Stability and preconditioning for a hybrid approximation on the sphere
This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting l...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2011
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| _version_ | 1797106816801308672 |
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| author | Le Gia, QT Sloan, IH Wathen, AJ |
| author_facet | Le Gia, QT Sloan, IH Wathen, AJ |
| author_sort | Le Gia, QT |
| collection | OXFORD |
| description | This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting linear system can be preconditioned by the block-diagonal preconditioner of Murphy, Golub and Wathen. Making use of a recently derived inf-sup condition and the Brezzi stability and convergence theorem for this approximation scheme, we show that in this context the Schur complement in the above preconditioner is spectrally equivalent to a certain non-constant diagonal matrix. Numerical experiments with a non-uniform distribution of data points support the theoretically proved quality of the new preconditioner. © 2011 Springer-Verlag. |
| first_indexed | 2024-03-07T07:06:19Z |
| format | Journal article |
| id | oxford-uuid:8376bb26-6c01-4605-9d3c-7fa201b24fb4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:06:19Z |
| publishDate | 2011 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:8376bb26-6c01-4605-9d3c-7fa201b24fb42022-05-13T14:19:36ZStability and preconditioning for a hybrid approximation on the sphereJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8376bb26-6c01-4605-9d3c-7fa201b24fb4EnglishSymplectic Elements at OxfordSpringer2011Le Gia, QTSloan, IHWathen, AJThis paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting linear system can be preconditioned by the block-diagonal preconditioner of Murphy, Golub and Wathen. Making use of a recently derived inf-sup condition and the Brezzi stability and convergence theorem for this approximation scheme, we show that in this context the Schur complement in the above preconditioner is spectrally equivalent to a certain non-constant diagonal matrix. Numerical experiments with a non-uniform distribution of data points support the theoretically proved quality of the new preconditioner. © 2011 Springer-Verlag. |
| spellingShingle | Le Gia, QT Sloan, IH Wathen, AJ Stability and preconditioning for a hybrid approximation on the sphere |
| title | Stability and preconditioning for a hybrid approximation on the sphere |
| title_full | Stability and preconditioning for a hybrid approximation on the sphere |
| title_fullStr | Stability and preconditioning for a hybrid approximation on the sphere |
| title_full_unstemmed | Stability and preconditioning for a hybrid approximation on the sphere |
| title_short | Stability and preconditioning for a hybrid approximation on the sphere |
| title_sort | stability and preconditioning for a hybrid approximation on the sphere |
| work_keys_str_mv | AT legiaqt stabilityandpreconditioningforahybridapproximationonthesphere AT sloanih stabilityandpreconditioningforahybridapproximationonthesphere AT wathenaj stabilityandpreconditioningforahybridapproximationonthesphere |