In our previous study we have demonstrated that an adaptive mesh refinement scheme can be effectively used to investigate the orientation profiles of a nematic liquid crystal around a spherical particle when the characteristic size of the defect core is much smaller than the particle radius. We have also shown that the dynamics of a nematic liquid crystal can be properly treated to reproduce the experimentally observed structural transition front the hedgehog to the Saturn ring configuration induced by an external field. As a preliminary study for the quantitative investigation of this transition, we study in this article how the choice of the criterion for mesh refinement/unrefinement influences the stability of the hedgehog under an external field in our numerical system. The severer is the criterion for mesh refinement to produce a larger number of fine grids, the weaker field can induce the transition. Nevertheless, the dependence of the critical field strength on the criterion for mesh refinement is weak and the criterion can be systematically controlled. Therefore our numerical scheme is shown to be safely applied to the quantitative study on the stability of the hedgehog configuration tinder an external field.