Structure of the Laplace operator, geometry of the Earth’s surface and successive approximations in the solution of the geodetic boundary value problem

The Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, in treating the geodetic boundary value problem the physical surface of the Earth substantially differs from a sphere or an oblate ellipsoid of revolution, even if optimally approximated. Therefore, an alternative between the boundary complexity and the complexity of the coefficients of the Laplace partial differential equation governing the solution is discussed. The situation is more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth (smoothed to a certain degree) is imbedded in the family of coordinate surfaces. The idea is close to concepts followed also in other branches of engineering and mathematical physics. A transformation of coordinates is applied. Subsequently, tensor calculus is used to express the Laplace operator in the system of new coordinates. The structure of the Laplacian is more complicated now, but in a sense it represents the topography of the physical surface of the Earth. Finally, the Green’s function method together with the method of successive approximations is used for the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and where useful and possible, modified by means of integration by parts. The iteration steps and their convergence are discussed and interpreted, numerically and in terms of functional analyses.