  A state-of-the-art approach for calculating the finite nuclear size
correction to atomic energy levels and the bound-electron $g$ factor is
introduced and demonstrated for a series of highly charged hydrogen-like ions.
Firstly, self-consistent mean-field calculations based on the Skyrme-type
nuclear interaction are employed in order to produce a realistic nuclear proton
distribution. In the second step, the obtained nuclear charge density is used
to construct the potential of an extended nucleus, and the Dirac equation is
solved numerically. The ambiguity in the choice of a Skyrme parametrization is
supressed by fine-tuning of only one parameter of the Skyrme force in order to
accurately reproduce the experimental values of nuclear radii in each
particular case. The homogeneously charged sphere approximation, the
two-parameter Fermi distribution and experimental nuclear charge distributions
are used for comparison with our approach, and the uncertainties of the
presented calculations are estimated. In addition, suppression of the finite
nuclear size effect for the specific differences of $g$ factors is
demonstrated.
