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

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.