On a spectral method of solving the Stokes equation

A method of solving the Stokes equation for a spherical mantle model by expansion in spherical harmonics was developed by Hager and O’Connell 1979]. However, this method is applicable only if the viscosity depends solely on depth. In this case, the Stokes equation reduces to a system of independent equations for each harmonic. Given lateral variations in viscosity, the Stokes equation contains terms in the form of products of harmonics, which invalidates all advantages of harmonic expansion. Zhang and Christensen 1993] developed a perturbation method for the case when terms containing products of lateral viscosity variations are small. These terms are first calculated from the preceding iteration and are then expanded in a series of harmonic functions. As a result, equations for harmonics remain independent. An evident advantage of the spectral method is the simplicity of the technique of incorporating the self-gravitation and compressibility effects. Moreover, this method partially eliminates difficulties related to the singularities at poles. As yet, it has not been applied in practice, possibly because the equations presented in Zhang and Christensen, 1993] contain misprints that have not been elucidated in the literature. In the present work, a system of equations is derived for the spectral-iterative method of solving the Stokes equation and the errata present in formulas of Zhang and Christensen 1993] and significantly affecting results of calculations are analyzed.