The finite element method for solving the oblique derivative boundary value problems in geodesy

We present a novel approach to the solution of the geodetic boundary value problem with an oblique derivative boundary condition by the finite element method. Namely, we propose and analyse a finite element approximation of a Laplace equation holding on a domain with an oblique derivative boundary condition given on a part of its boundary. The oblique vector in the boundary condition is split into one normal and two tangential components and derivatives in tangential directions are approximated as in the finite difference method. Then we apply the proposed numerical scheme to local gravity field modelling. For our two-dimensional testing numerical experiments, we use four nodes bilinear quadrilateral elements and for a three-dimensional problem, we use hexahedral elements with eight nodes. Practical numerical experiments are located in area of Slovakia that is given by grid points located on the Earth's surface with uniform spacing in horizontal directions. Heights of grid points are interpolated from the SRTM30PLUS topography model. An upper boundary is in the height of 240 km above a reference ellipsoid WGS84 corresponding to an average altitude of the GOCE satellite orbits. Obtained solutions are compared to DVRM05.