Numerical solutions to a boundary-integral equation with spherical radial basis functions

Abstract

The Laplace equation in the exterior of the unit sphere with a Dirichlet boundary condition arises from geodesy, oceanography and meteorology. This problem is reformulated into a weakly singular integral equation on the sphere. We study the use of spherical radial basis functions to find approximate solutions to this integral equation using collocation methods. Experiments with data collected by a NASA satellite are performed to clarify the method. Our results illustrate how scattered data can be handled when solving boundary value problems in the exterior of the sphere. References W. Freeden, T. Gervens, and M. Schreiner. Constructive approximation on the sphere with applications to geomathematics. Oxford: Clarendon Press, New York, 1998. T. M. Morton and M. Neamtu. Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory, 114 (2002), 242--268. doi:10.1006/jath.2001.3642 C. Mueller. Spherical Harmonics, volume 17 of Lecture Notes in Mathematics. Springer--Verlag, Berlin, 1966. F. J. Narcowich and J. D. Ward. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 33 (2002), 1393--1410. doi:10.1137/S0036141001395054 J.-C. Nedelec. Acoustic and Electromagnetic Equations. Springer--Verlag, New York, 2000. T. D. Pham and T. Tran. Collocation solutions to pseudodifferential equations of negative orders on the sphere using spherical radial basis functions. In preparation. I. J. Schoenberg. Positive definite function on spheres. Duke Math. J., 9 (1942), 96--108. doi:10.1215/S0012-7094-42-00908-6 S. Svensson. Pseudodifferential operators -- a new approach to the boundary problems of physical geodesy. Manuscr. Geod., 8 (1983), 1--40. H. Wendland. Meshless Galerkin methods using radial basis functions. Math. Comp., 68 (1999), 1521--1531. doi:10.1090/S0025-5718-99-01102-3 H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. Y. Xu and E. W. Cheney. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116 (1992), 977--981. doi:10.2307/2159477