Numerical solution of the oblique derivative BVP on the Earth’s surface topography using BEM
21/09/2016 | 11:45 | Session 3: Recent Development in Theory and Modelling
Author(s): Robert Čunderlík
The fixed gravimetric boundary value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. A direct formulation of the boundary element method (BEM) for the Laplace equation leads to a boundary integral equation (BIE) where a harmonic function is represented as a superposition of the single-layer and double-layer potential. Such a potential representation is applied to obtain a numerical solution of FGBVP. The oblique derivative problem is treated by a decomposition of the gradient of the unknown disturbing potential into its normal and tangential components. Our numerical scheme uses the collocation with linear basis functions. It involves a triangulated discretization of the Earth's surface as our computational domain considering its complicated topography. To achieve high-resolution numerical solutions, parallel implementations using the MPI subroutines as well as an iterative elimination of far zones' contributions are performed.
The first numerical experiments present a reconstruction of a harmonic function on the Earth’s surface topography given by the spherical harmonic approach, namely by the EGM2008 geopotential model up to degree 2160. The SRTM30 global topography model is used to approximate the Earth’s surface. The obtained BEM solutions are compared with EGM2008. The largest residuals are obviously in high mountainous regions. Our study shows that local refinements in these mountainous regions apparently improve the BEM numerical solutions despite the fact that the Earth’s topography is considered in more details.
The second numerical experiments focus on a very detailed local refinement in the area of Slovakia. The input gravity disturbances in collocation points are given by original terrestrial gravimetric measurements or they are generated from the GGMplus database. Achieved high precision of the local BEM solution together with the GNSS-Levelling test allow us to detect latent systematic tendencies that are included in input terrestrial gravimetric data.