### Table 10: Example (3.1), scaling of the coupled multigrid methods.

1998

"... In PAGE 11: ...11 computing times for a xed number of multigrid cycles. The results of Table10 were obtained with 100 multigrid cycles.... ..."

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### Table 1. Various multigrid and domain decomposition methods.

1995

"... In PAGE 4: ... Figure 1 shows the initial triangulation and a sample overlapped subdomain, and a sample solution. Table1 gives a key to the remaining gures. Figure 2 shows the performance of the methods, as a function of CPU time on a SPARC 10, for the aligned problem.... ..."

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### Table 1. Various multigrid and domain decomposition methods.

"... In PAGE 4: ... Figure 1 shows the initial triangulation and a sample overlapped subdomain, and a sample solution. Table1 gives a key to the remaining gures. Figure 2 shows the performance of the methods, as a function of CPU time on a SPARC 10, for the aligned problem.... ..."

### Table 4: Comparison of computational e ciency for solving Test 1 using multigrid method with FCS and UPS.

2000

"... In PAGE 10: ... For more details on this multigrid method and other multigrid techniques for solving convection di usion equations discretized by fourth order compact schemes, we refer readers to [8, 11, 24, 22, 23]. The test results for FCS and UPS for solving Test 1 with a few Re are presented in Table4 . CDS is not directly applicable in multigrid method since the relaxation scheme would not converge on the coarse grids where the e ective Reynolds number (with respect to the meshsize) is large.... ..."

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### Table 4: Comparison of computational e ciency for solving Test 1 using multigrid method with FCS and UPS.

2000

"... In PAGE 10: ... For more details on this multigrid method and other multigrid techniques for solving convection di usion equations discretized by fourth order compact schemes, we refer readers to [8, 11, 24, 22, 23]. The test results for FCS and UPS for solving Test 1 with a few Re are presented in Table4 . CDS is not directly applicable in multigrid method since the relaxation scheme would not converge on the coarse grids where the e ective Reynolds number (with respect to the meshsize) is large.... ..."

### Table 11: Number of iterations of the multigrid method with di erent orderings of the grid points for solving Test 3.

"... In PAGE 22: ... The convergence histories analogous to the cases showing in Figure 3 are depicted in Figure 4. Further, the results in Table11 show that, with quot;?1 = 100, the multigrid method does not converge with any of the ordering strategies tested. This is surprising, since test results with a slightly di erent governing convection-di usion equation show convergence of the multigrid method with at least the natural rowwise ordering when quot;?1 = 100, although divergence is observed for larger values of quot;?1 [40].... ..."

### Table 8: Number of iterations of the multigrid method with di erent orderings of the grid points for solving Test 2.

"... In PAGE 19: ... In addition to the ordering strategies used for Test 1, we add the FDPI ordering. Experimental data are listed in Table 7 for the point Gauss-Seidel, in Table8 for multigrid, and in Table 9 for GMRES/ILU(1). Table 7 shows that the performance of the point Gauss-Seidel method improves with... In PAGE 20: ... quot;?1 natural red-black four-color(a) four-color(b) FDPI 10 1552 1550 1548 1547 1532 50 711 707 703 703 665 100 368 364 361 361 327 200 190 187 183 184 159 500 176 163 161 158 135 1000 384 370 368 365 346 However, the good performance of the underlying relaxation method (point Gauss- Seidel in the current case) with the FDPI ordering does not necessarily carry over to the multigrid method. In Table8 , the multigrid method with FDPI ordering does not converge except in the di usion dominated case with quot;?1 = 10. All the other coloring strategies are shown to be better than the natural rowwise ordering with the multigrid method.... ..."

### Table 7: Example (3.1), the step length control (slc) in coupled multigrid methods.

1998

"... In PAGE 9: ...2) pays o . Results for the steady state case are presented in Table7 . They show the impor- tance of the step length control for the F{ and W{cycle.... In PAGE 11: ... The increase of the level has led to a reduction of the time on the coarsest grid which has compensated the communications on 4 and 16 processors. The nearly constant times for one V{cycle can be seen also in Table7 . These results demonstrate the very good performance of the nonconforming P1=P0{ nite element which comes from the absence of crosspoints.... In PAGE 11: ... The F{cycle shows a similar behaviour as the V{cycle. The losses due to the communication overhead are small, compare also Table7 . Only the W{cycle shows considerable losses of e ciency.... ..."

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