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arXiv · 1106.4351

Solving eigenvalue problems on curved surfaces using the Closest Point Method

Colin B. Macdonald·Jeremy Brandman·Steven J. Ruuth
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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.

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Colin B. Macdonald, Jeremy Brandman, Steven J. Ruuth. 2011-06-21. Solving eigenvalue problems on curved surfaces using the Closest Point Method. https://doi.org/10.1016/j.jcp.2011.06.021

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