The Mathematical Institute, University of Oxford, Eprints Archive

Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method

Macdonald, C. B. and Brandman, J. and Ruuth, S. J. (2011) Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method. Journal of Computational Physics . (Submitted)

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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.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:1169
Deposited By:Peter Hudston
Deposited On:14 May 2011 08:44
Last Modified:09 Feb 2012 14:04

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