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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.