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Why it is difficult to solve Helmholtz problems with classical iterative methods. (English) Zbl 1248.65128
Graham, Ivan G. (ed.) et al., Numerical analysis of multiscale problems. Selected papers based on the presentations at the 91st London Mathematical Society symposium, Durham, UK, July 5–15, 2010. Berlin: Springer (ISBN 978-3-642-22060-9/hbk; 978-3-642-22061-6/ebook). Lecture Notes in Computational Science and Engineering 83, 325-363 (2012).
Summary: 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.
For the entire collection see [Zbl 1234.65007].

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N85 Fictitious domain methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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