In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative method.

Banjai, L; Hackbusch, W (2008). *Hierarchical matrix techniques for low- and high-frequency Helmholtz problems.* IMA Journal of Numerical Analysis, 28(1):46-79.

## Abstract

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative method.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Helmholtz equation; boundary element method; hierarchical matrices |

Language: | English |

Date: | 2008 |

Deposited On: | 09 Nov 2009 00:01 |

Last Modified: | 05 Apr 2016 13:23 |

Publisher: | Oxford University Press |

ISSN: | 0272-4979 |

Publisher DOI: | https://doi.org/10.1093/imanum/drm001 |

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