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Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

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Abstract

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.

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Acknowledgments

The research of Bo Wang is partially supported by NSFC Grants (11341002, 11401206 and 11771137), the Construct Program of the Key Discipline in Hunan Province and a Scientific Research Fund of Hunan Provincial Education Department (No. 16B154). Ziqing Xie is partially supported by NSFC (91430107, 11171104 and 11771138) and the Construct Program of the Key Discipline in Hunan Province.

The research of Bo Wang and Li-Lian Wang is supported by a Singapore MOE AcRF Tier 1 Grant (RG 27/15).

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Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, People’s Republic of China

    Bo Wang & Ziqing Xie

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Bo Wang & Li-Lian Wang

Authors
  1. Bo Wang
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  2. Li-Lian Wang
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  3. Ziqing Xie
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Correspondence to Li-Lian Wang.

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Communicated by: Francesca Rapetti

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Wang, B., Wang, LL. & Xie, Z. Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids. Adv Comput Math 44, 951–985 (2018). https://doi.org/10.1007/s10444-017-9569-1

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  • DOI: https://doi.org/10.1007/s10444-017-9569-1

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