Costea, Adrian2; Gimperlein, Heiko4; Stephan, Ernst P.3
1 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet2 Leibniz University Hannover3 Leibniz Universitaet Hannover4 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet
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.