Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method
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 su...
| मुख्य लेखकों: | , , |
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| स्वरूप: | Journal article |
| प्रकाशित: |
2011
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| _version_ | 1826303013158912000 |
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| author | Macdonald, C Brandman, J Ruuth, S |
| author_facet | Macdonald, C Brandman, J Ruuth, S |
| author_sort | Macdonald, C |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T05:56:10Z |
| format | Journal article |
| id | oxford-uuid:ea9469d4-282d-4544-9f0e-3d8a8a45b9aa |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:56:10Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:ea9469d4-282d-4544-9f0e-3d8a8a45b9aa2022-03-27T11:03:18ZSolving Eigenvalue Problems on Curved Surfaces using the Closest Point MethodJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ea9469d4-282d-4544-9f0e-3d8a8a45b9aaMathematical Institute - ePrints2011Macdonald, CBrandman, JRuuth, SEigenvalue 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. |
| spellingShingle | Macdonald, C Brandman, J Ruuth, S Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title | Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title_full | Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title_fullStr | Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title_full_unstemmed | Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title_short | Solving Eigenvalue Problems on Curved Surfaces using the Closest Point Method |
| title_sort | solving eigenvalue problems on curved surfaces using the closest point method |
| work_keys_str_mv | AT macdonaldc solvingeigenvalueproblemsoncurvedsurfacesusingtheclosestpointmethod AT brandmanj solvingeigenvalueproblemsoncurvedsurfacesusingtheclosestpointmethod AT ruuths solvingeigenvalueproblemsoncurvedsurfacesusingtheclosestpointmethod |