## 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 - 2020 |

### Bibliographical note

Funding Information:∗Received by the editors November 21, 2018; accepted for publication (in revised form) by L. Giraud November 4, 2019; published electronically January 9, 2020. https://doi.org/10.1137/18M1228128 Funding: The work of the second and third authors was supported by National Science Foundation grant DMS-1521573 and the Minnesota Supercomputing Institute. The work of the fourth author was supported by the Simons Foundation under the MATH + X program, National Science Foundation grant DMS-1559587, the corporate members of the Geo-Mathematical Group at Rice University, and Total. †Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005 (xiao.liu@rice.edu, mdehoop@rice.edu). ‡Department of Mathematics, Emory University, Atlanta, GA 30322 (yxi26@emory.edu). §Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 (saad@umn.edu).

Publisher Copyright:

© 2020 Society for Industrial and Applied Mathematics

## Keywords

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