Abstract
After an introductory note reviewing the role and the treatment of boundary problems in physical geodesy, the explanation rests on the concept of the weak solution. The focus is on the linear gravimetric boundary value problem. In this case, however, an oblique derivative in the boundary condition and the need for a numerical integration over the whole and complicated surface of the Earth make the numerical implementation of the concept rather demanding. The intention is to reduce the complexity by means of successive approximations and step by step to take into account effects caused by the obliqueness of the derivative and by the departure of the boundary from a more regular surface. The possibility to use a sphere or an ellipsoid of revolution as an approximation surface is discussed with the aim to simplify the bilinear form that defines the problem under consideration and to justify the use of an approximation of Galerkin’s matrix. The discussion is added of extensive numerical simulations and tests using the ETOPO5 boundary for the surface of the Earth and gravity data derived from the EGM96 model of the Earth’s gravity field.
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Acknowledgements
The presentation of the paper at the VII Hotine-Marussi Symposium, Rome, Italy, July 6–10, 2009, was sponsored by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. The computations were performed in CINECA (Bologna) within the Project HPC-EUROPA\(++\) (RII3-CT-2003–506079), supported by the European Community – Research Infrastructure Action under the FP6 “Structuring the European Research Area” Programme. All this support as well as the possibility to discuss the topic with Prof. F. Sansò, Milano, is gratefully acknowledged. Thanks go also to anonymous reviewers for valuable comments.
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Holota, P., Nesvadba, O. (2012). Method of Successive Approximations in Solving Geodetic Boundary Value Problems: Analysis and Numerical Experiments. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds) VII Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 137. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22078-4_28
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