Reproducing kernel and Neumann’s function for the exterior of an oblate ellipsoid of revolution: application in gravity field studies

Abstract

The purpose of this paper is to discuss the construction of the reproducing kernel of Hilbert’s space of functions that are harmonic in the exterior of an oblate ellipsoid of revolution. The motivation comes from the weak solution concept applied to Neumann’s problem for Laplace’s partial differential equation in gravity field studies. The use of the reproducing kernel enables the construction of a function basis that is suitable for the approximation representation of the solution and offers a straightforward way leading to entries in Galerkin’s matrix of the respective linear system for unknown scalar coefficients. The serious problem, however, is the summation of the series that represents the kernel. It is difficult to reduce the number of summation indices since in the ellipsoidal case there is no straightforward analogue to the addition theorem known for the spherical situation. This makes the computation of the kernel and the set of the entries in Galerkin’s matrix rather demanding, even by means of high performance computer facilities. Therefore, the reproducing kernel and its series representation are analyzed. The apparatus of hypergeometric functions and series is used. The kernel is split into parts. Some of the resulting series may be summed relatively easily, except for some technical tricks. For the remaining series, however, the summation needs more complex tools. In particular, the summation was converted to elliptic integrals. This approach leads to an effective numerical treatment of the kernel. The results are presented. Finally, the relation of the reproducing kernel to Green’s function of the second kind (Neumann’s function) is discussed with a special view to physical geodesy applications.