Solving eigenvalue problems on curved surfaces using the Closest Point Method
Provides a practical, easy-to-implement method for computing eigenvalues on general curved surfaces, which is useful for scientists and engineers in geometry processing and physics.
The paper presents a simple algorithm based on the Closest Point Method for solving eigenvalue problems of the Laplace-Beltrami operator on curved surfaces, achieving second-order accuracy and handling open surfaces with boundary conditions. Convergence studies demonstrate its effectiveness.
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.