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Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

Published online by Cambridge University Press:  03 March 2015

Ira Livshits
Ira Livshits*
Affiliation:
Department of Mathematical Sciences, Ball State University, Muncie IN, 47306, USA
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Abstract

A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping. It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations — a Shifted Laplacian preconditioner for Krylov-type methods. Such preconditioning significantly accelerates Krylov iterations, much more so than the multigrid based on original Helmholtz equations. In this paper, we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift, and, based on our observations, propose a new hybrid approach that combines the two. Our analytical conclusions are supported by two-dimensional numerical results.

Keywords

65F1065N2265N55Indefinite Helmholtz operatormultigridshifted Laplacianray correction
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 1 , February 2015 , pp. 136 - 148
Copyright
Copyright © Global-Science Press 2015 

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