Abstract
In this paper, we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. Two variants of the collocation method are considered (sometimes called symmetric and unsymmetric, although here both are symmetric). A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fasshauer G.E.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)
Fasshauer G.E., Jerome J.W.: Multistep approximation algorithms: improved convergence rates through postconditioning with smooth kernels. Adv. Comput. Math. 10, 1–27 (1999)
Flyer N., Wright G.: Transport schemes on a sphere using radial basis functions. J. Comput. Phys. 226, 1059–1084 (2007)
Flyer N., Wright G.: A radial basis function method for the shallow water equations on a sphere. Proc. R. Soc. A 465, 1949–1976 (2009)
Franke C., Schaback R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 381–399 (1998)
Franke C., Schaback R.: Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput. 93, 73–82 (1998)
Franke, R.: A critical comparison of some methods for interpolation of scattered data. Technical Report NPS-53-79-003, Naval Postgraduate School (1979)
Giesl P., Wendland H.: Meshless collocation: error estimates with application to dynamical systems. SIAM J. Numer. Anal. 45, 1723–1741 (2007)
Hon Y.C., Mao X.Z.: An efficient numerical scheme for Burgers’ equation. Appl. Math. Comput. 95, 37–50 (1998)
Hon Y.C., Schaback R.: On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119, 177–186 (2001)
Kansa E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics i. Comput. Math. 19, 127–145 (1990)
Kansa E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics ii: solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. 19, 147–161 (1990)
Le Gia Q.T.: Galerkin approximation for elliptic PDEs on spheres. J. Approx. Theory 130, 123–147 (2004)
Le Gia Q.T., Narcowich F.J., Ward J.D., Wendland H.: Continuous and discrete least-square approximation by radial basis functions on spheres. J. Approx. Theory 143, 124–133 (2006)
Le Gia Q.T., Sloan I.H., Wendland H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48, 2065–2090 (2010)
Morton T.M., Neamtu M.: Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory 114, 242–268 (2002)
Müller, C.: Spherical Harmonics, Lecture Notes in Mathematics, vol. 17. Springer-Verlag, Berlin (1966)
Narcowich F.J., Ward J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33, 1393–1410 (2002)
Power H., Barraco V.: A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations. Comput. Math. Appl. 43, 551–583 (2002)
Saff E.B., Kuijlaars A.B.J.: Distributing many points on a sphere. Math. Intell. 19, 5–11 (1997)
Saff E.B., Rakhmanov E.A., Zhou Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1, 647–662 (1994)
Schoenberg I.J.: Positive definite function on spheres. Duke Math. J. 9, 96–108 (1942)
Wendland H.: Meshless Galerkin methods using radial basis functions. Math. Comput. 68, 1521–1531 (1999)
Wendland H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains. Numer. Math. (to appear) (2010)
Xu Y., Cheney E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977–981 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Le Gia, Q.T., Sloan, I.H. & Wendland, H. Multiscale RBF collocation for solving PDEs on spheres. Numer. Math. 121, 99–125 (2012). https://doi.org/10.1007/s00211-011-0428-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-011-0428-6