Domain decomposition iteration (10) is typically implemented as
where the right-hand side term 
represents the discretization
of the internal boundary conditions, which is always exact, and
the left-hand side term 
indicates solution of
the subdomain problems using some type of solver, which was assumed accurate
enough in the previous section.
In general, the stationary solution of (20) satisfies
the perturbed equations 
instead of Au = f.
Since with inaccurate subdomain solution, the difference between
and N may be quite large, the computed solution u
may have a very large error. Since the algorithm of the previous section
relies on (20), we may not use this
procedure with inaccurate
solution of subdomain problems. Instead we must use
for which the stationary solution u always satisfies Au = f.
With inaccurate subdomain solution, we have
with 
the Gauss-Seidel (sequential/multiplicative) version and
the Jacobi (parallel/additive) version of 
.
The matrices 
represent inaccurate subdomain solution.
The matrix vector product 
is computed like
where, for instance, 
represents an approximate
solution in subdomain 1 with a low accuracy.
Another possibility is to take
to be some incomplete LU factorization
of 
, see further on.
The GMRES subdomain solution implicitly constructs a polynomial 
of the subdomain matrix 
such that the final residual
is minimal in the Euclidean norm. Specifically, with
initial guess 
and right-hand side 
, we get for the final
subdomain solution 
.
Since the polynomial 
depends on both the required accuracy
and the right-hand side (initial residual), the matrix
can be different for each v.
Therefore, GMRES acceleration cannot be used since
the preconditioner 
varies in each step. Only for the case
we may apply GMRES acceleration, but we still apply
GCR in this case.
