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Numerical Algorithms

, Volume 50, Issue 3, pp 297–380 | Cite as

Equivalent operator preconditioning for elliptic problems

  • O. Axelsson
  • J. Karátson
Original Paper

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. In order to enable the solution of very large-scale systems, it is desirable that the total computational cost will be of optimal order, i.e. proportional to the degrees of freedom of the approximation used, which also induces 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 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 for both symmetric and nonsymmetric equations and systems, and some nonlinear examples in the context of Newton linearization.

Keywords

Elliptic problem Conjugate gradient method Preconditioning Equivalent operators Compact operators 

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© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Institute of Geonics AS CROstravaCzech Republic
  3. 3.Department of Applied AnalysisELTE UniversityBudapestHungary

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