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

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 erro...

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Bibliographic Details
Published in:Mathematics of computation 2009-10, Vol.78 (268), p.2157-2186
Main Authors: FUSELIER, EDWARD J., NARCOWICH, FRANCIS J., WARD, JOSEPH D., WRIGHT, GRADY B.
Format: Article
Language:English
Subjects:
Approximation
Approximations and expansions
Eigenvalues
Exact sciences and technology
Interpolation
Mathematical analysis
Mathematical functions
Mathematical vectors
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Sciences and techniques of general use
Sobolev spaces
Spherical harmonics
Stream functions
Vector fields
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-09-02214-5