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A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

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Abstract

We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces. The approach follows an embedding approach for solving the surface eikonal equation. We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood. Our proposed algorithm is easy to implement and efficient. We will give some two- and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their insightful comments and suggestions, which greatly improved the quality of this paper. Their careful reading and constructive feedback helped us to refine our ideas and clarify our arguments.

Funding

The work of Leung was supported in part by the Hong Kong RGC 16302223.

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Authors and Affiliations

  1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

    Young Kyu Lee & Shingyu Leung

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  1. Young Kyu Lee
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  2. Shingyu Leung
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Correspondence to Shingyu Leung.

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On behalf of all the authors, the corresponding author states that there is no conflict of interest.

Additional information

Dedicate to Professor Stanley Osher on the occasion of his 80th birthday.

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Lee, Y.K., Leung, S. A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00303-8

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  • DOI: https://doi.org/10.1007/s42967-023-00303-8

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