We propose a simple ₁embedding method for computing the eigenvalues and eigen-functions 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 and an extension operator on an outer layer of the computational tube. To observe a numerical convergence as the underlying mesh size approaches zero, we study different choices of the tube radius in the form of O(Δx^γ) for γ ∈ [0, 1]. 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....[ Read more ]

We propose a simple ₁embedding method for computing the eigenvalues and eigen-functions 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 and an extension operator on an outer layer of the computational tube. To observe a numerical convergence as the underlying mesh size approaches zero, we study different choices of the tube radius in the form of O(Δx^γ) for γ ∈ [0, 1]. 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.