Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements.

Abstract

Abstract The numerical approach for solving the fixed gravimetric boundary value problem (FGBVP) based on the finite element method (FEM) with mapped infinite elements is developed and implemented. In this approach, the 3D semi-infinite domain outside the Earth is bounded by the triangular discretization of the whole Earth’s surface and extends to infinity. Then the FGBVP consists of the Laplace equation for unknown disturbing potential which holds in the domain, the oblique derivative boundary condition (BC) given directly at computational nodes on the Earth’s surface, and regularity of the disturbing potential at infinity. In this way, it differs from previous FEM approaches, since the numerical solution is not fixed by the Dirichlet BC on some part of the boundary of the computational domain. As a numerical method, the FEM with finite and mapped infinite triangular prisms has been derived and implemented. In experiments, at first, a convergence of the proposed numerical scheme to the exact solution is tested. Afterwards, a numerical study is focused on a reconstruction of the harmonic function (EGM2008) above the Earth’s topography. Here, a special discretization of the Earth’s surface which is able to fulfil the conditions that arise from correct geometrical properties of finite elements, and it is suitable for parallel computing is implemented. The obtained solutions at nodes on the Earth’s surface as well as nodes that lie approximately at the altitude of the GOCE satellite mission have been tested.