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A Fixed-Point Iteration Method for High Frequency Helmholtz Equations

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Abstract

For numerically solving the high frequency Helmholtz equation, the conventional finite difference and finite element methods based on discretizing the equation on meshes usually suffer from the numerical dispersion errors (‘pollution effect’) that require very refined meshes (Babuska and Sauter in SIAM Rev 42(3):451–484, 2000). Asymptotic methods like geometrical optics provide an alternative way to compute the solutions without ‘pollution effect’, but they generally can only compute locally valid approximations for the solutions and fail to capture the caustics faithfully. In order to obtain globally valid solutions efficiently without ‘pollution effect’, we transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for certain prescribed accuracy requirement. And the Anderson acceleration is incorporated to accelerate the convergence. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.

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The datasets/codes generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

This work was partially supported by NSF DMS 1719907.

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Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Songting Luo

  2. Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, NC, 27708-0291, USA

    Qing Huo Liu

Authors
  1. Songting Luo
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  2. Qing Huo Liu
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Contributions

QHL: Conceptualization, Methodology, Writing—Review and Editing; SL: Conceptualization, Methodology, Formal analysis, Visualization, Validation, Writing—Original Draft , Writing—Review and Editing.

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Correspondence to Songting Luo.

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This work was partially supported by National Science Foundation Division of Mathematical Sciences 1719907.

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Luo, S., Liu, Q.H. A Fixed-Point Iteration Method for High Frequency Helmholtz Equations. J Sci Comput 93, 74 (2022). https://doi.org/10.1007/s10915-022-02039-8

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  • DOI: https://doi.org/10.1007/s10915-022-02039-8

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