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Some Topics Related to the Solution of Boundary-Value Problems in Geodesy

  • P. Holota
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 127)

Abstract

In geodesy the theory of boundary-value problems has a great number of important applications. It is hardly possible to approach all of them in one single paper. For this reason the exterior Neumann problem is discussed to demonstrate some of the topics which may be encountered in the solution of geodetic boundary-value problems in general. In addition, the problem is closely related to the gravimetric boundary value problem, which has a considerable practical importance. The tie to variational methods and the minimization of the respective quadratic functional are in the focus of the explanation. Within this concept Euler’s necessary condition has a form of an integral identity and represents a natural starting point for the interpretation of the solution in terms of function bases. The convergence of Galerkin’s approximations is also treated. For the solution domain given by the exterior of a sphere and also of an ellipsoid of revolution the particular attention is paid to the construction of kernels which have a reproducing property with respect to the inner product in the Sobolev space used. They generate suitable function bases for the approximation solution of the boundary-value problem considered. In particular, in the spherical case they allow for a closed expression of the elements in Galerkin’s matrix. A relation of the kernels to Dirichlet’s and Neumann’s function is also shown. Spherical and ellipsoidal harmonics are used as an efficient tool in constructing the respective integral kernels Finally, an approach to problems with data given on a more complicated boundary is discussed.

Keywords

Earth’s gravity field geodetic bound-ary-value problems Green’s functions variational methods reproducing kernels ellipsoidal harmonics 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • P. Holota
    • 1
  1. 1.Research Institute of GeodesyTopography and CartographyPraha-vychodCzech Republic

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