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

Adrian  Costea; Heiko Gimperlein; Ernst P. Stephan

doi:10.1007/s00211-013-0579-8

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

Adrian Costea, Heiko Gimperlein, Ernst P. Stephan

Institut for Matematiske Fag

2 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	Numerische Mathematik
Vol/bind	127
Sider (fra-til)	1-34
ISSN	0029-599X
DOI	https://doi.org/10.1007/s00211-013-0579-8
Status	Udgivet - maj 2014

Adgang til dokumentet

10.1007/s00211-013-0579-8

Citationsformater

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title = "A Nash-H{\"o}rmander iteration and boundary elements for the Molodensky problem",

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–H{\"o}rmander 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.",

author = "Adrian Costea and Heiko Gimperlein and Stephan, {Ernst P.}",

year = "2014",

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doi = "10.1007/s00211-013-0579-8",

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volume = "127",

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TY - JOUR

T1 - A Nash-Hörmander iteration and boundary elements for the Molodensky problem

AU - Costea, Adrian

AU - Gimperlein, Heiko

AU - Stephan, Ernst P.

PY - 2014/5

Y1 - 2014/5

N2 - 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.

AB - 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.

U2 - 10.1007/s00211-013-0579-8

DO - 10.1007/s00211-013-0579-8

M3 - Journal article

SN - 0029-599X

VL - 127

SP - 1

EP - 34

JO - Numerische Mathematik

JF - Numerische Mathematik

ER -