BEM, MFS and SBM applied for global gravity field modelling – comparison of their numerical solutions on an ellipsoid and the discretized Earth’s surface

This study presents high-resolution global gravity field modelling using the boundary element method (BEM), method of fundamental solutions (MFS) and singular boundary method (SBM). All three methods are applied to get numerical solutions of the fixed gravimetric boundary-value problem (FGBVP) which represents an exterior oblique derivative problem for the Laplace equation. In case of BEM, its direct formulation is applied to get the boundary integral equations that are numerically discretized using the collocation with linear basis functions. It involves a triangulated discretization of the Earth's surface considering its complicated topography.
   MFS is a mesh-free method which avoids a numerical integration of the singular fundamental solution introducing a fictitious boundary outside the domain, i.e. below the Earth's surface, where the source points are located. In MFS, the fundamental solution of the Laplace equation plays the role of its basis functions. We present how a depth of the fictitious boundary influences accuracy of the obtained MFS solution on the Earth's surface. In case that the source points are located directly on the Earth's surface, the ideas of SBM are applied to isolate singularities of the fundamental solution and its derivatives.
   Numerical experiments present high-resolution global gravity field modelling using BEM, MFS and SBM. All three methods are applied to reconstruct a harmonic function, namely the EGM2008 model up to degree 2160. At first, EGM2008 is reconstructed on the reference ellipsoid, and then on the discretized Earth’s surface. In all cases, the colocation/observation points are located with the same high-resolution of 0.075 deg. Comparisons of the obtained numerical solutions show that all three methods provide almost the same results when reconstructing EGM2008 on the ellipsoid.  When solving FGBVP on the discretized Earth’s surface, the BEM numerical solution gives the best result, then SBM and finally MFS. In all cases, the largest residuals are in high mountains of Himalayas and Andes, however, they are much smaller in the BEM solution due to a special treatment of the oblique derivative problem.