Advertisement

Numerical Methods for Solving the Oblique Derivative Boundary Value Problems in Geodesy

  • Róbert ČunderlíkEmail author
  • Marek Macák
  • Matej Medl’a
  • Karol Mikula
  • Zuzana Minarechová
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Der Beitrag beschftigt sich mit verschiedenen numerischen Verfahren zum schiefachsigen Randwertproblem der Geodsie. Zunchst wird eine numerische Lsung des Randwertproblems mittels Randelementmethoden beschrieben, welche die schiefachsigen Ableitungen in Normal- und Tangentialkomponenten zerlegt. Die sich ergebende Randintegralgleichung wird mittels Kollokationstechnik unter Verwendung linearer Basisfunktionen diskretisiert. Es folgt ein Lsungsvorschlag mittels Finite-Volumen-Technik auf und oberhalb der Erdoberflche. In diesem Fall wird eine schiefachsige Ableitung auf drei verschiedene Arten behandelt, nmlich (i) durch einen Zerlegungsansatz in Normal- und zwei Tangentialkomponenten, die dann mittels numerischer Lsungswerte Approximation finden (ii) durch einen Zugang, der auf ein erster und zweiter Ordnung basierendes upwind – Schema umsetzt (iii) durch eine Methodik der Konstruktion nicht-gleichfrmiger hexaedrischer 3D-Gitter oberhalb der Erdoberflche und einem upwind Schema hherer Ordnung. Jeder der vorgeschlagenen Zugnge wird numerisch auf ihre Effizienz untersucht.

Keywords

Geodetic boundary value problem Oblique derivative boundary condition Boundary element method Finite volume method Numerical solution Global gravity field modelling Local gravity field modelling Upwind method Advection equation Evolving surfaces 

Abstract

We present various numerical approaches for solving the oblique derivative boundary value problem. At first, we describe a numerical solution by the boundary element method where the oblique derivative is treated by its decomposition into the normal and tangential components. The derived boundary integral equation is discretized using the collocation technique with linear basis functions. Then we present solution by the finite volume method on and above the Earth’s surface. In this case, the oblique derivative in the boundary condition is treated in three different ways, namely (i) by an approach where the oblique derivative is decomposed into normal and two tangential components which are then approximated by means of numerical solution values (ii) by an approach based on the first order upwind scheme; and finally (iii) by a method for constructing non-uniform hexahedron 3D grids above the Earth’s surface and the higher order upwind scheme. Every of proposed approaches is tested by the so-called experimental order of convergence. Numerical experiments on synthetic data aim to demonstrate their efficiency.

Notes

Acknowledgements

This work was supported by Grants APVV-15-0522, VEGA 1/0608/15 and VEGA 1/0714/15.

References

  1. 1.
    Andersen, O.B., Knudsen, P., Berry, P.: The DNSC08 ocean wide altimetry derived gravity field. Presented at EGU-2008, CityplaceVienna, country-regionAustria (2008)Google Scholar
  2. 2.
    Aoyama, Y., Nakano, J.: RS/6000 SP: Practical MPI Programming. IBM, Poughkeepsie/New York (1999)Google Scholar
  3. 3.
    Baláš, J., Sládek, J., Sládek, V.: Stress Analysis by Boundary Element Methods. Elsevier, Amsterdam (1989)Google Scholar
  4. 4.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  5. 5.
    Bauer, F.: An alternative approach to the oblique derivative problem in potential theory. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. Shaker Verlag, Aachen (2004)Google Scholar
  6. 6.
    Becker, J.J., Sandwell, D.T., Smith, W.H.F., Braud, J., Binder, B., Depner, J., Fabre, D., Factor, J., Ingalls, S., Kim S.H., Ladner, R., Marks, K., Nelson, S., Pharaoh, A., Trimmer, R., Rosenberg, J. Von, Wallace, G., Weatherall, P.: Global bathymetry and elevation data at 30 arc seconds resolution: SRTM30 PLUS. Mar. Geod. 32(4), 355–371 (2009)CrossRefGoogle Scholar
  7. 7.
    Bitzadse, A.V.: Boundary-Value Problems for Second-Order Elliptic Equations. North-Holland, Amsterdam (1968)Google Scholar
  8. 8.
    Bjerhammar, A., Svensson, L.: On the geodetic boundary value problem for a fixed boundary surface. A satellite approach. Bull. Geod. 57(1–4), 382–393 (1983)CrossRefGoogle Scholar
  9. 9.
    Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques, Theory and Applications in Engineering. Springer, New York (1984)CrossRefGoogle Scholar
  10. 10.
    Čunderlík, R., Mikula, K., Mojzeš, M.: 3D BEM application to Neumann geodetic BVP using the collocation with linear basis functions. In: Proceedings of ALGORITMY 2002, Conference on Scientific Computing, Podbanské, pp. 268–275 (2002)Google Scholar
  11. 11.
    Čunderlík, R., Mikula, K., Mojžeš, M.: Numerical solution of the linearized fixed gravimetric boundary-value problem. J. Geod. 82, 15–29 (2008)CrossRefGoogle Scholar
  12. 12.
    Čunderlík, R., Mikula, K.: Direct BEM for high-resolution gravity field modelling. Stud. Geophys. Geod. 54(2), 219–238 (2010)CrossRefGoogle Scholar
  13. 13.
    Čunderlík, R., Mikula, K., Špir R.: An oblique derivative in the direct BEM formulation of the fixed gravimetric BVP. IAG Symp. 137, 227–231 (2012)Google Scholar
  14. 14.
    Eymard, R., Gallouet, T., Herbin, R.: Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37(1–2), 31–53 (2001)CrossRefGoogle Scholar
  15. 15.
    Fašková, Z.: Numerical methods for solving geodetic boundary value problems. PhD Thesis, SvF STU, Bratislava (2008)Google Scholar
  16. 16.
    Fašková, Z., Čunderlík, R., Mikula, K.: Finite element method for solving geodetic boundary value problems. J. Geod. 84(2), 135–144 (2010)CrossRefGoogle Scholar
  17. 17.
    Freeden, W.: Harmonic splines for solving boundary value problems of potential theory. In: Mason, J.C., Cox, M.G. (eds.) Algorithms for Approximation. The Institute of Mathematics and Its Applications, Conference Series, vol. 10, pp. 507–529. Clarendon Press, Oxford (1987)Google Scholar
  18. 18.
    Freeden, W., Gerhards, C.: Geomathematically Oriented Potential Theory. CRC press, Taylor & Francis Group, Boca Raton, Florida (2013)Google Scholar
  19. 19.
    Freeden, W., Kersten, H.: The Geodetic Boundary Value Problem Using the Known Surface of the Earth, Veröff. Geod. Inst. RWTH Aachen, 29 (1980)Google Scholar
  20. 20.
    Freeden, W., Kersten, H.: A constructive approximation theorem for the oblique derivative problem in potential theory. Math. Methods Appl. Sci. 3, 104–114 (1981)CrossRefGoogle Scholar
  21. 21.
    Freeden, W., Michel, V.: Multiscale Potential Theory (With Applications to Geoscience). Birkhauser, Boston (2004)CrossRefGoogle Scholar
  22. 22.
    Freeden W., Nutz H.: On the solution of the oblique derivative problem by constructive Runge-Walsh concepts. In: Pesenson I., Le Gia Q., Mayeli A., Mhaskar H., Zhou D.X. (eds.) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham (2017)Google Scholar
  23. 23.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulation. J. Comput. Phys. 73, 325–348 (1987)CrossRefGoogle Scholar
  24. 24.
    Gutting, M.: Fast multipole methods for oblique derivative problems. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern. Shaker Verlag, Aachen (2007)Google Scholar
  25. 25.
    Gutting, M.: Fast multipole accelerated solution of the oblique derivative boundary value problem. Int. J. Geomath. 3, 223–252 (2012)CrossRefGoogle Scholar
  26. 26.
    Gutting, M.: Fast Spherical/Harmonic Spline Modeling. Handbook of Geomathematics, 2nd edn., pp. 2711–2746. Springer, Berlin/Heidelberg (2015)CrossRefGoogle Scholar
  27. 27.
    Hartmann, F.: Introduction to Boundary Elements. Theory and Applications. Springer, Berlin (1989)CrossRefGoogle Scholar
  28. 28.
    Holota, P.: Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. J. Geod. 71, 640–651 (1997)CrossRefGoogle Scholar
  29. 29.
    Holota, P., Nesvadba, O.: Model refinements and numerical solution of weakly formulated boundary-value problems in physical geodesy. In: IAG Symposium, vol. 132, pp. 320–326 (2008)Google Scholar
  30. 30.
    Húska, M., Medľa, M., Mikula, K., Novysedlák, P., Remešíková, M.: A new form-finding method based on mean curvature flow of surfaces. In: Handlovičová, A., Minárechová, Z., Ševčovič, D., (eds.) ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanske, 9–14 Sept 2012, Proceedings of Contributed Papers and Posters. ISBN:978-80-227-3742-5. Publishing House of STU, pp. 120–131 (2012)Google Scholar
  31. 31.
    Klees, R.: Boundary value problems and approximation of integral equations by finite elements. Manuscr. Geodaet. 20, 345–361 (1995)Google Scholar
  32. 32.
    Klees, R., van Gelderen, M., Lage, C., Schwab, C.: Fast numerical solution of the linearized Molodensky problem. J. Geod. 75, 349–362 (2001)CrossRefGoogle Scholar
  33. 33.
    Koch, K.R., Pope, A.J.: Uniqueness and existence for the geodetic boundary value problem using the known surface of the Earth. Bull. Geod. 46, 467–476 (1972)CrossRefGoogle Scholar
  34. 34.
    Laursen, M.E., Gellert, M.: Some criteria for numerically integrated matrices and quadrature formulas for triangles. Int. J. Numer. Meth. Eng. 12, 67–76 (1978)CrossRefGoogle Scholar
  35. 35.
    Lehmann, R., Klees, R.: Numerical solution of geodetic boundary value problems using a global reference field. J. Geod. 73, 543–554 (1999)CrossRefGoogle Scholar
  36. 36.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics (2002). ISBN:978-0521009249Google Scholar
  37. 37.
    Lucquin, B., Pironneau, O.: Introduction to Scientific Computing. Wiley, Chichester (1998)Google Scholar
  38. 38.
    Macák, M., Čunderlík, R., Mikula, K., Minarechová, Z.: An upwind-based scheme for solving the oblique derivative boundary-value problem related to the physical geodesy. J. Geod. Sci. 5(1), 180–188 (2015)Google Scholar
  39. 39.
    Macák, M., Mikula, K., Minarechová, Z.: Solving the oblique derivative boundary-value problem by the finite volume method, In: ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanske, 9–14 Sept 2012, Proceedings of Contributed Papers and Posters, Publishing House of STU, pp. 75–84 (2012)Google Scholar
  40. 40.
    Macák, M., Mikula, K., Minarechová, Z., Čunderlík, R.: On an iterative approach to solving the nonlinear satellite-fixed geodetic boundary-value problem, In: IAGSymp, vol. 142, pp. 185–192 (2016)Google Scholar
  41. 41.
    Macák, M., Minarechová, Z., Mikula, K.: A novel scheme for solving the oblique derivative boundary-value problem. Stud. Geophys. Geo. 58(4), 556–570 (2014)CrossRefGoogle Scholar
  42. 42.
    Mantič, V.: A new formula for the C-matrix in the Somigliana identity. J. Elast. 33(3), 191–201 (1993)CrossRefGoogle Scholar
  43. 43.
    Mayer-Gurr, T., et al.: The new combined satellite only model GOCO03s. Presented at the GGHS-2012 in Venice (2012)Google Scholar
  44. 44.
    Medl’a, M., Mikula, K., Čunderlík, R., Macák, M.: Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography. J. Geod. 92(1), pp 1–19 (2017)CrossRefGoogle Scholar
  45. 45.
    Meissl, P.: The use of finite elements in physical geodesy. Report 313, Geodetic Science and Surveying, The Ohio State University (1981)Google Scholar
  46. 46.
    Mikula, K., Remešiková, M., Sarkóci, P., Ševčovič, D.: Manifold evolution with tangential redistribution of points. SIAM J. Sci. Comput. 36(4), A1384–A1414 (2014)CrossRefGoogle Scholar
  47. 47.
    Mikula, K., Ševčovič, D.: A direct method for solving an anisotropic mean curvature flow of planar curve with an external force. Math. Methods Appl. Sci. 27(13), 1545–1565 (2004)CrossRefGoogle Scholar
  48. 48.
    Minarechová, Z., Macák, M., Čunderlík, R., Mikula, K.: High-resolution global gravity field modelling by the finite volume method. Stud. Geophys. Geo. 59, 1–20 (2015)CrossRefGoogle Scholar
  49. 49.
    Miranda, C.: Partial Differential Equations of Elliptic Type. Springer, Berlin (1970)Google Scholar
  50. 50.
    Nesvadba, O., Holota, P., Klees, R.: A direct method and its numerical interpretation in the determination of the gravity field of the Earth from terrestrial data. In: IAG Symposium, vol. 130, pp. 370–376 (2007)Google Scholar
  51. 51.
    Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.: The development and evaluation of the Earth gravitational model 2008 (EGM2008). J. Geophys. Res. 117, B04406. https://doi.org/10.1029/2011JB008916 (2012)CrossRefGoogle Scholar
  52. 52.
    Schatz, A.H., Thomée, V., Wendland, W.L.: Mathematical Theory of Finite and Boundary Element Methods. Birkhauser Verlag, Basel/Boston/Berlin (1990)CrossRefGoogle Scholar
  53. 53.
    Shaofeng, B., Dingbo, C.: The finite element method for the geodetic boundary value problem. Manuscr. Geod. 16, 353–359 (1991)Google Scholar
  54. 54.
    Sleijpen, G.L.G., Fokkema, D.R.: Bicgstab (l) for Linear Equations Involving Unsymmetric Matrices with Complex Spectrum. http://dspace.library.uu.nl/handle/1874/16827 (1993)
  55. 55.
    Zhao, K., Vouvakis, M., Lee, J.-F.: The adaptive cross approximation algorithm for accelerated method of moment computations of EMC problems. IEEE Trans. Electromagn. Compat. 47, 763–773 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  • Róbert Čunderlík
    • 1
    Email author
  • Marek Macák
    • 1
  • Matej Medl’a
    • 1
  • Karol Mikula
    • 1
  • Zuzana Minarechová
    • 1
  1. 1.Faculty of Civil Engineering, Department of Mathematics and Descriptive GeometrySlovak University of TechnologyBratislavaSlovakia

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations