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

52 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

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The work of this author was supported by an NSERC postdoctoral fellowship, NSF grant No. CCF-0321917, and by Award No. KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST).The work of this author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship.The work of this author was partially supported by a Grant from NSERC Canada.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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