@TechReport{ it:2018-010,
author = {Owe Axelsson and Maya Neytcheva},
title = {Preconditioners for Two-by-Two Block Matrices with Square
Blocks},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2018},
number = {2018-010},
month = may,
abstract = {Two-by-two block matrices with square blocks arise in the
numerical treatment of numerous applications of practical
significance, such as optimal control problems, constrained
by a state equation in the form of partial differential
equations, multiphase models, solving complex linear
systems in real arithmetics, to name a few.a
Such problems lead to algebraic systems of equations with
matrices of a certain two-by-two block form. For such
matrices, a number of preconditioners has been proposed,
some of them with tight eigenvalue bounds. In this paper it
is shown that in particular one of them, referred to as
PRESB, is very efficient, not only giving robust,
favourable properties of the spectrum but also enabling an
efficient implementation with low computational complexity.
Various applications and generalizations of this
preconditioning technique, such as in time-harmonic
parabolic and Stokes equations, eddy current
electromagnetic problems and problems with additional
box-constraints, i.e. upper and/or lower bounds of the
solution, are also discussed.
The method is based on the use of coupled inner-outer
iterations, where the inner iteration can be performed to
various relative accuracies. This leads to variable
preconditioners, thus, a flexible version of a Krylov
subspace iteration method must be used. Alternatively, some
version of a defect-correction iterative method can be
applied. }
}