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

Young Kyu Lee, Shingyu Leung

Communications on Applied Mathematics and Computation ›› 2023, Vol. 6 ›› Issue (2) : 1189-1216. DOI: 10.1007/s42967-023-00303-8

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.

Keywords

Laplace-Beltrami operator / Level set method / Implicit representation / Eigenvalues / Numerical PDEs

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Young Kyu Lee, Shingyu Leung. A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces. Communications on Applied Mathematics and Computation, 2023, 6(2): 1189‒1216 https://doi.org/10.1007/s42967-023-00303-8

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Funding
University Grants Committee(16302223)

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