5 Figure Captions

Figure 1. Convergence factors for various discretizations of the Laplacian (Case 1). The lines show smoothing factors of optimally weighted point Jacobi (solid line), line Jacobi (dashes), and unweighted point RB (alternating dashes). The symbols correspond to numerical calculations using V(1,0) cycles with point Jacobi (), line Jacobi (+), and point RB (2), and also two-level (1,0) cycles using point RB (4). Figure 2. Convergence factors for Case 2. The lines show smoothing factors of optimally weighted point Jacobi (solid line), and unweighted point RB (dashes). The symbols correspond to numerical calculations using V(1,0) cycles with point Jacobi () and point RB (+), and also two-level (1,0) cycles using point RB (2). Figure 3. Convergence factors for Case 3. The lines show smoothing factors of optimally weighted x-line Jacobi (solid line), and unweighted (zebra) x-line RB (dashes). The symbols correspond to numerical calculations with optimally damped line Jacobi, using V(1,0) cycles () and two-level (1,0) cycles (+), and also with unweighted zebra, using V(1,0) cycles (2) and two-level (1,0) cycles (4). Figure 4. Convergence factors for Case 3. The lines show smoothing factors of optimally weighted x-line Jacobi (solid line), and unweighted (zebra) x-line RB (dashes), employing y-partial coarsening.. The symbols correspond to numerical calculations with optimally damped line Jacobi, using V(1,0) cycles () and two-level (1,0) cycles (+), and also with un-weighted zebra, using V(1,0) cycles (2) and two-level (1,0) cycles (4), all employing y-partial coarsening. 21 tain elementary multigrid matters. We nd, for example, that point Jacobi is better than line Jacobi (both with optimal damping) for virtually any nine-point discretization of the Laplacian, and it is even better than point red-black for many discretizations. We nd that undamped point Jacobi is an excellent smoother for the discretization of the Laplacian that is the average of the Skew Laplacian and the usual ve-point discretization, even though it is a useless smoother for either of the latter two discretizations. We also connrm, as expected, that employing line relaxation (RB or damped Jacobi) in one direction and partial coarsening in the other yields a robust smoother, so long as the Skew-Laplacian component is not dominant 6, 13]. For nine-point operators, four-color relaxation is a natural choice, and future investigation may provide formulae for this case as well. Another interesting question is what is the optimal weighting for RB relaxation of nine-point operators. For ve-point operators overrelaxation proves to be quite …