@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.}
}