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

MULTIGRID AS A SOLVER AND A PRECONDITIONER 103FIG. 6. The convergence of the MG solvers (a), and GMRES(20) with MG preconditioners (b),for the rotating convection-diffusion equation on a 33 2 grid.TABLE 4Solutions 1 − λ ∗ k related to the polynomials p k chosen by GMRES with MG1 and MG2 for theconvection-diffusion equation (ɛ = 10 −5 ).k: 5 6 7 8 90.32 0.33 0.34 0.34 0.34MG1 0.25 0.28 0.24 + 1.9.10 −2 i 0.27 0.260.14 0.18 0.24 − 1.9.10 −2 i 0.21 0.211 − λ ∗ k 2.4.10 −2 −4.3.10 −2 −5.4.10 −2 −6.4.10 −2 −8.5.10 −3−1.7.10 −2 6.6.10 −2 0.12 0.12 0.131.6.10 −2 2.5.10 −2 1.7.10 −2 1.7.10 −2−1.1.10 −2 5.9.10 −2 6.4.10 −2−5.3.10 −2−5.9.10 −2 + 2.3.10 −2 i−5.9.10 −2 − 2.3.10 −2 ik: 4 5 6 78.5.10 −2 + 1.8.10 −2 i 9.0.10 −2 + 1.6.10 −2 i 9.1.10 −2 0.14MG2 8.5.10 −2 − 1.8.10 −2 i 9.0.10 −2 − 1.6.10 −2 i 9.1.10 −2 + 2.2.10 −2 i 9.5.10 −2 + 3.5.10 −2 i−2.8.10 −2 −2.4.10 −2 9.1.10 −2 − 2.2.10 −2 i 9.5.10 −2 − 3.5.10 −2 i1 − λ ∗ k 9.0.10 −3 4.4.10 −2 −1.0.10 −2 + 1.6.10 −2 i −1.3.10 −2 + 1.7.10 −2 i7.9.10 −3 −1.0.10 −2 − 1.6.10 −2 i −1.3.10 −2 − 1.7.10 −2 i3.3.10 −3 6.6.10 −28.4.10 −3MILU preconditioner. The MILU(α 1 ) preconditioner converges satisfactorily withα 1 = 0.95. Here, GMRES(50) did not lead to convergence. For BiCGSTAB, thenumber of iterations to reduce the initial residual with eight orders of magnitudewere as follows: for 129 2 grid, 32 iterations; for 257 2 grid, 58 iterations; for 513 2 grid,133 iterations.Problem III: The rotated anisotropic diffusion equation. The third equation investigatedis the well-known rotated anisotropic diffusion equation(47)−(cos 2 β + ɛ sin 2 β) ∂2 φ∂x 2 − 2(ɛ − 1) cos β sin β ∂2 φ∂x∂y − (ɛ cos2 β + sin 2 β) ∂2 φ∂y 2 = 1on (0, 1) × (0, 1).