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.
Note: A preliminary version of the same article is published as Preprint 2007-04, ELTE Dept. Appl. Anal. Comp. Math., http://www.cs.elte.hu/applanal/preprints
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