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Boundary Complexity and Kernel Functions in Classical and Variational Concepts of Solving Geodetic Boundary Value Problems

  • Petr HolotaEmail author
  • Otakar Nesvadba
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 149)

Abstract

In gravity field studies the complex structure of the Earth’s surface makes the solution of geodetic boundary value problems quite challenging. This equally concerns classical methods of potential theory as well as modern methods often based on a (variational or) weak solution concept. Aspects of this nature are reflected in the content of the paper. In case of a spherical Neumann problem the focus is on the classical Green’s function method and on the use of reproducing kernel and elementary potentials in generating function bases for Galerkin’s approximations. Similarly, the construction of Neumann’s function – Green’s function of the second kind and of entries in Galerkin’s matrix for basis functions generated by the reproducing kernel and by elementary potentials is also highlighted when solving Neumann’s problem in the exterior of an oblate ellipsoid of revolution. In this connection the role of elliptic integrals is pointed out. Finally, two concepts applied to the solution of the linear gravimetric boundary value problem are mentioned. They represent an approach based on variational methods and on the use of a transformation of coordinates offering an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. Successive approximations are involved in both the cases.

Keywords

Elliptic integrals Galerkin’s system Green’s function Laplace’s operator Reproducing kernel Transformation of coordinates 

Notes

Acknowledgements

This work was supported by the Czech Science Foundation through Project No. 14-34595S and by the Ministry of Education, Youth and Sports of the Czech Republic through Project No. LO1506. This support is gratefully acknowledged. Sincere thanks go also to two anonymous reviewers for valuable comments.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Research Institute of Geodesy, Topography and CartographyPrague-EastCzech Republic
  2. 2.Land Survey OfficePrague 8Czech Republic

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