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Solving eigenvalue problems on curved surfaces using the Closest Point Method
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DOI：10.1016/j.jcp.2011.06.021.png)
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Abstract
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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. (C) 2011 Elsevier Inc. All rights reserved.
Keywords:
Eigenvalues
Eigenfunctions
Laplace-Beltrami operator
Closest Point Method
Surface computation
Implicit surfaces
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