Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions
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Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions Q. T. Le Gia · E. P. Stephan · T. Tran
Received: 5 April 2009 / Accepted: 23 January 2010 / Published online: 17 June 2010 © Springer Science+Business Media, LLC 2010
Abstract We consider the exterior Neumann problem of the Laplacian with boundary condition on a prolate spheroid. We propose to use spherical radial basis functions in the solution of the boundary integral equation arising from the Dirichlet–to–Neumann map. Our approach is particularly suitable for handling of scattered data, e.g. satellite data. We also propose a preconditioning technique based on domain decomposition method to deal with ill-conditioned matrices arising from the approximation problem. Keywords Exterior Neumann problem · Boundary integral equation · Prolate spheroid · Radial basis function Mathematics Subject Classifications (2010) 65N30 · 65N38 · 65N55
Communicated by Juan Manuel Peña. Dedicated to R.S. Anderssen on the occasion of his 70th birthday. Q. T. Le Gia · T. Tran (B) School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia e-mail: [email protected] Q. T. Le Gia e-mail: [email protected] E. P. Stephan Institut für Angewandte Mathematik and QUEST (Centre for Quantum Engineering and Space-Time Research), Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: [email protected]
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1 Introduction In geophysical applications [4, 5], one is interested in the Neumann problem exterior to a spheroid where the orbits of satellites are located. The satellite creates data which amount to boundary conditions in scattered points. The consideration of spheroids provides a more realistic setting than spheres. It should be noted that satellite orbits are not located on one spheroid. As a simple model problem, we consider here the Neumann problem for the Laplacian exterior to a prolate spheroid. A key tool of our approach is the use of the Dirichlet–to–Neumann map which directly converts the boundary value problem into a pseudodifferential equation on the spheroid. This integral equation is then handled with Fourier techniques by expansion into appropriate spherical harmonics. This approach was originally taken by Huang and Yu [6] who solved this pseudodifferential equation numerically with standard boundary elements on a regular grid on the angular domain of the spherical coordinates. Our approach uses spherical radial basis functions instead, allowing for better handling of scattered data. As the main result, we prove that if the solution is smooth then a high rate of convergence of the approximate solution can be achieved by choosing appropriate radial basis functions. The paper is organised as follows. In Section 2, we report from [6] existence of the weak solution of the underlying boundary integral equation. In Section 3 we introduce the space of radial basis functions and prove an optimal a priori error estimate for the Galerkin approximation of t
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