Solving eigenvalue problems on curved surfaces using the Closest Point Method

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

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

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. © 2011 Elsevier Inc.
Original languageEnglish (US)
Pages (from-to)7944-7956
Number of pages13
JournalJournal of Computational Physics
Volume230
Issue number22
DOIs
StatePublished - Jun 2011
Externally publishedYes

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