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Quadrature by Two Expansions for Evaluating Helmholtz Layer Potentials

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Abstract

In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. The QB2X method explicitly includes nonlinear dependency of the boundary geometry in the plane wave expansions, thereby providing for higher-order representations of both the boundary geometry and density functions in the integrand, with its convergence following standard FMM error analysis. Numerical results are presented to demonstrate the performance of the QB2X method for Helmholtz layer potentials using one expansion center for the entire FMM-leaf box for both flat and curved boundaries with various densities.

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Data Availability

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

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Funding

M.H. Cho was supported by NSF grant DMS 2012382 and a grant from the Simons Foundation (No. 404499). J. Huang was supported by NSF grant DMS 2012451.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA 01854, USA

    Jared Weed & Min Hyung Cho

  2. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA

    Lingyun Ding & Jingfang Huang

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  1. Jared Weed
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Correspondence to Min Hyung Cho.

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Weed, J., Ding, L., Huang, J. et al. Quadrature by Two Expansions for Evaluating Helmholtz Layer Potentials. J Sci Comput 95, 96 (2023). https://doi.org/10.1007/s10915-023-02222-5

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