Solving eigenvalue problems on curved surfaces using the Closest Point Method

TL;DR

This paper introduces a straightforward algorithm using the Closest Point Method to compute eigenvalues and eigenfunctions of the Laplace--Beltrami operator on various curved surfaces, enabling accurate solutions through embedding techniques.

Contribution

The paper presents a novel, simple algorithm leveraging surface embedding and the Closest Point Method for eigenvalue problems on complex curved surfaces.

Findings

Effective second-order accuracy for boundary conditions

Convergence demonstrated through multiple examples

Applicable to general curved and open surfaces

Abstract

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.