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Error and stability estimates for surface-divergence free RBF interpolants on the sphere


Authors: Edward J. Fuselier, Francis J. Narcowich, Joseph D. Ward and Grady B. Wright
Journal: Math. Comp. 78 (2009), 2157-2186
MSC (2000): Primary 41A05, 41A63; Secondary 76M25, 86-08, 86A10
Published electronically: January 22, 2009
MathSciNet review: 2521283
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Abstract: Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in $ \mathbb{R}^3$. In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, $ \mathbb{S}^2$. In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.


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Additional Information

Edward J. Fuselier
Affiliation: Department of Mathematical Sciences, United States Military Academy, West Point, New York 10996
Email: edward.fuselier@usma.edu

Francis J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: fnarc@math.tamu.edu

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: jward@math.tamu.edu

Grady B. Wright
Affiliation: Department of Mathematics, Boise State University, Boise, Idaho 83725-1555
Email: wright@diamond.boisestate.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02214-5
Keywords: Sphere, vector fields, incompressible fluids, radial basis functions, numerical modeling, stream function.
Received by editor(s): February 8, 2008
Received by editor(s) in revised form: August 25, 2008
Published electronically: January 22, 2009
Additional Notes: The second author’s research was supported by grant DMS-0504353 from the National Science Foundation.
The third author’s research was supported by grant DMS-0504353 from the National Science Foundation.
The fourth author’s research was supported by grant ATM-0801309 from the National Science Foundation.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.