AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

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94 C. W. OOSTERLEE AND T. WASHIOThe second algorithm evaluated, denoted by MG2, adopts the “upwind” prolongationoperator by de Zeeuw ([25]). This operator is also presented in [24] and isspecially designed for problems with jumping coefficients and for second-order differentialequations with a dominating first-order derivative. As already indicated in [4],it is appropriate for the construction of transfer operators for unsymmetric matricesto split a matrix A into a symmetric and an antisymmetric part:(23)S = 1 2 (A + AT ), T = A − S = 1 2 (A − AT ).The investigated transfer operators for MG2 are also based on this splitting. Thediagonal elements of matrices S and T are numbered similarly as the elements of Ain (2). One now defines the following:(24)(25)(26)(27)(28)(29)(30)d w = max(|s 1 2i,2j−1 + s 4 2i,2j−1 + s 7 2i,2j−1|, |s 1 2i,2j−1|, |s 7 2i,2j−1|),d e = max(|s 3 2i,2j−1 + s 6 2i,2j−1 + s 9 2i,2j−1|, |s 3 2i,2j−1|, |s 9 2i,2j−1|),d n = max(|s 7 2i−1,2j + s 8 2i−1,2j + s 9 2i−1,2j|, |s 7 2i−1,2j|, |s 9 2i−1,2j|),d s = max(|s 1 2i−1,2j + s 2 2i−1,2j + s 3 2i−1,2j|, |s 1 2i−1,2j|, |s 3 2i−1,2j|),σ 1 = 1 ∑ 9 )(1,2 min ∣ 1 − k=1 sk 2i,2j−1a 5 ,2i,2j−1∣σ 2 = 1 ∑ 9 )(1,2 min ∣ 1 − k=1 sk 2i−1,2j,∣a 5 2i−1,2jc 1 = t 3 2i,2j−1 + t 6 2i,2j−1 + t 9 2i,2j−1 − (t 1 2i,2j−1 + t 4 2i,2j−1 + t 7 2i,2j−1),(31)(32)(33)(34)(35)c 2 = t 7 2i−1,2j + t 8 2i−1,2j + t 9 2i−1,2j − (t 1 2i−1,2j + t 2 2i−1,2j + t 3 2i−1,2j),[w w = σ 1 1 + d ]w − d e c 1+,d w + d e d w + d e + d n + d sw e = 2σ 1 − w w ,[w n = σ 2 1 + d ]s − d n c 2+,d s + d n d w + d e + d n + d sw s = 2σ 2 − w n .Weights w1 and w2 are now found as follows:• for (2i, 2j − 1) ∈ Q GL+1w1 2i,2j−1 = min(2σ 1 , max(0, w w )); w2 2i,2j−1 = min(2σ 1 , max(0, w e ));• for (2i − 1, 2j) ∈ Q GL+1w1 2i−1,2j = min(2σ 2 , max(0, w s )); w3 2i−1,2j = min(2σ 2 , max(0, w n )).Equations (21) and (22) are used for the remaining grid points.Analysis of this operator and the numerical experiments by de Zeeuw [25] showthe very interesting behavior of MG2 as a solver for several problems.Restriction. As a restriction operator in MG1 and MG2, the transpose of theprolongation is used. Restriction R L : u L+1 ∈ Q GL+1 ↦→ u L ∈ Q GL is defined as(36)R L = (P L+1 ) T , 1 ≤ L ≤ M − 1.
MULTIGRID AS A SOLVER AND A PRECONDITIONER 95Grid:finestcoarsestV-cycleF-cyclefinestcoarsestW-cycleFIG. 4. Different ways of cycling through fine and coarse grids.Coarse grid discretization. Coarse grid matrix A L on grid G L is defined withGalerkin coarsening ([6], [24], [25])(37)(38)A M = A,A L = R L A L+1 P L+1 , 1 ≤ L ≤ M − 1.A drawback of the multigrid preconditioners, compared with, for example, ILU preconditioners,is that additional storage for the matrices on coarse grids as well as forthe matrix-dependent transfer operators is needed.Smoother. The smoother in MG1 and MG2 is the alternating zebra line Gauss–Seidel smoother. First, all odd (white) lines are processed in one direction, afterwhich all even (black) lines are processed. This procedure takes place in the x-and y-directions. Fourier smoothing analysis for model equations, presented in [24],indicates the robustness of this smoother. The parallel implementation by means ofgrid partitioning is explained in detail in [23]. Care has been taken that the singleblock and the multiblock convergences are identical.Multigrid cycles. The multigrid correction scheme is used for solving the linearequations. In [23] we chose the V-cycle as iteration cycle. Here, the robustness andefficiency of more expensive cycles, like the W- and F-cycles, are also evaluated. Thecycles are shown in Figure 4.The multigrid F-cycle is a cycle between the cheap V-cycle and the expensiveW-cycle. The algorithm is written down for convenience in a recursive way.MG F-cycleMGF(A L , f L , u L , ν 1 , ν 2 , ν 3 ) {if (L = 1) {u 1 := smoother(A 1 , f 1 , u 1 , ν 3 );return;}u L := smoother(A L , f L , u L , ν 1 );
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AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

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