Additive Schwarz preconditioners for interpolation of divergence-free vector fields on spheres

Abstract

The linear system arising from the interpolation problem of surface divergence-free vector fields using radial basis functions tends to be ill-conditioned when the separation radius of the scattered data is small. When the surface under consideration is the unit sphere, we introduce a preconditioner based on the additive Schwarz method to accelerate the solution process. Theoretical estimates for the condition number of the preconditioned matrix are given. Numerical experiments using scattered data from the MAGSAT satellite show the effectiveness of our preconditioner. References E. Fuselier, F. Narcowich, J. D. Ward, and G. Wright. Error and stability estimates for surface-divergence free {RBF} interpolants on the sphere. Math. Comp., 78:2157--2186, 2009. doi:10.1090/S0025-5718-09-02214-5. J. R. Holton. An Introduction to Dynamic Meteorology. Academic Press, San Francisco, 3rd ed., 1992. doi:10.1119/1.1987371. Q. T. {Le Gia}, I. H. Sloan, and T. Tran. Overlapping additive {S}chwarz preconditioners for elliptic {PDE}s on the unit sphere. Math. Comp., 78:79--101, 2009. doi:10.1090/S0025-5718-08-02150-9. Q. T. {Le Gia} and T. Tran. An overlapping additive {S}chwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions. J. of Complexity, 26:552--573, 2010. doi:10.1016/j.jco.2010.06.003. F. J. Narcowich, J. D. Ward, and G. B. Wright. Divergence-free {RBFs} on surfaces. J. Fourier Anal. Appl., 13:643--663, 2007. doi:10.1007/s00041-006-6903-2. H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. doi:10.2277/0521843359.