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The exact solves involving 
and 
in
can be replaced by inexact solves
and 
,
which can be standard elliptic preconditioners themselves
(e.g. multigrid, ILU, SSOR, etc.).
For the Schwarz methods, the modification is straightforward
and the Inexact Solve Additive Schwarz Preconditioner
is simply:
The Schur Complement methods require more changes to accommodate
inexact solves.
By replacing 
by 
in
the definitions of 
and 
, we can easily obtain
inexact preconditioners 
and 
for 
.
The main difficulty is, however, that the evaluation of the product
requires exact subdomain solves in 
.
One way to get around this
is to use an inner iteration using 
as a preconditioner for 
in order to compute the action
of 
.
An alternative is to perform the iteration on the larger system
(
) and construct a preconditioner from the
factorization in () by replacing the terms
by 
respectively,
where 
can be either 
or 
.
Care must be taken to scale 
and 
so that they are as close to 
and as possible respectively -
it is not sufficient that the condition number of 
and 
be close to unity, because
the scaling of the coupling matrix 
may be wrong.