HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A

Model problem 1, standard triangulation

In the second column of Table 3 we consider the special case of exact solvers ( $B_I = C_I = K_I$). The bounded iteration numbers confirm the spectral equivalence of $C_C$ to $S_C$.

In all other columns the three components $B_I, C_I$, and $C_C$ are chosen as described above. The iteration numbers are compared for the MG and FMG basis transformation, respectively, and for different smoothing parameters $\omega$ of $B_I$. We observe that the cg iteration numbers depend heavily on the proper choice of $\omega$. The best iteration numbers are obtained with $\omega \approx 1.2 \ldots 1.5$. This corresponds to the theoretical results of the full two-grid operator (cf. Figure 3). Thus the theoretical results obtained there can be generalized (to some extent) to the full multigrid operator.

For a fixed $\omega \le 1.5$ the comparison reveals that FMG requires less (or no more) iterations than multigrid. Finally, Table 4 summarized the degrees of freedom and the computational time for $\omega = 1.5$.

 

Table: Model problem 1, standard triangulation, # cg iterations

 

Table: Model problem 1, standard triangulation, # cg iterations and time for $\omega=1.5$

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A