Mathematics – Numerical Analysis
Scientific paper
2011-06-21
J. Comput. Phys. 230 (2011) 7944-7956
Mathematics
Numerical Analysis
Scientific paper
10.1016/j.jcp.2011.06.021
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.
Brandman Jeremy
Macdonald Colin B.
Ruuth Steven J.
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