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

  
Verification of the model problem analysis

Here we only verify the model problem analysis of section 4 by applying the ppcg algorithm to model problem 1. The three components of the preconditioner $C$ of (1) are as follows:

• The basis transformation $B_I$ is defined by one full two-grid step. Three different parameters of the $\omega$-Jacobi smoother are tested.

• The well-known Dryja preconditioner serves as $C_C$.

• We set $C_I := K_I$.

Let $h$ be the coarse grid mesh size. The last two components ensure $\, \underline{\gamma} \;/\; \overline{\gamma} \,=\, O(1)$. $B_I$ has been investigated in section 4, and the theoretical results are comprised in Table 1, in conjecture 1, in Figure 3, and in theorem 2. They strongly suggest $\mu\,=\,\varrho \,(S_C^{-1} T_C^{\phantom{\!\!\!\!-1}}) = O(h^{-1})$ as $h \to 0$. According to (2) we expect the number of cg-iterations to be $\, O( \sqrt{ \kappa ( C^{-1} K ) } = O( \mu^{1/2} )$.

The computed values of $\, \mu = 2 \delta_i \,$ are contained in Table 1. Hence for $\omega = 0.5$ and $\omega = 1.0$ the number of iterations should be $O( h^{-1/2} ) \,$ as $h \to 0$.. For $\omega = 1.5$ we expect bounded iteration numbers for $h \ge 1/256$; for smaller $h$ the iteration numbers should grow as $O( h^{-1/2} )$. Table 2 presents the numerical results.

 

Table 2: Number of cg-iterations for the full two-grid operator

2.5ex

$n = 1/h$	4	8	16	32	64	128	256	512
$\omega=0.5$	5	7	8	10	12	18	23	37
$\omega=1.0$	5	6	6	7	7	9	10	13
$\omega=1.5$	5	7	7	8	7	7	7	8

For a fixed $\omega$ the iteration numbers show the anticipated behaviour and thus verify the model problem analysis.

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