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Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

  • O. G. Ernst
  • M. J. Gander
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 83)

Abstract

In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.

Keywords

Coarse Grid Helmholtz Equation Multigrid Method Domain Decomposition Method Schwarz Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

The authors would like to acknowledge the support of the Swiss National Science Foundation Grant number 200020-121828.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Section of MathematicsUniversity of GenevaGeneva 4Switzerland
  2. 2.TU Bergakademie Freiberg, Institut für Numerische Mathematik und OptimierungFreibergGermany

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