Solving eigenvalue problems on curved surfaces using the Closest Point Method
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.
Cite
@article{arxiv.1106.4351,
title = {Solving eigenvalue problems on curved surfaces using the Closest Point Method},
author = {Colin B. Macdonald and Jeremy Brandman and Steven J. Ruuth},
journal= {arXiv preprint arXiv:1106.4351},
year = {2011}
}