The talk will focus on solution methods for augmented linear systems of
the form
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$[ A B ][x] = [b] [ B' 0 ][y] [0]$.
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Augmented linear systems of this type arise in several areas of
numerical applied mathematics including mixed finite element / finite
difference discretisations of flow equations (Darcy flow and Stokes
flow), electrical network simulation and optimisation. The general
properties of such systems are that they are large, sparse and
symmetric, and efficient solution techniques should make use of the
block structure inherent in the system as well as of these properties.
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Iterative linear solution methods will be described that
attempt to take advantage of the structure of the system, and
observations on augmented
systems, in particular the distribution of their eigenvalues, will be
presented which lead to further iterative methods and also to
preconditioners for existing solution methods. For the particular case
of Darcy flow, comments on properties of domain decomposition methods
of additive Schwarz type and similarities to incomplete factorisation
preconditioners will be made.