@TechReport{ it:2007-033,
author = {Owe Axelsson and Radim Blaheta and Maya Neytcheva},
title = {A Black-Box Generalized Conjugate Gradient Minimum
Residual Method Based on Variable Preconditioners and Local
Element Approximations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2007},
number = {2007-033},
month = dec,
abstract = {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.}
}