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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.