000193452 001__ 193452 000193452 000__ $$92017 000193452 005__ 20181127114100.0 000193452 041__ $$aeng 000193452 1001_ $$aHolota, Petr 000193452 24510 $$aWeak Solution Concept and Galerkin’s Matrix for the Exterior of an Oblate Ellipsoid of Revolution in the Representation of the Earth’s Gravity Potential by Buried Masses 000193452 5203_ $$9eng$$aThe paper is motivated by the role of boundary value problems in Earth’s gravity field studies. The discussion focuses on Neumann’s problem formulated for the exterior of an oblate ellipsoid of revolution as this is considered a basis for an iteration solution of the linear gravimetric boundary value problem in the determination of the disturbing potential. The approach follows the concept of the weak solution and Galerkin’s approximations are applied. This means that the solution of the problem is approximated by linear combinations of basis functions with scalar coefficients. The construction of Galerkin’s matrix for basis functions generated by elementary potentials (point masses) is discussed. Ellipsoidal harmonics are used as a natural tool and the elementary potentials are expressed by means of series of ellipsoidal harmonics. The problem, however, is the summation of the series that represent the entries of Galerkin’s matrix. It is difficult to reduce the number of summation indices since in the ellipsoidal case there is no analogue to the addition theorem known for spherical harmonics. Therefore, the straightforward application of series of ellipsoidal harmonics is complemented by deeper relations contained in the theory of ordinary differential equations of second order and in the theory of Legendre’s functions. Subsequently, also hypergeometric functions and series are used. Moreover, within some approximations the entries are split into parts. Some of the resulting series may be summed relatively easily, apart from technical tricks. For the remaining series the summation was converted to elliptic integrals. The approach made it possible to deduce a closed (though approximate) form representation of the entries in Galerkin’s matrix. 000193452 5203_ $$9cze$$aThe paper is motivated by the role of boundary value problems in Earth’s gravity field studies. The discussion focuses on Neumann’s problem formulated for the exterior of an oblate ellipsoid of revolution as this is considered a basis for an iteration solution of the linear gravimetric boundary value problem in the determination of the disturbing potential. The approach follows the concept of the weak solution and Galerkin’s approximations are applied. This means that the solution of the problem is approximated by linear combinations of basis functions with scalar coefficients. The construction of Galerkin’s matrix for basis functions generated by elementary potentials (point masses) is discussed. Ellipsoidal harmonics are used as a natural tool and the elementary potentials are expressed by means of series of ellipsoidal harmonics. The problem, however, is the summation of the series that represent the entries of Galerkin’s matrix. It is difficult to reduce the number of summation indices since in the ellipsoidal case there is no analogue to the addition theorem known for spherical harmonics. Therefore, the straightforward application of series of ellipsoidal harmonics is complemented by deeper relations contained in the theory of ordinary differential equations of second order and in the theory of Legendre’s functions. Subsequently, also hypergeometric functions and series are used. Moreover, within some approximations the entries are split into parts. Some of the resulting series may be summed relatively easily, apart from technical tricks. For the remaining series the summation was converted to elliptic integrals. The approach made it possible to deduce a closed (though approximate) form representation of the entries in Galerkin’s matrix. 000193452 655_4 $$aanotace 000193452 653_0 $$aEarth’s gravity field 000193452 653_0 $$aboundary value problems 000193452 653_0 $$avariational methods 000193452 653_0 $$aGalerkin approximations 000193452 653_0 $$aelementary potentials 000193452 653_0 $$aellipsoidal harmonics 000193452 7001_ $$aNesvadba, Otakar 000193452 7730_ $$9 2017 000193452 85642 $$uhttps://www.rvvi.cz/riv?s=jednoduche-vyhledavani&ss=detail&h=RIV%2F00025615%3A_____%2F17%3AN0000041%21RIV18-GA0-00025615 000193452 85640 $$uhttp://meetingorganizer.copernicus.org/EGU2017/EGU2017-15962.pdf 000193452 943__ $$aRIV:O 000193452 980__ $$aclanky_vugtk 000193452 985__ $$aanotace 000193452 985__ $$ariv 000193452 985__ $$aholota 000193452 985__ $$anesvadba