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A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains

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Abstract

We extend the fictitious domain spectral method presented in Gu and Shen (SIAM J Sci Comput 43:A309–A329, 2021) for elliptic PDEs in bounded domains to the Helmhotlz equation in exterior domains. We first reduce the problem in an exterior domain to a bounded domain using the exact Dirichlet-to-Neumann operator. Next, we formulate the reduced problem into an equivalent problem in an annulus by using a fictitious domain approach. Then, we apply the Fourier-spectral method in the radial direction to reduce the problem in an annulus to a sequence of 1-D Bessel-type equations, each with a one-sided open boundary condition that are to be determined by the boundary condition of the original Helmholtz equation. We solve these 1-D Bessel-type equations by the Legendre-spectral method, and determine the open boundary conditions with a least square approach. We derive a wave number explicit error estimate for the special case of a circular obstacle, and provide ample numerical results to show the effectiveness of the proposed method.

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Funding

Funding was provided by Division of Mathematical Sciences under Grand Number DMS-2012585.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong

    Yiqi Gu

  2. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Jie Shen

  3. Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo, 315200, Zhejiang, People’s Republic of China

    Jie Shen

Authors
  1. Yiqi Gu
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  2. Jie Shen
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Correspondence to Jie Shen.

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This work is supported in part by NSF Grant DMS-1720442 and AFOSR Grant FA9550-16-1-0102.

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Gu, Y., Shen, J. A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains. J Sci Comput 94, 46 (2023). https://doi.org/10.1007/s10915-023-02098-5

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  • DOI: https://doi.org/10.1007/s10915-023-02098-5

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