The basic Schwarz domain decomposition iteration converges slowly and is not always convergent for the Navier-Stokes equations. Therefore, we use Krylov subspace acceleration, which is frequently used to accelerate domain decomposition methods, see for example [5] and many of the papers on iterative substructuring methods in [24,14,15,25,33]. The acceleration procedure used with accurate solution of subdomain problems is GMRES applied to the interface equations (19) and is described in detail in [11,10]. This section describes the procedure used with inaccurate subdomain solution.
The GCR [23] method for solving Ax = f
can be easily adapted to cope with
variable preconditioners. Because of its simplicity and for
completeness, we describe the GCR
method here. GCR is based on maintaining two
subspaces, a subspace 
for storing the
search directions 
and a subspace
with 
. In every operation
of GCR the property 
is preserved.
For simplicity we take the initial guess 
, in which case
GCR minimizes the residual
over 
. Clearly, if the 
form an orthonormal basis, we can obtain the solution by projecting onto
the space 
. So we must find 
such that
for 
, therefore,
Since 
we have
and by substituting (29)
into (28)
we get 
so that
Since 
, we have
so that 
.
This gives 
and
with 
we get
.
The GCR algorithm proceeds by choosing a new search direction 
(preferably such that 
approximates the residual 
) and computes the vector
. A modified Gram-Schmidt procedure is used to
make 
orthogonal to 
(
). The same linear
combinations of vectors are applied to the space of search directions 
to ensure that 
still holds for all i.
Figure 4 shows the resulting GCR algorithm.
Figure 4: The GCR algorithm with general search directions without
restart and with a relative stopping criterion [46]
For the special case of the search direction 
, we
obtain the classical GCR algorithm, which is equivalent to GMRES [41].
For this choice of search direction, the space 
is called the
Krylov space. The difference between GCR and GMRES is that, with the
benefit of allowing more general search directions, GCR requires
twice the storage of GMRES and 
times the number of floating
point operations for orthogonalization.
However, GCR can be combined with truncation
strategies, for instance the Jackson & Robinson [31] strategy,
whereas GMRES can only be restarted. Because of this, truncated GCR may converge
faster than GMRES, see for example Section 6.2.
Furthermore, restarted GCR can be optimized [49],
which makes GCR just as efficient as GMRES. Both optimized restarted
GCR and truncated GCR will be considered in our numerical experiments.
Recent developments have led to a more flexible GMRES algorithm which allows more general search directions, so called FGMRES [40]. The FGMRES method is used in [6] to investigate the Neumann-Dirichlet method with inexact subdomain solution. The emphasis in [6] is on restrictions on subdomain solution accuracy to retain the h-independent convergence of the Neumann-Dirichlet algorithm rather than on reduction of computing time. Optimized restarted GCR is just as efficient as FGMRES, both in memory requirements and work.
In the present paper, we use 
, which corresponds
to a single iteration of (21) with initial guess 
.
The case of multiple iterations of (21) to determine 
is not considered in this paper.
If the subdomain problems are solved (inaccurately) using GMRES, this method reduces
to GMRESR [46]
for the single domain case.
In case 
is the (relaxed) incomplete LU factorization of
, we obtain a blocked version of the subdomain RILU(
) [27]
postconditioner (with parameter 
), here
called RIBLU(
) (Relaxed Incomplete Block LU). The parameter 
may be varied to improve convergence.
The RIBLU(
) preconditioner is investigated for parallel
implementation in for example [21,32,18,19].
The present paper also investigates the multiplicative version of the
RIBLU(
) postconditioner.
The GMRES acceleration procedure may be applied with RIBLU(
), which is
equivalent to GCR acceleration in this case.
The stopping criterion for accurate solution of subdomain problems
differs from that for inaccurate solution. With accurate solution,
the stopping criterion is based on the preconditioned residual
of
only the interface unknowns. On the
other hand, with inaccurate solution, it is based on the unpreconditioned
residual r = f - A u of all unknowns. Therefore, a comparison between
the two methods is difficult.
Nevertheless, we assume that the final
solution obtained with both methods is equally accurate if the
relative stopping criterion 
is used.
This assumption was confirmed in [9].
With inaccurate subdomain solution, the results for different subdomain
solution accuracies can be compared since the stopping criterion does
not depend on the way subdomain problems are solved.
