A Nash-Hormander iteration and boundary elements for the Molodensky problem

TL;DR

This paper presents a novel numerical method combining a smoothed Nash-Hormander iteration with boundary elements to accurately solve the nonlinear Molodensky problem of Earth's surface reconstruction from gravitational data.

Contribution

It introduces a regularization technique based on a higher-order heat equation within a Nash-Hormander iteration framework for improved surface reconstruction accuracy.

Findings

Quantitative error estimates after multiple iteration steps

Validation of heat equation-based smoothing operators

Numerical results demonstrate improved approximation accuracy

Abstract

We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after m steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.