## Abstract

We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n^{1}/^{3}) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.

Original language | English (US) |
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Pages (from-to) | 58-82 |

Number of pages | 25 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 41 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2020 |

## Keywords

- Cauchy integral
- Helmholtz preconditioner
- Polynomial iteration
- Shifted Laplacian