Inf-sup condition for spherical polynomials and radial basis functions on spheres

Sloan, Ian; Wendland, Holger

doi:10.1090/S0025-5718-09-02207-8

Inf-sup condition for spherical polynomials and radial basis functions on spheres
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by Ian H. Sloan and Holger Wendland PDF

Math. Comp. 78 (2009), 1319-1331 Request permission

Abstract:

Interpolation by radial basis functions and interpolation by polynomials are both popular methods for function reconstruction from discrete data given on spheres. Recently, there has been an increasing interest in employing these function families together in hybrid schemes for scattered data modeling and the solution of partial differential equations on spheres. For the theoretical analysis of numerical methods for the associated discretized systems, a so-called inf-sup condition is crucial. In this paper, we derive such an inf-sup condition, and show that the constant in the inf-sup condition is independent of the polynomial degree and of the chosen point set, provided the mesh norm of the point set is sufficiently small. We then use the inf-sup condition to derive a new error analysis for the hybrid interpolation scheme of Sloan and Sommariva.

References

Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179–192. MR 359352, DOI 10.1007/BF01436561
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
Jean Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^{m}$-splines, RAIRO Anal. Numér. 12 (1978), no. 4, 325–334, vi (French, with English summary). MR 519016, DOI 10.1051/m2an/1978120403251
Howard C. Elman, David J. Silvester, and Andrew J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2155549
Q. T. Le Gia, F. J. Narcowich, J. D. Ward, and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory 143 (2006), no. 1, 124–133. MR 2271729, DOI 10.1016/j.jat.2006.03.007
Simon Hubbert and Tanya M. Morton, A Duchon framework for the sphere, J. Approx. Theory 129 (2004), no. 1, 28–57. MR 2070179, DOI 10.1016/j.jat.2004.04.005
Simon Hubbert and Tanya M. Morton, $L_p$-error estimates for radial basis function interpolation on the sphere, J. Approx. Theory 129 (2004), no. 1, 58–77. MR 2070180, DOI 10.1016/j.jat.2004.04.006
Kurt Jetter, Joachim Stöckler, and Joseph D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp. 68 (1999), no. 226, 733–747. MR 1642746, DOI 10.1090/S0025-5718-99-01080-7
Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), no. 250, 743–763. MR 2114646, DOI 10.1090/S0025-5718-04-01708-9
Ian H. Sloan and Alvise Sommariva, Approximation on the sphere using radial basis functions plus polynomials, Adv. Comput. Math. 29 (2008), no. 2, 147–177. MR 2420870, DOI 10.1007/s10444-007-9048-1
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724