Abstract
Spherical radial basis functions are used to approximate the solution of a boundary integral equation on the unit sphere which is a reformulation of a geodetic boundary value problem. The approximate solution is computed with a corresponding meshless Galerkin scheme using scattered data from satellites. Numerical experiments show that this meshless method is superior to standard boundary element computations with piecewise constants. If we increase the element order, BEM might be competitive but then we also have to approximate appropriately the surface otherwise the convergence rate will be spoiled.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices of the AMS 51, 1186–1194 (2004)
Heck, B.: Integral Equation Methods in Physical Geodesy. In: Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.) Geodesy - The Challenge of the 3rd Millenium, pp. 197–206. Springer, Heidelberg (2002)
Le Gia, Q.T., Tran, T., Stephan, E.P.: Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions. Advances in Computational Mathematics (submitted)
Le Gia, Q.T., Tran, T., Sloan, I.H., Stephan, E.P.: Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis. Applied Numerical Mathematics (in print), http://dx.doi.org/10.1016/j.apnum.2008.12.033
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)
Stephan, E.P., Wendland, W.L.: Remarks to Galerkin and Least Squares Methods with Finite Elements for General Elliptic Problems. Manuscripta geodaetica 1, 93–123 (1976)
Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Preconditioners for Pseudodifferential Equations on the Sphere with Radial Basis Functions. Numer. Math. (to appear)
Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics 4, 389–396 (1995)
Wendland, W.L.: Asymptotic accuracy and convergence. In: Brebbia, C.A. (ed.) Progress in Boundary Element Methods, vol. 1, pp. 289–313. Pentech Press, London (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Berlin Heidelberg
About this chapter
Cite this chapter
Stephan, E.P., Tran, T., Costea, A. (2010). A Boundary Integral Equation on the Sphere for High-Precision Geodesy. In: Kuczma, M., Wilmanski, K. (eds) Computer Methods in Mechanics. Advanced Structured Materials, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05241-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-05241-5_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05240-8
Online ISBN: 978-3-642-05241-5
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)