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.

Mon Nov 20 08:52:54 EST 1995