Solving eigenvalue problems on curved surfaces using the Closest Point Method

Authors:
Colin B. Macdonald;Jeremy Brandman;Steven J. Ruuth
Affiliations:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK;Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, USA;Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Venue:
Journal of Computational Physics
Year:
2011

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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.