Numerical simulations of the nonlinear Molodensky problem

TL;DR

This paper introduces a boundary element method combined with Nash-Hörmander iteration and heat equation smoothing to numerically solve the nonlinear Molodensky problem, reconstructing Earth's surface from gravitational data.

Contribution

It develops a novel numerical approach integrating boundary element methods and iterative smoothing for the nonlinear Molodensky problem.

Findings

Numerical results demonstrate the method's accuracy in model problems.

The approach effectively handles the nonlinearity and derivative loss.

Error analysis confirms the method's convergence and stability.

Abstract

We present a boundary element method to compute numerical approximations to the non-linear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. Our solution procedure solves a sequence of exterior oblique Robin problems and is based on a Nash-H\"{o}rmander iteration. We apply smoothing with the heat equation to overcome a loss of derivatives in the surface update. Numerical results compare the error between the approximation and the exact solution in a model problem.