Global gravity field modelling by solving the infinite nonlinear fixed geodetic boundary value problem
Abstract
The aim of presented paper is to solve the nonlinear geodetic boundary value problem (BVP) by the finite element method ({FEM}) involving the mapped infinite elements (MIE).
In comparison to our previous works,
see [\textsc{{Mac\'ak, M. et al.}}:
\textit{On an iterative approach to solving the nonlinear satellite-fixed geodetic boundary-value problem}.
In: IAG Symp. Vol. 142 (2016), pp. 185--192.]
and
[\textsc{Mac\'ak, M. et al.}:
\textit{A gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method},
Acta Geodaetica et Geophysica, \textbf{58} (2023), 305--320.]
dealing with bounded domains, in this paper we propose and study numerical concept on unbounded domains, given as an exterior BVP for the Laplace equation outside the gravitating body, e.g. Earth, with the nonlinear boundary condition ({BC}) prescribed on the Earth's surface and considering the solution regularity condition at infinity. This concept can be found in many scientific disciplines being also the most natural from physical geodesy point of view, see, e.g.,
[\textsc{Backus, G. E.}:
\textit{Application of a non-linear boundary-value problem for Laplace's equation to gravity and geomagnetic intensity surveys},
Q. J. Mech. Appl. Math. \textbf{2} (1968), 195--221.]
and
of large practical importance when we are not able to prescribe {BC}s
on a bounded domain.
The proposed concept is based on the iterative procedure, and as the numerical method we have implemented the {FEM} with the {MIE} to take
into account the regularity of the disturbing potential at infinity.
Since the boundary of the computational domain is the discretized real Earth’s surface considering its topography, as finite and infinite elements we have chosen the triangular prisms. We study and verify this numerical approach by a testing experiment with a homogeneous sphere, by the experiment using {EGM}2008, and finally, we present one detailed numerical experiment with DTU21GRA data.
In comparison to our previous works,
see [\textsc{{Mac\'ak, M. et al.}}:
\textit{On an iterative approach to solving the nonlinear satellite-fixed geodetic boundary-value problem}.
In: IAG Symp. Vol. 142 (2016), pp. 185--192.]
and
[\textsc{Mac\'ak, M. et al.}:
\textit{A gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method},
Acta Geodaetica et Geophysica, \textbf{58} (2023), 305--320.]
dealing with bounded domains, in this paper we propose and study numerical concept on unbounded domains, given as an exterior BVP for the Laplace equation outside the gravitating body, e.g. Earth, with the nonlinear boundary condition ({BC}) prescribed on the Earth's surface and considering the solution regularity condition at infinity. This concept can be found in many scientific disciplines being also the most natural from physical geodesy point of view, see, e.g.,
[\textsc{Backus, G. E.}:
\textit{Application of a non-linear boundary-value problem for Laplace's equation to gravity and geomagnetic intensity surveys},
Q. J. Mech. Appl. Math. \textbf{2} (1968), 195--221.]
and
of large practical importance when we are not able to prescribe {BC}s
on a bounded domain.
The proposed concept is based on the iterative procedure, and as the numerical method we have implemented the {FEM} with the {MIE} to take
into account the regularity of the disturbing potential at infinity.
Since the boundary of the computational domain is the discretized real Earth’s surface considering its topography, as finite and infinite elements we have chosen the triangular prisms. We study and verify this numerical approach by a testing experiment with a homogeneous sphere, by the experiment using {EGM}2008, and finally, we present one detailed numerical experiment with DTU21GRA data.
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Subscribers OnlyDOI: https://doi.org/10.2478/tmmp-2025-0016