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.