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

MULTIGRID AS A SOLVER AND A PRECONDITIONER 1050.01Spectrum MG2, F(0,2), 33 x 33 grid0.005Im[z]0-0.005-0.01-0.2 0 0.2 0.4 0.6 0.8 1Re[z]0.01Spectrum MG2, F(0,2), 65 x 65 grid0.005Im[z]0-0.005-0.01-0.2 0 0.2 0.4 0.6 0.8 1Re[z]FIG. 7. The eigenvalue spectra for the rotated anisotropic diffusion problem, ɛ = 10 −5 , β = 135 oon a 33 2 and a 65 2 grid, with MG2 F (0, 2).Spectrum analysis. For these parameters we also calculated the eigenvalue spectrumof the Richardson iteration matrix (5) for a 33 2 and a 65 2 problem. The spectrapresented in Figure 7 are obtained with MG2 and the F(0,2)-cycle. They are almostidentical to the spectra obtained with MG1. With the two eigenvalue picturesin Figure 7 it is possible to observe the mesh dependency of the multigrid convergence.It can be seen, for example, that the spectral radius is increasing as the gridgets finer. The spectral radius is, for these coarse grid problems, already larger than0.6. Therefore, the multigrid convergence slows down more dramatically than theconvergence of the preconditioned Krylov methods. Many eigenvalues are clusteredaround zero and only a limited number of eigenvalues are larger than 0.4 for bothgrid sizes, so eigenvalues of the preconditioned matrix (AK −1 ) are clustered aroundone, which is advantageous for the Krylov methods. The convergence of MG1, MG2,and MG3 as solution methods (Figure 8(a)) and as preconditioners for GMRES(20)(Figure 8(b)) is presented for the 33 2 grid. Also, for this problem, it is interesting toexamine the way GMRES minimizes the residual to find the solutions of the different