Inexact Solves

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Inexact Solves

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


in the definitions of


, we can easily obtain inexact preconditioners



. 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 (gif) and construct a preconditioner from the factorization in (gif) by replacing the terms


respectively, where

can be either


. Care must be taken to scale


so that they are as close to


as possible respectively - it is not sufficient that the condition number of


be close to unity, because the scaling of the coupling matrix

may be wrong.

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995