It is possible to obtain significant reductions in computing time by inaccurate solution of subproblems. When the subdomain solution accuracy is lowered, the number of iterations increases only slightly, which is confirmed by Theorem 2. Especially for small numbers of blocks (equivalently: large subdomains) the reduction in computing time can be quite large.
For small subdomain problems, superlinear convergence for the subdomain GMRES solver can occur earlier so that a reduction in subdomain solution accuracy can lead to only a very small reduction of work needed to solve subdomain problems, but may still cause a more significant increase in the number of iterations needed by the outer GCR iteration.
The sensitivity of convergence in the outer GCR loop stays approximately the same as the number of subdomains is enlarged. This was shown to be true for the additive algorithm in Section 3.4 (Theorems 1 and 2), but it probably also holds for the multiplicative algorithm. Convergence of the multiplicative algorithm seems to be more sensitive to the subdomain solution accuracy than the additive algorithm.
The actual reductions obtained by inaccurate subdomain solution are very much problem dependent. Significant reductions in computing time can be obtained for the momentum equations. For the pressure equations these reductions are typically much less.
The RIBLU(
) postconditioned GCR method does not perform well
for 
close to 1. For such methods, the 
parameter
only improves convergence of single-block solution but its effect
on convergence of multi-block solution is much less.
As shown in [9] and Figure 7,
the number of iterations needed with multiplicative RIBLU(0) postconditioners
is only slightly larger than with single-block solution.
Generalizations of the RIBLU(
) postconditioner to more subdomains
that preserve this property are therefore of interest.
Furthermore, overheads in the implementation can be quite important
and are especially to the disadvantage of the RIBLU(
) algorithms.
These disadvantages of the current
RIBLU(
) postconditioner prevent a reduction of computing time to
almost that
of single-block solution. The optimized restarted GCR method does not give
significant reductions in computing time because of an increased
number of iterations compared to Jackson & Robinson truncation.
Parallel implementation of the GCR based algorithm is attractive,
because convergence of the outer GCR loop does not depend sensitively
on the subdomain solution accuracy. Therefore, the number of iterations
will in general be approximately the same as with very accurate subdomain
solution, so that reduced computing time is almost certain.
Only for very inaccurate
subdomain solution, for instance when the RIBLU(
) postconditioner
is used, we get a significant increase in the number of iterations and
therefore communication.
Inaccurate solution of subdomain problems combined with GCR acceleration removes the restriction inherent in GMRES solution of interface equations (19) that subdomain problems should be solved accurately (enough). The GCR based algorithm is therefore in general more reliable than the GMRES algorithm for solving interface equations.