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A Robust Multilevel Preconditioner Based on a Domain Decomposition Method for the Helmholtz Equation

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Abstract

In this paper, we present a robust multilevel preconditioner for the algebraic system resulting from the continuous interior penalty finite element method for the approximation of the Helmholtz equation. The key idea in this work is the replacement of traditional smoothers by the one level overlapping domain decomposition method on coarse grids. The proposed multilevel method then serves as a preconditioner in the outer GMRES iteration. Numerical results show that for fixed wave numbers, the convergence of our multilevel method is independent of the mesh size. Furthermore, the performance of the algorithm depends relatively mildly on the wave number.

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Acknowledgements

We thank the editor and the anonymous referees, who meticulously read through the paper and made many helpful suggestions which led to an improved presentation of this paper.

Funding

The work of the Xuejun Xu was supported by National Natural Science Foundation of China (Grant Nos. 11671302 and 11871272).

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Authors and Affiliations

  1. School of Mathematics Sciences, Soochow University, Suzhou, 215006, People’s Republic of China

    Peipei Lu

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200442, People’s Republic of China

    Xuejun Xu

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O.Box 2719, Beijing, 100190, People’s Republic of China

    Xuejun Xu

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  1. Peipei Lu
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  2. Xuejun Xu
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Correspondence to Peipei Lu.

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Lu, P., Xu, X. A Robust Multilevel Preconditioner Based on a Domain Decomposition Method for the Helmholtz Equation. J Sci Comput 81, 291–311 (2019). https://doi.org/10.1007/s10915-019-01015-z

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