In order to control the accuracy of a preconditioner for an outer iterative process one often involves variable preconditioners. The variability may for instance be due to the use of inner iterations in the construction of the preconditioner. Both the outer and inner iterations may be based on some conjugate gradient type of method, e.g. generalized minimum residual methods.
A background for such methods, including results about their computational complexity and rate of convergence, is given. It is then applied for a variable preconditioner arising for matrices partitioned in two-by-two block form. The matrices can be unsymmetric and also indefinite. The aim is to provide a black-box solver, applicable for all ranges of problem parameters such as coefficient jumps and anisotropy.
When applying this approach for elliptic boundary value problems, in order to achieve the latter aim, it turns out to be efficient to use local element approximations of arising block matrices as preconditioners for the inner iterations.
It is illustrated by numerical examples how the convergence rate of the inner-outer iteration method approaches that for the more expensive fixed preconditioner when the accuracies of the inner iterations increase.
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