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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.