Costea, Adrian^{2}; Gimperlein, Heiko^{4}; Stephan, Ernst P.^{3}

Affiliations:

^{1} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet^{2} Leibniz University Hannover^{3} Leibniz Universitaet Hannover^{4} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet

DOI:

10.1007/s00211-013-0579-8

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–Hörmander 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.Aboundary 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.