where ‖ · ‖2 is the Euclidean norm. The new residual sequence with βk satisfying (2) obviouslyhas a nonincreasing Euclidean norm; i.e.,

‖rk‖2 ≤ ‖rk−1‖2, and ‖rk‖2 ≤ ‖rk‖2,

for each k. In [7, 10], we used the MRS procedure to smooth the residual before it is projectedto the coarse grid, and the multigrid iteration is continued with the new approximate solutionuk and the new residual rk. The original approximate solution uk and its associated residualrk are discarded after the presmoothing and the MRS procedure. There are some inexact MRSacceleration techniques [14] that compute the MRS parameter βk on a coarse grid and may reducethe MRS acceleration cost by 40%. Note that we apply the MRS acceleration only on the finestlevel; the reasons for applying or not applying MRS on the coarse levels are discussed in [9]. Thecost of such an acceleration is roughly equivalent to 9 floating-point operations (FPO) per gridpoint on the fine grid. This cost is independent of the underlying component operators used inthe multigrid method. In other words, the more expensive the original multigrid method is, therelatively cheaper the MRS acceleration is.

For example, one sweep of point Gauss–Seidel relaxation using the fourth-order 9-point com-pact scheme has 17 FPO. One V (1, 1)-cycle (with two relaxation sweeps and one residual eval-uation on each level) has about 98 FPO. If the intergrid transfer operators add 20% more to thiscost, the total number of FPO for a multigrid V (1, 1)-cycle is about 118. One application of theMRS acceleration procedure costs only about 7.6% more to add the total cost of 127 FPO to theaccelerated multigrid method with the point Gauss–Seidel relaxation. For other multigrid cyclingalgorithms, the relative cost of the MRS acceleration is even lower. The MRS acceleration cost ina similar V (1, 1)-cycle algorithm using alternating Gauss–Seidel line relaxation is less than 5%of the total cost.

ACCELERATED HIGH-ACCURACY MULTIGRID SOLUTION. . . 5

B. Residual Scaling

Although discussions and numerical experiments in [7, 10] are concerned with Eq. (1), they donot specifically address high Reynolds number problems, which are the concern of this article. Inour new implementation, we use the MRS technique to smooth the residual on the finest grid, andthe smoothed residual is denoted by r as above. We then explicitly scale r by a scaling parameter(factor) α before it is projected to the coarse grid (without any weighting scheme).

The actual scaling factor is determined by the absolute values of the convection coefficientsat the reference point, and the scaling factor is a function of the grid point index (i, j). In [10],we found by numerical experiments that the optimal residual scaling factor seems to lie in aninterval [3.9, 4.0] if |p(x, y)| > 103 or |q(x, y)| > 103. However, for a problem containingstagnation points in its computational domain, it is essential that the residual be set to zero at thestagnation points, so that the feature of the stagnation points may be represented on the coarse grid;otherwise, the multigrid method could diverge [4]. For small but nonzero values of the convectioncoefficients, a somewhat smaller scaling factor may be beneficial. Hence, we propose a residualinjection operator with a variable scaling factor as:

Let Rei,j = |pi,j |+ |qi,j |, set

ri,j = 4ri,j if Rei,j > 103,

ri,j = ri,j if 102 ≤ Rei,j ≤ 103,

ri,j = 3ri,j/4, if 10 ≤ Rei,j < 102,

ri,j = ri,j/4, if 10−14 ≤ Rei,j < 10,ri,j = 0, if Rei,j ≤ 10−14,

where ri,j is the residual value projected to the coarse grid (after the application of the MRSacceleration).

Here is the summary of the components of our new multigrid implementation:

• Presmoothing with ν1 sweeps of the alternating line Gauss–Seidel relaxation in lexico-graphic order is used to damp oscillating components of the errors on the fine grid.

• Minimal residual smoothing technique is used to smooth the residual after presmoothing.The new iterate generated by the MRS acceleration and its residual replace the originalmultigrid iterate and its residual in further computations.

• The smoothed residual is scaled explicitly by a parameter α determined by the magnitudeof the convection coefficients. The coarse grid problem is formed by this scaled residualwith the same fourth-order discretization scheme (on the coarse grid).

• The coarsest possible grid is utilized (h = 1/2) and one iteration is used to obtain the exactsolution on the coarsest grid (with only one unknown).

• The coarse grid solution is transferred back to the fine grid by the standard bilinear inter-polation operator.

• Postsmoothing using ν2 sweeps of the alternating line Gauss–Seidel relaxation in lexico-graphic order is applied to finish one complete multigrid cycle.

de Zeeuw [5] used a somewhat more powerful smoother called incomplete line LU factorizationwith matrix-dependent intergrid transfer operators for solving Eq. (1). His method belongs to thecategory of algebraic multigrid methods. However, as Bandy [15] remarked, the efficiency of deZeeuw’s solver is based on the efficiency of the incomplete line LU factorization smoother, whichis complicated and costly, but not on the efficiency of the complicated intergrid transfer operatorsthat were employed. Consequently, an easier and cheaper alternating line relaxation-based solver

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has been shown [15] to achieve similar performance as de Zeeuw’s solver, when combined withBandy’s new intergrid transfer operators.

III. NUMERICAL EXPERIMENTS

The test problems were chosen as:

Test Problem 1

p(x, y) = 106x(1− y),q(x, y) = −106y(1− x),u(x, y) = sinπx+ sinπy + cosπx+ cosπy.

Test Problem 2

p(x, y) = −106xy(1− x2),q(x, y) = 1062xy(1− y2),u(x, y) = sin(x+ y) + exp(x+ y).

Test Problem 3

p(x, y) = −106 sinπx cosπy,q(x, y) = 106 cosπx sinπy,u(x, y) = xy(1− x)(1− y) exp(x+ y).

The Reynolds number is about 106 and is referred to as the high Reynolds number limit, since theconvergence of iterative methods changes only slightly beyond that limit [7]. The fourth-order9-point compact discretization uses a uniform mesh size on the unit square Ω = (0, 1)×(0, 1) (see[1, 7] for the detailed discretization formula). Although the computed accuracy of the fourth-ordercompact scheme reduces to the second order at the high Reynolds number limit, it is still moreaccurate than the traditional first-order upwind scheme. Dirichlet boundary values and the forcingfunctions were chosen to satisfy the given exact solutions. Several multigrid cycling algorithmswere tested. The initial guess was u(x, y) = 0 for all tests. The computations were done in doubleprecision on a workstation. Convergence was reached when the residual in discrete L2-norm wasreduced by a factor of 1010. The number of iterations (with and without the MRS acceleration)are given in Table I.

TABLE I. Number of iterations with different multigrid cycling algorithms.

Problem 1 Problem 2 Problem 3

Mesh size (h) No-MRS MRS No-MRS MRS No-MRS MRS

V (1, 1)-cycle

1/32 27 19 47 25 23 261/64 44 29 79 44 34 381/128 65 40 112 63 58 68

W (1, 1)-cycle

1/32 25 18 33 23 18 161/64 35 25 46 34 29 221/128 47 34 59 45 43 36

V (2, 2)-cycle

1/32 17 13 28 18 12 141/64 27 18 46 26 21 191/128 44 28 76 54 137 47

ACCELERATED HIGH-ACCURACY MULTIGRID SOLUTION. . . 7

TABLE II. Number of iterations as a function of scaling factor for Problem 1.

α 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

No-MRS 549 326 212 71 58 50 44 40 81 no conv.MRS 308 195 118 41 35 31 29 34 40 46

It is clear from the results of Table I that the new solution method converged faster thanthe method proposed in [7]. We point out that, except for some cases with the stagnation pointproblem (Problem 3), the MRS technique did provide significant convergence acceleration witha negligible cost (less than 5%). Roughly speaking, a W (1, 1)-cycle is one and a half times, anda V (2, 2)-cycle is about twice as expensive as a V (1, 1)-cycle. It seems that the W (1, 1)-cyclealgorithm is the best among the three algorithms tested. This agrees with the observation madein [4].

We also note that the convergence rate of our multigrid method deteriorated as the numberof levels increased. This may be a result of the decreasing accuracy of the intergrid transferoperators as the number of levels increases. The same problem has been reported with othermultigrid methods [4]. To the best knowledge of the author, there is no well-known multigridmethod based on geometric approach that has indeed achieved the grid-independent convergencefor convection-dominated problems. Under this condition, a higher-order discretization scheme,which requires a coarser grid to produce an approximate solution with competitive accuracy, hasthe advantage of grid-dependent convergence.

A. Influence of the Scaling Factor

We stated in the previous section that the residual of the fine grid must be scaled by a factor αbefore it is projected to the coarse grid. In general, the value of this scaling factor is obtained bynumerical experiments. In [10], our numerical experiments with a constant coefficient problemfound that the optimal scaling factor α seems to be in the interval [3.9, 4.0]. For the current threevariable coefficient problems, Tables II–IV tabulate the number of iterations as a function of α,as it varies in some useful range (found by numerical experiments). The algorithm used was theV (1, 1)-cycle with h = 1/64. The smallest numbers of iterations achieved with and without theMRS acceleration are highlighted.

For the first two problems without a stagnation point, the smallest number of iterations wasachieved with α between 4.5 and 5.0, which is slightly larger than what we reported in [10] witha constant coefficient problem. However, for Problem 3, the smallest number of iterations wasachieved with α = 0.7, and divergence occurred with α ≥ 0.8 without the MRS acceleration.

The results of Tables II–IV indicate that there is no single residual scaling factor that canyield good convergence for both problems with and without a stagnation point in their domains.Therefore, a variable scaling factor is necessary, as used in our new implementation outlined inSection II.

TABLE III. Number of iterations as a function of scaling factor for Problem 2.

α 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

No-MRS 870 465 268 171 126 99 79 66 56 74 >1000MRS 436 211 142 86 63 48 44 40 48 57 >1000

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TABLE IV. Number of iterations as a function of scaling factor for Problem 3.

α 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

No-MRS >1000 824 200 148 138 128 diverge diverge divergeMRS >1000 313 106 79 72 47 62 132 125

These experimental results also support the idea of employing an MRS acceleration procedureto increase the robustness of the underlying multigrid method. For many cases, the MRS techniquesignificantly reduced the iteration counts, and, in some cases, it turned a divergent multigridmethod to a convergent one.

B. Diagonal Dominance

The results in Table I in terms of iteration counts should not be compared literally with the resultsof Brandt and Yavneh [4], de Zeeuw [5], and Reusken [6]; since different solution methodsand different discretization schemes were employed. The results of [5, 6] were obtained usingupwind (or similar) discretization schemes, so that the coefficient matrices were weakly diagonallydominant. This is only true for the fourth-order compact scheme, when the convection coefficientsare constant, and when the cell Reynolds number h · Re/2 is less than one [16].

For large Reynolds number problems, even a matrix generated by the upwind discretizationscheme may not be 100% weakly diagonally dominant, due to numerical rounding. To give an ideaof some algebraic properties of the coefficient matrices generated by the fourth-order compactscheme and the upwind scheme, we list in Table V information about the percentage of weaklyrow- and column-diagonal dominance and the Frobenius norm of each coefficient matrix. It canbe seen that the coefficient matrices generated by the fourth-order compact scheme have far worsealgebraic properties (lower weakly row- and column-diagonal dominance and higher Frobeniusnorm) than those generated by the upwind scheme. Furthermore, these algebraic properties usuallystrongly influence the performance of iterative methods. Hence, in some sense, the current testproblems (linear systems) are more difficult to solve than those used by other multigrid methodsand reported by other authors. It is not a surprise that the iteration counts are generally largerthan those reported. Another reason for the relatively large iteration counts is that we reduced

TABLE V. Percentage of weakly diagonal dominance and the Frobenius norm of the coefficient matrices.9.99(9) stands for 9.99 × 109.

Mesh size h = 1/32 h = 1/64 h = 1/128

Scheme Row Col. Norm Row Col. Norm Row Col. Norm

Problem 1

Compact 6.04% 6.45% 9.99(9) 3.07% 3.17% 5.23(9) 1.55% 1.57% 2.67(9)Upwind 53.7% 67.7% 6.71(5) 52.1% 67.4% 6.89(5) 51.0% 62.0% 6.98(5)

Problem 2

Compact 0.00% 3.64% 6.70(9) 0.00% 1.76% 3.41(9) 0.02% 0.86% 1.72(9)Upwind 52.2% 67.3% 5.60(5) 11.7% 71.6% 3.27(6) 50.9% 68.8% 5.68(5)

Problem 3

Compact 11.2% 12.5% 2.00(10) 5.95% 6.25% 1.03(10) 3.05% 3.12% 5.21(9)Upwind 93.9% 51.1% 1.05(6) 96.9% 69.2% 1.08(6) 98.4% 74.8% 1.09(6)

ACCELERATED HIGH-ACCURACY MULTIGRID SOLUTION. . . 9

the residual norm by a factor of 1010, while some other reported results were obtained when theresidual norm was reduced only by a factor of 106 or 107 [5].

IV. CONCLUDING REMARKS

We have proposed a new implementation strategy for an accelerated multigrid solution method.It consists of a variably scaled residual injection operator, a robust alternating line Gauss–Seidelrelaxation, and a minimal residual smoothing acceleration technique. Numerical experimentsdemonstrated that our method converges fast for the high Reynolds number convection-diffusionequations discretized by the fourth-order 9-point compact scheme. The method was also shown toconverge satisfactorily for a problem containing a stagnation point in the computational domain.

Although our W (1, 1)-cycle algorithm with the alternating line Gauss–Seidel relaxation isabout twice as expensive as the W (1, 1)-cycle algorithm with the point Gauss–Seidel relaxationused in [7] (assuming that the LU matrices at all levels are prefactored and stored), the number ofiterations is reduced by about 30–50 times. The disadvantage of the alternating line Gauss–Seidelrelaxation is its lack of parallelization potential. We did some numerical experiments to show thatthe alternating zebra line relaxation did not provide a robust solver, partly because it is difficultto determine the residual scaling factor, because the residual is not locally smooth with a patternrelaxation. However, this disadvantage should not put a severe limit on the application of ourmethod, since the two-dimensional fourth-order method may not necessitate parallel computers.In three-dimensional computations, plane relaxations are beneficial for a robust smoother, andour efficient two-dimensional solver may be used to approximate the needed plane relaxations.

For solving linear systems arising from the upwind discretization of high Reynolds numberconvection-diffusion equations, whether or not to use a scaling factor (α = 1.4) is not that critical,although 50% reduction in iteration counts may sometimes happen. However, in the context ofthe fourth-order compact scheme, the use or not use of a scaling factor means much more thanthat; up to 90% reduction in iteration counts is usual [14].

One shortcoming of the geometric multigrid methods is that different methods are neededfor solving diffusion-dominated and convection-dominated problems. For example, the methodproposed in this article cannot be used to solve a Poisson equation. One may argue that thisis because completely different problems are solved; i.e., elliptic problems when diffusion isdominant and almost hyperbolic problems when convection is dominant. Algebraic multigridmethods usually do not have this shortcoming, but their constructions generally incur an additionalcost. There is no definite answer as to which class of methods is better; it all depends on theapplication.

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