@TechReport{ it:2008-002,
author = {Owe Axelsson and Janos Karatson},
title = {Equivalent Operator Preconditioning for Linear Elliptic
Problems},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2008},
number = {2008-002},
month = jan,
note = {A preliminary version of the same article is published as
Preprint 2007-04, ELTE Dept. Appl. Anal. Comp. Math.,
\url{http://www.cs.elte.hu/applanal/preprints}},
abstract = {The numerical solution of linear elliptic partial
differential equations most often involves a finite element
or finite difference discretization. To preserve sparsity,
the arising system is normally solved using an iterative
solution method, commonly a preconditioned conjugate
gradient method. Preconditioning is a crucial part of such
a solution process. It is desirable that the total
computational cost will be optimal, i.e. proportional to
the degrees of freedom of the approximation used, which
also includes mesh independent convergence of the
iteration. This paper surveys the equivalent operator
approach, which has proven to provide an efficient general
framework to construct such preconditioners. Hereby one
first approximates the given differential operator by some
simpler differential operator, and then one chooses as
preconditioner the discretization of this operator for the
same mesh. In this survey we give a uniform presentation of
this approach, including theoretical foundation and several
practically important applications.}
}