A generalized Rayleigh-Ritz (RR) method combined with the Galerkin (RRG) functional space approximation is elaborated by using the extended and modified Laguerre functions manifold for the weak solution of the asymmetric grain-boundary thermal grooving problem with the Dirichlet boundary. This new hybrid RRG approach, which resembles the front-tracking method, reveals the fine features of the grain boundary groove-root topography (rough or faceted regions) more accurately than the previous approach under the severe nonanalyticity of the surface stiffness anisotropy, and showing almost excellent in accord with the experimental observations made by atomic force microscopy and scanning tunneling microscopy. The large deviations from Mullins' t(1/4) scaling law combined with the self-trapping (quasifaceting) are observed especially at low values of the normalized longitudinal mobilities, where the kinetics rather than the energetic considerations are found to be the dominating factor for the whole topographic appearances. For very high longitudinal mobilities, the smooth and symmetric groove profiles (no faceting) are found to be represented by the Mullins' function for the fourfold symmetry in the stationary state with great precision, if one modifies the rate parameter by the anisotropy constant and simultaneously utilizes the anisotropic complementary dihedral angle in the calculation of the slope parameter. A recently developed analytical theory fully supports this observation rigorously, and furnishes the quantitative determination of the threshold level of the anisotropy constant for the ridge formation, and as well as the penetration depth evaluation. (C) 2007 American Institute of Physics.