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On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling

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Abstract

The paper presents local gravity field modelling in spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). For such a problem, we derive a new numerical scheme where the oblique derivative BC are considered directly at computational nodes on the discretized Earth’s topography. Then, the developed FEM approach is tested in several artificial testing experiments as well as by a reconstruction of a known harmonic function above the extremely complicated Earth’s topography in the Himalayas. A main numerical experiment is focused on very detailed local gravity field modelling in Slovakia using terrestrial gravity data. The high horizontal resolution 100 \(\times \) 100 m and non-uniform resolution in the radial direction has resulted in a 3D unstructured mesh of finite elements with 5,287,500,000 unknowns. Large-scale parallel computations were performed on a parallel cluster using 1.5 TB of distributed memory. The obtained local quasigeoid model is tested at 403 GNSS-levelling benchmarks. The standard deviation of residuals 2.77 cm, which decreases to 2.54 cm after excluding 7 outliers, indicates its high precision. However, depicted residuals show their low-frequency character with amplitudes about ± 3 cm. As a by-product, the first and second derivatives of the obtained disturbing potential in the radial direction are also evaluated in several altitude levels as well as on the Earth’s surface. Finally, the paper presents a comparison of the obtained FEM solution with the recent local quasigeoid models in Slovakia computed in the spatial as well as spectral domain. It illustrates a practical contribution of the presented FEM approach for precise local gravity field modelling, especially in high mountains.

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Data availability

All datasets generated and/or analysed within the testing experiments (Sects 4.1 and 4.2) are available from the corresponding author. The input data used in Sect. 4.3 are not open for public as the authors do not have the rights to provide them to a third party. The output data of Sect. 4.3 are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by Grants APVV-15-0522, APVV-19-0460 and VEGA 1/0486/20. We would like express our thanks to Pavol Zahorec (SAV Banská Bystrica) and Juraj Papčo (STU Bratislava) for providing input gravity disturbances as well as measured vertical gravity gradients. We also thank the Geodetic and Cartographic Institute Bratislava for providing the GNSS/levelling data and to Blažej Bucha (STU Bratislava) for providing the local quasigeoid model in Slovakia published in Bucha et al. (2016).

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Authors and Affiliations

  1. Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05, Bratislava, Slovakia

    Zuzana Minarechová, Marek Macák, Róbert Čunderlík & Karol Mikula

  2. Algoritmy:SK s.r.o., Šulekova 6, 81106, Bratislava, Slovakia

    Karol Mikula

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  1. Zuzana Minarechová
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  2. Marek Macák
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  3. Róbert Čunderlík
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  4. Karol Mikula
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Contributions

Zuzana Minarechová contributed to develop the theory and wrote the manuscript with support from Róbert Čunderlík. Marek Macák developed the theory and performed the numerical simulations. Róbert Čunderlík carried out the experiment and analysed the data. Karol Mikula devised the project. All authors discussed the results and contributed to the final manuscript.

Corresponding author

Correspondence to Zuzana Minarechová.

Appendix

Appendix

See Fig. 18.

Fig. 18
figure 18

The first and second derivatives of the disturbing potential in the radial direction evaluated in 5 altitude levels (approximately at 500 m, 1 km, 2 km, 5 km and 10 km above the reference ellipsoid)

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Minarechová, Z., Macák, M., Čunderlík, R. et al. On the finite element method for solving the oblique derivative boundary value problems and its application in local gravity field modelling. J Geod 95, 70 (2021). https://doi.org/10.1007/s00190-021-01522-8

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