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Construction of Galerkin’s Matrix for Elementary Potentials and an Ellipsoidal Solution Domain Based on Series Developments and General Relations between Legendre’s Functions of the first and the Second Kind: Application in Earth’s Gravity Field Studies
Holota, Petr ; Nesvadba, Otakar

Publication type: anotace

Annotation:
The paper 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. Galerkin’s approximations are applied, which means that the solution of the problem is approximated by linear combinations of basis functions with scalar coefficients. Our aim is to discuss the construction of Galerkin’s matrix for basis functions generated by elementary potentials. Ellipsoidal harmonics are used as a natural tool and 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 Legendre’s functions. Subsequently, 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 needs more complex tools. It was converted to elliptic integrals. In essence the result rests on concepts and methods of mathematical analysis. In the paper it is confronted with a direct numerical approach concerning the implementation and use of Legendre’s functions. The computation of the entries is more demanding in this case. Nevertheless, conceptually it avoids approximations. The discussion illustrates some specific features associated with function bases generated by elementary potentials in case of problems formulated for the ellipsoidal solution domain.
Keywords: Earth’s gravity field ; boundary value problems ; ellipsoidal harmonics

Citation: HOLOTA, Petr a Otakar NESVADBA. Construction of Galerkin’s Matrix for Elementary Potentials and an Ellipsoidal Solution Domain Based on Series Developments and General Relations between Legendre’s Functions of the first and the Second Kind: Application in Earth’s Gravity Field Studies. 2016.
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Focus on VÚGTK > VÚGTK Departments > Geodesy and Geodynamics
Focus on VÚGTK > Researchers > Petr Holota
Documents of VÚGTK > Articles VÚGTK
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